Group approximation in Cayley topology and coarse geometry, Part II: Fibered coarse embeddings

The objective of this series is to study metric geometric properties of disjoint unions of amenable Cayley graphs by group properties of the Cayley accumulation points in the space of marked groups. In this Part II, we prove that a disjoint union admits a fibred coarse embedding into a Hilbert space (in a generalized sense) if and only if the Cayley boundary of the sequence in the space of marked groups is uniformly a-T-menable. We furthermore extend this result to ones with other target spaces. By combining our main results with constructions of Arzhantseva--Osajda and Osajda, we construct two systems of markings of a certain sequence of finite groups with two opposite extreme behaviors of the resulting two disjoint unions: With respect to one marking, the space has property A. On the other hand, with respect to the other, the space does not admit fibred coarse embeddings into Banach spaces with non-trivial type (for instance, uniformly convex Banach spaces) or Hadamard manifolds; the Cayley limit group is, furthermore, non-exact.

The main topics of this paper are the fibred coarse embeddings of disjoint unions of Cayley graphs and equivariant coarse embeddings of groups. Before proceeding to these two concepts, we first recall the definition of (genuine) coarse embeddings. By generalized metrics, we mean metrics that possibly take the value +∞. A basic example of generalized metric spaces is constructed as follows. For a sequence of metric spaces (X m , d m ) m∈N , we define a generalized metric d on m∈N X m by d(x, y) = d m (x, y) if x, y ∈ X m for some m and d(x, y) = +∞ otherwise. We call the resulting generalized metric space ( m∈N X m , d) the disjoint union, and simply write it as m∈N X m .
Definition 1.1. Let (X, d X ) be a generalized metric space and M be a non-empty class of (genuine) metric spaces.
(2) We say that X admits a coarse embedding into M if there exist M ∈ M and a coarse embedding f : X → M. The notion of fibred coarse embeddings was introduced by Chen-Wang-Yu [CWW14a]. This is a weakening of the (genuine) coarse embeddability; see Remark 2.6. In this paper, since we consider the disjoint union of possibly infinite graphs, we relax the condition on exceptional sets, and call the modified notion that of fibred coarse embeddings in a generalized sense; see Definition 2.4. This new notion coincides with the original notion of [CWW14a] for a coarse disjoint union of finite graphs; see Remark 2.5. In [CWW14a], they proved that if a coarse disjoint union X of finite graphs of uniformly bounded degree admits a fibred coarse embedding into a Hilbert space, then the maximal Baum-Connes conjecture holds for X. Furthermore, Chen-Wang-Wang [CWW14b] proved that if X above admits a fibred coarse embedding into a complete, connected and simply connected Riemannian manifold with non-positive sectional curvature, then the coarse Novikov conjecture holds for X.
The concept of equivariant coarse embedding is defined for finitely generated groups in terms of isometric actions. It relates to Gromov's a-T-menability if the target space is a Hilbert space, and to a-M-menability in general cases; see Definition 2.11.
We employ the space of (k-)makred groups G(k) to study a relationship between these two notions. This space was introduced by R. I. Grigorchuk [Gri84,Section 6], and it is the space of (equivalence classes of) all pairs of a group and a k-generating ordered set. The space G(k) is equipped with the topology of local convergence as rooted diagrams. This topology is sometimes called the Cayley topology, and it is compact and metrizable. We will briefly recall G(k) in Subsection 2.1. For a sequence (G m ) m∈N , we consider the following two objects: • The disjoint union m∈N Cay(G m ) of Cayley graphs, which is a generalized metric space without group structure. • The Cayley boundary ∂ Cay (G m ) m∈N (⊆ G(k)), defined as follows.
Definition 1.2. The Cayley boundary ∂ Cay (G m ) m∈N is defined as the set of all accumulation points of (G m ) m∈N in G(k) in the Cayley topology.
It forms a non-empty compact set, consisting of marked groups G ∞ ∈ ∂ Cay (G m ) m∈N . The following, Proposition 1.4 and Theorem A, are our main results; they provide quantitative relationships between control pairs for each side of embeddings, as well as qualitative aspects. Our results employ several operations on classes of metric Then for every such q in the first case (respectively, for every q ∈ [1, ∞) in the second case) the following holds true: If m∈N Cay(G m ) admits a fibred coarse embedding into M in a generalized sense, then ∂ Cay (G m ) m∈N is uniformly a-F S q (M)-menable. Moreover, it holds that for every such q above, If (G m ) m∈N is a convergent sequence, then we may replace ℓ q (M) with the original class M in the assertions above; in that case, it holds that where G ∞ is the Cayley limit group of (G m ) m .
As a corollary, we obtain the following.
Corollary B. Let (G m ) m∈N a sequence of amenable marked groups in G(k).
(1) The disjoint union m∈N Cay(G m ) admits a fibred coarse embedding into a Hilbert space in a generalized sense if and only if ∂ Cay (G m ) m∈N is uniformly a-T-menable. Note that (1) in Corollary B is essentially a special case of (2-1) with q = 2. Some work has been done by other researchers before our results in the context of box spaces for RF (Residually Finite) groups. If a finitely generated infinite group G with a finite generating set S admits a chain (N m ) m∈N , N m+1 N m , of normal subgroups of finite index in G such that m∈N N m = {e G }, then the box space of G is defined by where m denotes a coarse disjoint union (see [MS13,Definition 2.15.(2)] and Subsection 2.2). Chen-Wang-Wang [CWW13] showed that G admits a fibred coarse embedding into a Hilbert space if and only if G is a-T-menable. They also showed that for a metric space M, if G is a-M-menable, then G admits a fibred coarse embedding into M. It supplies several examples that admit fibred coarse embeddings into Hilbert spaces, but that do not admit genuine coarse embeddings; compare with Example 7.12.
Here we stress that the following points are visible only after extending our framework from the class of box spaces to our general class; see the definitions of RF/LEF/LEA groups in Definition 2.2. To illustrate point (c) above, we study the following example.
(1) Fix a prime p. For n ∈ N ≥1 , denote by F p n the finite field of order p n . It is well-known that the multiplicative group F × p n is cyclic; for each p and each n, we fix a generator t n = t p,n ∈ F p n of it. Fix a sequence (n m ) m∈2N+1 ≥3 of positive integers such that lim m→∞ n m = +∞.
For (odd) m ∈ 2N + 1 ≥3 , set H m = SL(m, Z/l m Z) and take two markings P m , Q m as follows: • Set P m = (σ (m) , τ (m) ), where σ (m) and τ (m) are the matrices with exactly the same entries of 0 and 1 as in, respectively, σ (m) and τ (m) as in (1) above. Define V ′ = V ′ (lm) = m∈2N+1 ≥3 Cay(H m ; P m ). • Set Q m = (σ (m) , σ ′(m) , τ (m) ), where σ (m) and τ (m) are the same as P m , and σ ′(m) = t σ (m) . Define W ′ = W ′ (lm) = m∈2N+1 ≥3 Cay(H m ; Q m ). In our Part I paper [MS13,Remark 5.9] (see also Corollary B and Proposition 2.11 therein), we proved that X ′ and V ′ above have property A of G. Yu [Yu00]. We do not recall the definition of property A here; see [Yu00] or [MS13, Definition 2.17]. Yu [Yu00] showed that property A implies the coarse embeddability into a Hilbert space. By the Dvoretzky theorem [BL00, Chapter 12] and a theorem of M. I. Ostrovskii [Ost12], it then follows that a locally finite generalized metric space with property A admits a coarse embedding into every infinite dimensional Banach space. At the other end of the spectrum, we obtain the following. For every prime p, set (∈ (0, 1 2 )); see Remark 7.4 for the background of this constant. For δ 0 ∈ (0, 1], let CAT (0) <δ 0 denote the class of all complete CAT(0) spaces whose Izeki-Nayatani invariants are strictly less than δ 0 ; see also (1) of Example 2.19.
(1) The spaces Y ′ , V ′ , W ′ do not admit a fibred coarse embedding in a generalized sense into ( <ℵ 0 QT ) ℓ 1 or into ( <ℵ 0 M) ℓ 2 , where QT denotes the class of all quasi-trees and M is the class of all finite dimensional, complete, connected and simply connected Riemannian manifolds with strictly negative sectional curvature that are cocompact; see Remark 2.20 for the definitions.
p,(nm)m does not admit a fibred coarse embedding in a generalized sense into B type>1 , the class of all Banach spaces with (linear, also known as Rademacher) type > 1; see (4) of Example 2.18. Neither does Y ′ admit a fibred coarse embedding in a generalized sense into CAT (0) <δ(p) .
(3) The space W ′ does not admit a fibred coarse embedding in a generalized sense into B β<1/2 ; see (5) of Example 2.18.
For every prime p, the class CAT (0) <δ(p) as in (2) above includes CAT (0) ≤0 , and the subclass CAT (0) ≤0 contains all (possibly infinite dimensional) complete, connected and simply connected Riemannian manifolds with non-positive sectional curvature. Hence, the space Y ′ above does not admit a fibred coarse embedding in a generalized sense into such spaces. Neither does it admit a fibred coarse embedding (in a generalized sense) into uniformly convex Banach spaces (see (7) of Example 2.18 for the definition) because every uniformly convex Banach space is contained in B type>1 . See [TJ89] and [BL00] on geometry of Banach spaces. After work [BBF15] of Bestvina-Bromberg-Fujiwara, study of actions on finite products of quasi-trees has been paid an intensive attention.
Remark 1.7. If we hope to allow even integers m as well as odd ones, then it will be more natural to consider unimodular linear groups SL ± rather than special linear groups SL; see [MS13,Subsection 5.3] for the discussion in that framework and compare with examples [MS13, Example 1.5 and Example 2.10] in our Part I paper.
Directions concerning point (c) above are closely related to the following two questions: • For a fixed (infinite) group G and a fixed k ∈ N ≥1 , to which (marked) groups can a sequence of (G; S m ) m∈N converge, where (S m ) m∈N is a system of kmarkings of G? This problem is studied in the paper of L. Bartholdi [Sha06]. In Section 6, we explain several gadgets which can be used to study the former question. One simple but striking tool is an absorption lemma (Lemma 6.4), which we learned from a paper of Bartholdi and Erschler [BE15, Lemma 6.13]. A standard (restricted) wreath product G ≀ H is a key tool for this; see Subsection 6.1 for the definition. We will deal with the latter question in a distinct paper [Mim18] from the present one, because the focus may be distinct from coarse geometry of disjoint unions.
We apply constructions described in Section 6 to an example of D. Osajda [Osa16] of an RF group, or that of Arzhantseva-Osajda [AO14] of a LEF group, without property A. It provides us with the following extreme examples; see also Remark 7.5.
Theorem C. (i) (See Proposition 7.6 for the detailed statement.) There exist a sequence of finite groups (G n ) n∈N and d ∈ N such that the following holds true: For every prime p and for every sequence (l n ) n∈N of integers at least 2 such that lim n→∞ l n = ∞, there exist three systems (S n ) n , (T n ) n and (U n ) n of d-markings of (H n,p (= H n,p,(ln)n ) = G n ≀ SL(2n + 3, F p ln )) n∈N such that the following hold true: • The sequence ((H n,p ; S n )) n∈N converges in the Cayley topology to an amenable group. • The sequence ((H n,p ; T n )) n∈N converges in the Cayley topology to a group without property A (in other words, it is a non-exact group), but the Cayley limit group is a-T-menable. • The sequence ((H n,p ; U n )) n∈N converges in the Cayley topology to a group without property A; in addition, the Cayley limit group is not a-Mmenable for M = B type>1 or M = CAT (0) <δ(p) , where δ(p) is described as above Corollary 1.6. (ii) (See Proposition 7.7 for the detailed statement.) In (i), for every prime p, we may replace (H n,p ) n with (S([h n,p ])) n , a sequence of finite symmetric groups of specified degrees. Here (h n,p ) n∈N ≥1 is a sequence of integers at least 2 that satisfies that lim n→∞ h n,p = ∞, which depends on p. The symbol [h n,p ] denotes the set {1, 2, . . . , h n,p }; see our notation of symmetric groups at the end of Introduction.
By Theorem A of our Part I paper [MS13], the disjoint union n∈N Cay(H n,p ; S n ) has property A. By (ii) of Theorem A, n∈N Cay(H n,p ; T n ) admits a fibred coarse embedding into a Hilbert space. At the other end of the spectrum, by (i) of Theorem A, n∈N Cay(H n,p ; U n ) does not admit a fibred coarse embedding into B type>1 or CAT (0) <δ(p) . We also are able to take S n = (s are all conjugate (but group elements in H n,p which conjugate them depend on i).
T. Pillon introduced a notion of fibred property A and showed that a box space of a group has this property if and only if the group has property A. In this aspect, it is furthermore plausible that n∈N Cay(H n,p ; T n ) and n∈N Cay(H n,p ; U n ) both fail to enjoy fibred property A. These examples may indicate that, beyond the world of box spaces, it is no longer reasonable to write disjoint unions as n G n without expressing markings.
We, moreover, observe that point (a) above is striking in the study of fibred coarse embeddings: Unlike amenability and property (T), uniformity is not automatic for a-M-menability; compare with [MS13, Proposition 3.4] and [MOSS15, Proposition 5.1]. Owing to this observation, we answer the question of Yu (in private communication) which asks whether the fibred coarse embeddability into a Hilbert space is closed under taking finite direct products. The answer is that it is almost never true for (coarse) disjoint unions: Corollary 1.8. Let (Γ m ) m∈N and (Λ n ) n∈N be two sequence of connected graphs of uniformly bounded degree. Let X 1 = m∈N Γ m and X 2 = n∈N Λ n . Endow X 1 × X 2 with the structure of a disjoint union where Γ m × Λ n is equipped with the ℓ 1 -metric from d Γm and d Λn . Let M be a nonempty class of metric spaces such that UP(M) ⊆ M.
Then X 1 × X 2 admits a fibred coarse embedding in a generalized sense into M only if X 1 and X 2 both admit (genuine) coarse embeddings into M. In particular, this assertion applies to the case where M = Hilbert.
If all Γ m and Λ n are finite, then we may replace disjoint unions above with coarse disjoint unions. In this case, the product above is equivalent to the product as metric spaces and fibred coarse embeddings may be taken in the original sense.
Finally, we construct some generalized metric spaces that have very exotic features with respect to fibred coarse embeddings; see also point (c) above for (ii).
Theorem D. (i) There exists a sequence (Γ l ) l∈N of finite graphs of uniformly bounded degree such that all of the following hold true.
• The sequence (Γ l ) l forms an expander family; see Definition 7.8.
• The disjoint union l∈N Γ l does not admit a fibred coarse embedding in a generalized sense into CAT (0) <1 or the class of Banach spaces that are sphere equivalent (see below) to Hilbert spaces. • There exists a complete CAT(0) space M such that l∈N Γ l admits a biLipschitz embedding into M, namely, it admits a coarse embedding with control pair (ρ, ω), where ρ and ω are both linear functions. (ii) There exist (k l ) l∈N of a sequence of natural numbers at least 2 with lim l→∞ k l = ∞ and two (ordered) systems of generators (Ξ l ) l∈N , (Ω l ) l∈N of symmetric groups (S([k l ])) l∈N over [k l ] = {1, 2, . . . , k l } that satisfy all of the following.
(3) The disjoint union l∈N Cay(S([k l ]); Ω l ) does not admit a fibred coarse embedding in a generalized sense into any of these spaces: • Banach spaces of non-trivial type, and Banach spaces that are sphere equivalent to Banach spaces of non-trivial type.
Here two Banach spaces are said to be sphere equivalent if there exists a bijection Φ between the unit spheres such that Φ and Φ −1 are both uniformly continuous; see [Mim15] for details. Note that many reasonable CAT(0) spaces, including all Euclidean buildings associated with simple algebraic groups, belong to the class CAT (0) <1 ; see a paper of T. Toyoda [Toy10] for other examples of elements in CAT (0) <1 . For the proof of Theorem D, we utilize the notion of embedded expanders; see Definition 7.8 and Proposition 7.14 for details.
Our proof of Theorem A is inspired by a trick by Gromov, [dCTV07, Proposition 4.4] for Hilbert spaces and [NP11, Section 9] for general Banach spaces, as we will explain in Sections 3 and 4. Independently to our results, S. Arnt [Arn16] applied this trick in a special situation where the coarse disjoint union is a box space (in particular, all G m , m ∈ N, are finite) and the target class consists only of Banach spaces. For the case where M = Hilbert, V. Alekseev and M. Finn-Sell [AFS16] extended the framework of Theorem A for the case where (G m ) m is a LEF approximation of G ∞ , see Definition 2.2, to a sofic approximation of a sofic group. However, in that generality, only one direction (the direction of (i) in Theorem A) can be deduced; see the construction of a counterexample to the other direction by T. Kaiser [Kai17], which is explained below Theorem 5.3 in the concerning reference [Kai17]. Compare also with our points (a), (b) with LEA approximations, and the case where M is general.
Notation and Conventions: We use G for a (non-marked) group and G for a marked group. We write the group unit of G as e G . A finite generating set S of G is regarded as an ordered set (sometimes an ordered multi-set) S = (s 1 , s 2 , . . . , s k ) so that (G; S) is seen as a marked group. A marked group G = (G; S) is said to be finite (respectively, amenable, and a-T-menable) if so is G. For k ∈ N ≥1 , we denote by F k the free k-marked group, namely, F k = (F k ; a 1 , . . . , a k ). Here (a 1 , . . . , a k ) denotes a free basis of F k . For a non-empty set B, denote by S(B) the full symmetric group, and by S <ℵ 0 (B) the symmetric group with finite support, namely, the group of all permutations on B that fix all but finitely many elements in B. For R ∈ R ≥0 , let ⌊R⌋ denote the integer part of R. For m ∈ N ≥1 , let [m] = {1, 2, . . . , m}. We use the terminology isometries for surjective ones; we use geodesics for minimal ones, namely, a geodesic from y ∈ M to z ∈ M is an isometric embedding c : [0, d(y, z)] → M. We always exclude the empty-set from metric spaces. For a metric space M, we write the class {M} consisting only of M as M for short.
Organization of the paper: In Section 2, we briefly recall the space of marked groups and the Cayley topology, and the definition of fibred coarse embeddings (in a generalized sense). In Subsections 2.4 and 2.5, we respectively, define several operations to classes of metric spaces and provide examples. In section 3, we explain the key idea to non-linear version of Gromov's trick in relation to (pointed) metric ultraproducts. There we prove Proposition 1.4. Section 4 is devoted to the proof of (i) of Theorem A. It is done by the non-linear version of Gromov's trick. In Section 5, we prove (ii) of Theorem A and Corollary B. Section 6 is for description of several gadgets that may be used in the space of marked groups, such as the absorption lemma (Lemma 6.4). In Section 7, we discuss various examples to apply Theorem A (and Proposition 3.3), including the proofs of Corollary 1.6 and Corollary 1.8. Theorem C (Propositions 7.6 and 7.7) is proved in Subsection 7.2; Theorem D is verified in Subsections 7.5 and 7.6.

Preliminaries
2.1. Space of k-marked groups and Cayley topology. We recall basic facts of the Cayley topology from our Part I paper [MS13]; see Subsection 2.1 there for more details. Fix k ∈ N ≥1 . A k-marked group G = (G; S) = (G; s 1 , s 2 , . . . , s k ) is a pair of a finitely generated group G and an ordered k-tuple S = (s 1 , . . . , s k ) of generators of G (as a group). From a k-marked group G, we construct two combinatorial objects, the Cayley diagram CayD(G) and the Cayley graph Cay(G) of G as follows. The former is defined as a diagram (edge-colored and edge-oriented graph), with the edge coloring set [k], by setting the vertex set as G and by putting edges of the form (g, s j g) with orientation from g to s j g in color j(∈ [k]) for every j ∈ [k] and every g ∈ G. The latter is the graph (with no edge colorings or no edge orientations) constructed by forgetting the edge-colorings/orientations of CayD(G). Both of them are endowed with the shortest path metric d G (in CayD(G), we ignore the edge-orientation to consider d G ) on the vertex set G. In this way, we regard CayD(G) and Cay(G) as geometric objects. We also consider G itself as a metric space with this metric d G ; in other words, d G on G is the right-invariant word metric with respect to S.
For ∅ = Y ⊆ G and for R ∈ N ≥1 , denote by ∂ G (Y, R) the R-neighborhood of Y in d G , namely, the set of all h ∈ G such that there exists g ∈ Y such that d G (g, h) ≤ R. If Y = {g}, then we simply write ∂ G ({g}, R) as B G (g, R) (closed ball of radius R centered at g). In this setting, we define B CayD(G) (g, R) by restricting the vertex set of CayD(G) to B G (g, R) and by taking the induced sub-diagram (more precisely, we collect all edges connecting vertices in B G (g, R) with remembering its edge-colorings/orientations). By declaring g to be the root, B CayD(G) (g, R) has the structure of a rooted diagram. Note that B CayD(G) (e G , R) completely remembers the multiplication table of G up to word length ⌊R/2⌋.
Denote by G(k) the set of all k-marked groups (up to marked group isomorphisms). This space is equipped with a natural topology, the Cayley topology, which is metrizable and compact. One definition of that topology is the induced topology of the product topology on {0, 1} F k to the set of all normal subgroups in F k ; there is a natural one-to-one correspondence between that subset of {0, 1} F k and G(k) by the standard marked quotient map F k ։ G. Another characterization of this topology is the topology of local convergence (also known as the Gromov-Hausdorff convergence in this setting) among rooted diagrams, as stated in the following lemma (Lemma 2.4 in [MS13]). Here for two groups G, H and for two subsets e G ∈ K 1 ⊆ G and e H ∈ K 2 ⊆ H, a map β : K 1 → K 2 is called a partial homomorphism if for all g 1 , g 2 ∈ K 1 such that g 1 g 2 ∈ K 1 , β(g 1 g 2 ) = β(g 1 )β(g 2 ) holds true. The map β is called a partial isomorphism if it is furthermore bijective. Here an isomorphism of rooted diagrams means a graph automorphism that preserves edge-colorings (in [k]) and edge-orientations and that sends the root of the former diagram to the root of the latter.
In other words, for G = (G; s 1 , . . . , s k ) ∈ G(k), if we define for each R ∈ N, We also recall the definitions of RF/LEF/LEA groups; recall these abbreviations from Introduction.
Definition 2.2. Let G be a finitely generated group.
(1) The group G is said to be RF if there exists a sequence (N m ) m∈N of finite index normal subgroups of G such that (2) The group G is said to be LEF if for some (equivalently, every) marking G of G, there exists a Cayley convergent sequence consisting of finite marked groups that converges to G . (3) The group G is said to be LEA if for some (equivalently, every) marking G of G, there exists a Cayley convergent sequence consisting of amenable marked groups that converges to G . We say a sequence (G m ) m is a LEF (respectively, LEA) approximation of G if it consists of finite (respectively, amenable) marked groups converging to G in the Cayley topology. A LEF approximation is moreover called an RF approximation if it consists of marked group quotients; namely, for every m, there exists a group quotient map ϕ m : G ։ G m that sends the marking S = (s 1 , . . . , s k ) of G to that S m = (s In (1) in the definition above, we may relax the condition of liminf However, if we hope to have an RF approximation out of (N m ) m by taking marked group quotients, then the right condition is the former one, not the latter.
Remark 2.3. If a marked group G is finitely presented (this is independent of the choice of markings), then the set of all marked group quotients of G forms an open set. Hence, in that case, every LEF approximation eventually is an RF approximation; see [MS13, Subsection 2.1].
2.2. Fibred coarse embeddings. Recall from Introduction the construction of the disjoint union m∈N X m out of a sequence of metric spaces (X m , d m ) m∈N . If every X m has finite diameter (the diameter is defined as the supremum of the distances between two points in the metric space), then we may construct a coarse disjoint union m∈N X m , which is a (genuine) metric space. However, we do not go into details in this paper; instead, we refer the readers to [MS13, Definition 2.15.(2)] on this notion.
In this paper, we study fibred coarse embeddings from the disjoint union constructed above. For this purpose, we relax the definition of the fibred coarse embeddings as follows. For a generalized metric space X, we say that X is uniformly locally finite if for every R ∈ R ≥0 , there exists C ∈ N such that every closed Rball (for every center x ∈ X) has cardinality at most C. For a sequence of metric spaces (X m ) m∈N , we say that it is equi-uniformly locally finite if every X m is uniformly locally finite and if moreover C = C(R) is taken uniformly on m ∈ N for every R ∈ R ≥0 . If (X m ) m∈N is equi-uniformly locally finite, then the disjoint union X = m∈N X m is uniformly locally finite.
Definition 2.4. Let M be a non-empty class of metric spaces. Let (X, d) = m∈N X m be the disjoint union of a sequence of metric spaces (X m ) m∈N that is equiuniformly locally finite. Let ρ, ω : [0, ∞) → [0, ∞) be two non-decreasing proper functions.
(i) We say that X admits a (ρ, ω)-fibred coarse embedding into M in a generalized sense if there exists M ∈ M such that the following holds true: There exist • a field of metric spaces (M x ) x∈X over X such that each M x is isometric to M, • a section s : X → x∈X M x , (namely, s(x) ∈ M x for every x ∈ X), such that for every R ∈ R ≥0 there exists m (R) ∈ N ≥1 such that for each nonempty subset C ⊆ X \ N <m (R) X m of diameter at most R, there exists a "trivialization" t C,R : (M x ) x∈C → C × M such that the following holds. The restriction of t C,R to the fibre M x , x ∈ C, is an isometry t C,R (x) : M x → M that satisfies (1) for every x 1 , x 2 ∈ C, ρ(d(x 1 , x 2 )) ≤ d M (t C,R (x 1 )(s(x 1 )), t C,R (x 2 )(s(x 2 )) ≤ ω(d(x 1 , x 2 )); (2) for every two subsets C 1 , C 2 ⊆ X \ m∈N <m (R) X m of diameter at most R with C 1 ∩ C 2 = ∅, there exists an isometry t C 1 ,C 2 ,R : M → Y such that We say that X admits a fibred coarse embedding into M if it admits a (ρ, ω)fibred coarse embedding into M for some (non-decreasing, proper) pair (ρ, ω).
(ii) We say (ρ, ω) is a control pair for fibred coarse embeddings in a generalized sense for X into M if there exists a (ρ, ω)-fibred coarse embedding from X to M in a generalized sense. Denote by CP fib M (X) the set of all control pairs above. The functions ρ and ω are, respectively, called a compression function and an expansion function in the setting above.
Note that if a non-empty set C of X = m∈N X m is of bounded diameter, then there exists a unique m ∈ N such that C ⊆ X m .
Remark 2.5. In the original formulation in [CWW14a, Definition 2.1] (for the case M being the class of all Hilbert spaces), for each R ∈ N, we are allowed to choose a bounded exceptional set K, and consider C of diameter at most R from X \ K. In our definition in generalized sense, we relax this process and allow to take K = m∈N <m (R) X m , the disjoint union of finitely many components in X = m∈N X m . Therefore, in Definition 2.4, if all X m , m ∈ N, are finite, then our notion of the fibred coarse embeddability in a generalized sense coincides with that of the fibred coarse embeddability in the original sense from a coarse disjoint union m∈N X m .
In this paper, we discuss quantitative aspects (control pairs) for fibred coarse embeddings (in a generalized sense) as well as qualitative aspects (the property itself). For this purpose, disjoint unions are more suited than coarse ones.
Remark 2.6. The fibred coarse embeddability into M (in a general sense) is weaker than the the (genuine) coarse embeddability. Indeed, if f : X = m X m → M is a coarse embedding with control pair (ρ, ω), then set m (R) = 0 for all R and M x = M for all x ∈ X. Let s : X → x∈X M be s(x) = f (x), and t C,R = id M for all R and for all C of diameter at most R. This gives rise to a (ρ, ω)-fibred coarse embedding in a generalized sense into M.
Remark 2.7. Strictly speaking, if M = E consists of Banach spaces, then we furthermore assume that all isometries in the definition above are affine. However, by the Mazur-Ulan theorem [BL00, Chapter 14.1], the restriction on isometries in the case M = E is automatic if E consists of real Banach spaces. If E consists of complex Banach spaces, then we need to care the affine property. However, in some case, this problem is automatically fixed in the following way: Consider a fibred coarse embedding into some complex Banach space E with respect to which t C,R (x) above are isometries, not necessarily affine. Then, regard E as a real Banach space and write it as E R ; by the Mazur-Ulam theorem, t C,R (x) are real affine. Take the Taylor complexification E R of E R as follows: As a real vector space, E R = E R ⊕ E R , and complex multiplication is defined by (We may regard (ξ, η) ∈ E R as "ξ + √ −1η".) The norm is defined as This gives rise to a fibred coarse embedding into E R with the same control pair as the original one. By construction, now these t C,R (x) are complex affine. Thus if the class E is closed under the procedure E → ( E R , · T ), then the issue of the affine property is essentially automatic. This closeness holds for some classes of our concern in this paper, for instance B type>1 and B β<1/2 ; see Example 2.18.
For this reason, in what follows, we do not go into details of the issue between isometries and affine isometries.
Remark 2.8. Though it was implicit in the original formulation [CWW14a, Definition 2.1], the "trivialization" t C = t C,R in Definition 2.4 is allowed to be inconsistent on changing R. More precisely, for 0 ≤ R 1 < R 2 and for C ⊆ X \ N <m (R 2 ) X m of diameter at most R 1 , we do not require that t C,R 1 = t C,R 2 . This observation is important in our proof of (ii) of Theorem A.
We observe the following two lemmata. Here for a metric space X, x ∈ X and R ∈ R >0 , denote by B X (x, R) the closed ball of radius R centered at x. Lemma 2.9. Let (X m ) m∈N be a sequence of metric spaces that is equi-uniformly locally finite. Let X = m∈N X m . Let M be a non-empty class of metric spaces. Let ρ, ω : [0, ∞) → [0, ∞) be two non-decreasing proper functions. Then, X admits a (ρ, ω)-fibred coarse embedding into M in a generalized sense if and only if there exists M ∈ M such that the following holds true: There exist • a field of metric spaces (M x ) x∈X which are all isometric to M, • there exists a section s : for each m ∈ N and each g ∈ X m , such that the following hold.
(1) For every n ∈ N, for every g ∈ X m and every Lemma 2.10. In the setting of Lemma 2.9, let Y = n∈N Y mn be such that (m n ) n∈N is a subsequence of (m) m∈N and for each n ∈ N, Y mn is a non-empty subset of X mn equipped with the induced metric. Then, if X admits a fibred coarse embedding into M in a generalized sense with control pair (ρ, ω), then so does Y .
Proofs of Lemma 2.9 and Lemma 2.10. Lemma 2.10 is obvious.
To show that the (ρ, ω)-fibred coarse embeddability in a generalized sense implies the conditions as in Lemma 2.9, take R → m (R) as in Definition 2.
There is an ambiguity on the choice of g; however, if we fix the choices for all C, then condition (3) as in Lemma 2.9 ensures condition (2) as in Definition 2.4. Recall also Remark 2.8.

Equivariant coarse embeddings and a-M-menability.
In Section 4 in our Part I paper [MS13], we recall the definition of a-T-menability for finitely generated groups. Here we generalize this concept in terms of other target spaces. The following property should be stated as a-F M -menability in the strict sense. However, through communications with Arnt, we have agreed to use the terminology of a-M-menability to avoid messes on notation. In the following definition, recall that a marked group G is naturally equipped the metric d G ; see Subsection 2.1.
Definition 2.11. Let G be a marked group and M be a non-empty class of metric spaces.
(1) The marked group G is said to be a-M-menable if there exist M ∈ M and a coarse embedding f : (G, d G ) → M such that the following condition is satisfied: The map f is of the form where α : M G is a right action by isometries and y ∈ M. We say that a coarse embedding f is (G-)equivariant if it satisfies the condition above.
(2) We say a finitely generated group G is a-M-menable if for some (equivalently, In this case, we call ρ and ω, respectively, an equivariant compression function and an equivariant expansion function from G into M. (4) We denote by CP ♯ M (G) be the set of all equivariant control pairs for G into M. In the definition above, we take a right action, not a left action, because we equip marked groups with right-invariant metrics.
Remark 2.12. Similarly to Remark 2.7, strictly speaking, if M = E consists of Banach spaces, then we require that the action α above is affine. However, for the same reason as one in Remark 2.7, we do not go into details of this issue in this paper.
Let Hilbert denote the class of all Hilbert spaces. Then the notion of a-Hilbertmenablity coincides with that of a-T-menability.
Remark 2.13. We warn that, unlike some other literature, the control pair (ρ, ω) is regarded as the pair of concrete functions, not only as growth orders. In particular, . If we consider a class M that is not necessarily closed under rescaling, this remark applies even when C 1 = C 2 . A similar issue to above applies to CP M (X) and CP fib M (X). Definition 2.14. Let q ∈ [1, ∞). Let B be a non-empty set that is at most countable. Let (r j ) j∈B be such that r j ∈ (0, ∞) for all j ∈ B. For a sequence (M j , d j , y j ) j∈B of pointed metric spaces, define the (pointed) ℓ p -product with scaling (r j ) j , denoted by ( j∈B (M j , y j , r j )) ℓp , by and with the base point (y j ) j . If the scaling factor (r j ) j is all 1 (r j = 1 for all j ∈ B), then we simply write ( j∈B (M j , y j , 1)) ℓq as ( j∈B (M j , y j )) ℓq . This space is called the (pointed) ℓ q -product of (M j , y j ) j . (If M j are Banach spaces, then it is usually called the pointed ℓ q -sum.) If ♯(B) < ∞, then (the isometry type of) the resulting space ( j∈B (M j , y j , r j )) ℓq does not depend on the choice (y j ) j of base points. In that case, we write it as ( j∈B (M j , r j )) ℓq for short.
We now switch our subject to (pointed) metric ultraproducts. A ultrafilter U over N has a one-to-one correspondence to a probability mean ν (finitely additive measure with ν(N) = 1) on N that is {0, 1}-valued and is defined over all subsets of N. The correspondence is given by setting that A ∈ U if and only if ν(A) = 1. The cofinite filter U cofin = {A ⊆ N : ♯(N \ A) < ∞} is a filter, but not an ultrafilter. A nonprincipal ultrafilter U is an ultrafilter that includes U cofin (as a subfilter). In what follows, fix a non-principal ultrafilter U over N.
For a sequence (r m ) m∈N in R and for r ∞ ∈ R, we say that By local compactness and Hausdorff property of R, it is standard to show that every bounded real sequence (r m ) m∈N has a unique U-limit. The limit in general depends on the choice of a non-principal ultrafilter U. However, if lim m→∞ r m exists, then lim U r m coincides with the limit above. We now consider a sequence ((M m , d m , y m )) m∈N of pointed metric spaces. Set This is a pseudo-metric, namely, d U does not separate points in general. To obtain a genuine metric space,introduce an equivalence relation is equipped with a genuine metric d U . We call the resulting space the (pointed) metric ultraproduct of (M m , y m ) with respect to U. We write the equivalence class (1) We define ℓ q (M) as the class of all metric spaces (that is isometric to ones) of the form ( j∈B (M j , y j )) ℓq (after forgetting the base point) for a non-empty at most countable sets B and for M j ∈ M and y j ∈ M j for j ∈ B.
(2) We define F q (M) as the class of all metric spaces (that is isometric to ones) that are constructed by the following three steps.
for non-empty finite sets F . Here ( 1 (♯(F )) 1/q ) f ∈F means that we take the constant scaling factor 1 (♯(F )) 1/q . • (Step 3.) Take an arbitrary sequence ((N m , y m )) m∈N , where for all m ∈ N, N m = N m (F (m) ) lies in the class of all metric spaces constructed in Step 2 that is associated with a finite set F (m) such that lim m→∞ ♯(F (m) ) = ∞ and y m ∈ N m . Construct all metric spaces of the form lim U (N m , y m ) (after forgetting the base points) for non-principal ultrafilters U of N.
space. Namely, for every z, w ∈ L, there exists a geodesic c : [0, d(z, w)] → L connecting z and w; recall our notation from Introduction. If this is the case, then we construct F S q (M) in the following way.
• If M consists only of Banach spaces, then every element L in F q (M) has a structure of affine Banach spaces. Then set F S q (M) as the class of all Banach spaces isometrically affinely isomorphic to non-empty closed affine subspaces of L for all L ∈ F q (M). • Otherwise, define F S q (M) to be the class of all metric spaces isometric to non-empty closed convex subsets L 0 of L (equipped with the induced metric from L) for all L ∈ F q (M). Here a non-empty subset L 0 ⊆ L is said to be convex if for every z, w ∈ L 0 and for every geodesic c : Note that unlike the construction of ℓ q (M), in Step 1 of the construction of F q (M), we use a single M ∈ M to take the ℓ q -product with scaling. (Similarly for UP(M).) The symbol F in (2) stands for finite and Følner. The symbol F S in (3) stands for Følner and subspaces (or subsets).
Remark 2.17. The scaling factor ( 1 (♯(F )) 1/q ) f ∈F in Step 2 above is chosen exactly in order to ensure that the diagonal embedding M ֒→ f ∈F (M, In particular, by letting r = 2, we observe that for M = Hilbert (the class of all Hilbert spaces), ℓ 2 (M) ⊆ M and F 2 (M) ⊆ M; the proof of the latter item is much easier than that for the general L r -case. In that case, moreover, F S 2 (M) ⊆ M holds.
(3) More generally to (2), let M = B NCLr denote the class of all non-commutative L r -spaces (associated with all von-Neumann algebras). Then, ℓ r (M) ⊆ M and UP(M) ⊆ M; the latter follows from work of Raynaud [Ray02]. It also holds that F r (M) ⊆ M. (4) A Banach space E is said to be of non-trivial (linear or Rademacher) type if there exists r ∈ (1, 2] and a constant C > 0 such that the following holds true: For every m ∈ N ≥1 and for every (ξ i ) i∈ [m] in E, If the inequality above is satisfied for fixed r and C, we say that E has a type r constant C.
where E ′ runs over all (complex) linear subspaces of E with the condition above. Here ℓ m 2,C denotes the m-dimensional complex ℓ 2 -space for m ∈ N. The class B β<1/2 is defined as the class of all complex Banach spaces E for which there exist 0 < β < 1/2 and C > 0 such that the condition for all k ∈ N ≥1 , d k (E) ≤ Ck β is satisfied. Then, it follows that UP(B β<1/2 ) ⊆ B β<1/2 and that F 2 (B β<1/2 ) ⊆ F S 2 (M) ⊆ B β<1/2 . Moreover, for fixed β ∈ (0, 1/2) and C > 0, if we denote by B β,C the class of all complex Banach spaces such that the condition above holds for that pair (β, C), then ℓ 2 (B β,C ) ⊆ B β,C holds.
A fact states that a complex Banach space E is of non-trivial type if and only if lim k→∞ k −1/2 d k (E) = 0; see [TJ89]. In particular, B β<1/2 ⊆ B type>1 . It is not known whether the inclusion above is strict.
de Laat-Mimura-de la Salle [dLMdlS16] studied fixed point properties with respect to B β<1/2 ; see (3) of Theorem 7.3. (6) Similarly to (4), for each r ∈ [2, ∞) and each C > 0, we define the class B cotype r,C as that of all Banach spaces that satisfy the cotype r inequality with constant C: Here the expected value in the left-hand side is defined as one in (4) Here S(E) denotes the unit sphere of E. For a fixed r ∈ [2, ∞), if there exists C > 0 such that ∆ above satisfies that ∆(ǫ) ≥ C r ǫ r for all ǫ ∈ (0, 2], then we say that E is uniformly convex with modulus of convexity of power type r.
Ball-Carlen-Lieb [BCL94, Proposition 7] showed that the condition above is equivalent to saying that there exists C ′ > 0 such that for all ξ, η ∈ X and for all t ∈ [0, 1], the following inequality holds true: ( They also made estimate between C and C ′ above. In this paper, we say a Banach space E is r-uniformly convex with constant C ′ if the inequality above is satisfied; this terminology is consistent with that of r-uniformly convex metric spaces in more general framework; see (2) of Example 2.19. A Banach space E is said to be superreflexive if every (equivalently, some) metric ultrapower lim U (E, 0) is reflexive. Enflo's characterization states that E is superreflexive if and only if E is isomorphic to a uniformly convex Banach space. A theorem of G. Pisier [Pis75] shows that, moreover, for every superreflexive Banach E, there exists r ∈ [2, ∞) such that E is isomorphic to a uniformly convex Banach space with modulus of convexity of power type r. For r ∈ [2, ∞), for C ′ > 0 and for D ≥ 1, we define the class B sr r,C ′ ,D as that of all Banach spaces whose Banach-Mazur distance at most D to r-uniformly convex Banach spaces with constant C ′ . Then, for M = B sr r,C ′ ,D , it holds that ℓ r (M) ⊆ M, UP(M) ⊆ M, and F r (M) ⊆ F S r (M) ⊆ M. By aforementioned theorems in [BCL94] and [Pis75], the union B sr = r,C ′ ,D B sr r,C ′ ,D coincides with the class of all superreflexive Banach spaces. It is known that B sr ⊆ B type>1 ⊆ B cotype<∞ and that both of the inclusions are strict; see [TJ89] and [BL00].
Example 2.19. We furthermore provide classes of non-linear metric spaces.
(1) Let M be a geodesic space. Then, M is CAT(0) if and only if for every x ∈ M and for every geodesic c : [0, d(y, z)] → M with c(0) = y and c(d(y, z)) = z and for every 0 ≤ t ≤ 1, the following inequality Here Dirac p means the Dirac mass at p. Note that for such µ, there exists a unique point µ ∈ M that minimizes the function this point µ is called the barycenter of µ. For such µ, define Here f runs over all maps from supp(µ) to L 2 = L 2 ([0, 1]) that satisfies the two We refer the readers to [IN05] and [Toy09] for discussions on the Izeki-Nayatani invariants.
For δ 0 ∈ (0, 1], if we set M as then it holds that Note that all CAT(0) spaces are uniquely geodesic; indeed, by setting t = 1/2, we observe that for every pair (y, z), a geodesic midpoint of it is unique. Izeki and Nayatani [IN05] studied fixed point properties with respect to CAT (0) <δ 0 for certain δ 0 ; see (2) of Theorem 7.3.
(2) Fix r ∈ [2, ∞). Then, some ℓ r -analogue of item (1) may be defined as follows: Let C ∈ (0, 1]. A geodesic space M is said to be r-uniformly convex with constant C if for every x ∈ M and for every geodesic c : [0, d(y, z)] → M with c(0) = y and c(d(y, z)) = z and for every 0 ≤ t ≤ 1, the following inequality holds true, where c t denotes c(td(y, z)); see also [NS11]; compare with the inequality of r-uniformly convex Banach spaces in (7) of Example 2.18. The Clarkson inequality (see for instance [BL00]) shows that L r is r-uniformly convex with a certain constant C r . For fixed C ∈ (0, C r ], we write the class of all r-uniformly convex (geodesic) spaces with constant C as UC r,C . Then, for every C ∈ (0, C r ], M = UC r,C satisfies that (2) t . Apply the inequality of r-convexity with t = 1/2 and c = c ′ respectively for (x, y, z) = (y, c (1) t , c (2) t ) and for (x, y, z) = (z, c a contradiction. Therefore, c (1) ≡ c (2) , and we are done.
However, the class M = UC r,C itself seems too enormous. We need to restrict it to a reasonably smaller subclass, and some variant of the Izeki-Nayatani invariant would be demanded in the current framework.
(3) Since we choose our scaling factor in Step 2 in F q (M) in a specific manner, a certain quasification of classes in (1) and (2) satisfies some closeness property. For instance, for a given ǫ > 0, we may define that a geodesic space M is ǫ-coarse CAT(0) if the following holds true: For every x ∈ M and for every geodesic c : [0, d(y, z)] → M with c(0) = y and c(d(y, z)) = z and for every 0 ≤ t ≤ 1, the following inequality holds: where c t = c(td(y, z)). Then, for every fixed ǫ, the class M of all ǫ-coarse CAT(0)-spaces satisfies that However, similarly to (2) and (3), these classes are big and we may need to refine them to certain subclasses. Note that there is a notion of the Markov cotype of a metric space, but that it may not be suited for the study of F S r (M). This is because in the definition, we need to take extra points according to given points in the metric space, and the existence of such extra points may fail if we consider a convex subset, rather than the original space. For treatises on these concepts, we refer to [MN08] and [MN13].
Remark 2.20. The main difference between F q (M) and UP(M) is that the latter does not take (finite) ℓ q -products (or rescaling) before taking metric ultraproducts. Therefore, the latter procedure may preserve some "dimension" under certain conditions. First we consider the class RT of all R-trees (namely, geodesic 0-hyperbolic metric spaces). By the four-point condition of Gromov-hyperbolicity [BH99, Chapter III. Remark 1.21], it follows that UP(RT ) ⊆ RT . Even if we consider a smaller class T of all simplicial trees (considered as geodesic spaces, possibly with uncountably many vertices), then UP(T ) ⊆ T . This is because we may endow a metric ultraproduct with a simplicial structure by declaring vertices to be (equivalence classes of) bounded sequence of vertices; we draw edges between those with the limit distance 1.
We consider the class QT of quasi-trees, namely, graphs (considered as geodesic spaces, possibly with uncountably many vertices) that are quasi-isometric ([BH99, Chapter I. Definition 8.14]) to simplicial trees. By the argument above, we see that UP(QT ) ⊆ QT ; recall that we fix a single element of M and take pointed metric ultraproducts of it to construct UP(M).
Definition 2.21. Let M be a non-empty class of metric spaces and q ∈ [1, ∞). Denote ( <ℵ 0 M) ℓq by the class of all metric spaces (isometric to) finite ℓ q -products Since for a fixed m ∈ N ≥2 , taking an m-fold product is compatible with taking a metric ultraproduct, we conclude that UP(( <ℵ 0 QT ) ℓ 1 ) ⊆ ( <ℵ 0 QT ) ℓ 1 . (We may replace ℓ 1 simultaneously with ℓ q for each q ∈ (1, ∞).) Another construction is the following. Let M = M be a proper metric space that is cocompact. Here the properness means that all closed bounded balls are compact; M is said to be cocompact if the full isometry group of M acts on M cocompactly. Then, UP(M) = M. Here the cocompactness assumption is needed in order to make control on choices of base points (y m ) m to take a pointed metric ultraproduct.  with control pair (ρ, ω). For every m ∈ N, take R m as above. Now, for each g ∈ G ∞ , we associate the following sequence (y(g) m ) m∈N of points in M: otherwise. By (⋆), we observe the following: • For every g ∈ G ∞ , sup m∈N d M (y(g) m , f (e Gm )) ≤ ω(d G∞ (e G∞ , g))(< ∞).
Finally, fix a non-principal ultrafilter U over N and take the pointed metric ultraproduct M U = lim U (M, d M , f (e Gm )); we define the following map By the two observations above, we conclude that this f ∞ is well-defined, and that it is a coarse embedding with the same control pair (ρ, ω) as one for the original f . If M = E is a Banach space, then the arguments in the paper of Ostrovskii [Ost12] indicate a way to construct a coarse embedding from Cay(G ∞ ) to the original E out of the metric ultraproduct construction above; this procedure will affect the control pair by some multiplicative and additive factors.
Proof of Proposition 1.4. Let M ∈ M. Suppose there exists a fibred coarse embedding from m∈N Cay(G m ) into M with control pair (ρ, ω). Let G ∞ ∈ ∂ Cay (G m ) m∈N . By definition, there exists a subsequence (G mn ) n of (G m ) that converges to G ∞ in G(k). By Lemma 2.10, there exists a fibred coarse embedding from n Cay(G mn ) into M with control pair (ρ, ω). Thus, we may assume that (G m ) m∈N itself converges to G ∞ .
For every m ∈ N ≥1 , take R m as in (⋆) and take R ′ m as in Lemma 2.9. Let R ′′ m be the minimum of R m and R ′ m . By construction, lim m→∞ R ′′ m = +∞. Take the local trivialization t e Gm ,R ′ m : Define a map ). By (2) in Lemma 2.9, this f m is a coarse embedding with compression pair (ρ, ω).
Then, we may modify the construction of (y(g) m ) as in the proof of Lemma 3.1 by setting for every m ∈ N, otherwise.
Then it will complete our proof of Proposition 1.4.
Remark 3.2. To prove these lemma and proposition, we do not use the property that β Gm,G∞, is an isomorphism as rooted diagrams; what we needed above is this map is an isomorphism as rooted (non-labelled, non-oriented) graphs. From this point of view, we consider the space of rooted graphs with bounded degree and generalize Proposition 1.4 in the following manner; see Proposition 3.3 for the conclusion. Fix k ∈ N ≥2 , We set R(k) as the space of all connected graphs (without labellings/orientations) (Γ, r Γ ) with roots r Γ (∈ V (Γ)) such that the degrees of all vertices are at most k. We say φ : (Γ 1 , r Γ 1 ) ≃ → (Γ 2 , r Γ 2 ) is an isomorphism as rooted graphs if φ(r Γ 1 ) = r Γ 2 and if φ is a graph isomorphism. In R(k), we identify two rooted graphs that are isomorphic in the sense above. We endow R(k) with the topology of local convergence as rooted graphs. This means, ((Γ m , r Γm )) m∈N converges to (Γ ∞ , r Γ∞ ) if for every R ∈ N ≥1 , there exists m R ∈ N such that for every m ≥ m R , the R-balls B Γm (r Γm , R) and B Γ∞ (r Γ∞ , R), centered at roots, are isomorphic as rooted graphs. The space R(k), equipped with this topology, is a compact metrizable space.
Consider a sequence (Γ m ) m∈N of connected graphs with all degrees at most k. Then, each Γ m forms a (possibly, non-singleton) subset Γ m = {(Γ m , v) : v ∈ V (Γ m )} of R(k); we define the rooted graph boundary of (Γ m ) m by the set of all possible accumulation points of m∈N Γ m in R(k) as m → ∞. We write it as ∂ r (Γ m ) m∈N . If M consists only of Banach spaces, then the following holds true: If m∈N Γ m admits a fibred coarse embedding into M in a generalized sense, then ∂ r (Γ m ) m∈N admits equi-coarse embeddings into M.

3.2.
Metric ultraproducts of fragmentary actions. In Subsection 3.1, we see how to recover (non-equivariant) coarse embeddings from Cayley limit groups out of a (fibred) coarse embeddings of the disjoint union. See for instance a survey of Y. Stalder [Sta09, Theorem 3.12] for the standard argument.
On this recovery procedure, what we need is not the global actions of the whole groups G m , but local actions of balls; compare with the proof of Lemma 3.1. Here we give the definition of a fragmentary action of a subset of a group, which is a local version of the action of the whole group.
Definition 3.4. Let M be a metric space. Let G be a group and e G ∈ K ⊆ G be a subset. A partial homomorphism from K to the isometry group Isom(M) is called a fragmentary action of K on M. In other words, a right fragmentary action α : M K (where for all g ∈ K, α(g) is an isometry on M) satisfies the following property: For every g 1 , g 2 ∈ K such that g 1 g 2 ∈ K, z · α(g 1 g 2 ) = (z · α(g 1 )) · α(g 2 ) for all z ∈ M.
We use the word "fragmentary" because the terminology "partial action" is referred to a quite different concept in the literature. Proof. For every m ∈ N, take R m ∈ N and β Gm,G∞,Rm as in (⋆). otherwise.
By construction, the restriction of α ′ m on B G∞ (e G∞ , R ′′ m ) is a fragmentary action. Finally, for every g ∈ G ∞ , define α U (g) : It is straightforward to check that this is well-defined. Since lim m→∞ R ′′ m = +∞, this α U is a (global ) action of G ∞ on M U (by isometries). By assumption, it furthermore holds that for every g 1 , g 2 ∈ G ∞ , ρ(d G∞ (g 1 , g 2 )) ≤ d M U (y U · α U (g 1 ), y U · α U (g 2 )) ≤ ω(d G∞ (g 1 , g 2 )), as desired; compare with the proofs of Lemma 3.1 and Proposition 1.4.

3.3.
Key to the non-linear version of Gromov's trick. Proposition 3.5 will be used for the proof of (i).(1) of Theorem A. To deal with (i).
Definition 3.6. Let G be a group and e G ∈ K ⊆ G be a subset. Let M be a metric space and y ∈ M. Let ǫ ≥ 0. We say that a map α : K → Isom(M) is an ǫ-almost fragmentary (right) action at y if the following condition is fulfilled: For every g 1 , g 2 ∈ K such that g 1 g 2 ∈ K, d(y · α(g 1 g 2 ), (y · α(g 1 )) · α(g 2 )) ≤ ǫ.
If K = G and α is a 0-almost fragmentary at y, then α : G → Isom(M) gives rise to a genuine action on the G-orbit {y · α(g) : g ∈ G} of y.
On the proposition above, recall from Theorem A the definitions of convex subsets of geodesic spaces and of the unique geodesic condition. • For every m ∈ N and for every g 1 , g 2 ∈ B Gm (e Gm , r m ), it holds that ρ m (d Gm (g 1 , g 2 )) ≤ d Mm (y m · α m (g 1 ), y m · α m (g 2 )) ≤ ω m (d Gm (g 1 , g 2 )).
Assume that there exists a non-principal ultrafilter U over N such that M U = lim U (M m , y m ) is uniquely geodesic.
Then, for every such U over N, there exist a closed convex subset L 0 of M U and an isometric right (genuine)action (α U , L 0 ) of G ∞ that satisfy all of the following conditions: • For y U = [(y m ) m ] U , it holds that {y U · α U (g) : g ∈ G ∞ } ⊆ L 0 .
• The orbit map of y U by α U is an (equivariant) coarse embedding of (G ∞ , d G∞ ) (into L 0 ) with equivariant control pair (ρ, ω).
Here we equip L 0 with the induced metric from that of M U .
Proof. For each g ∈ G ∞ , the construction of α U (g) : M U → M U is exactly the same as one in the proof of Proposition 3.5. Indeed, since each α m (h), for h ∈ B Gm (e Gm , r m ), is isometric, α m is ǫ-almost fragmentary action at y and the "orbit map" of y m by α m is a coarse embedding with control pair (ρ m , ω m ), it follows that for each g ∈ G ∞ , recall that ρ m and ω m are non-decreasing. The construction of α U (g) above is welldefined, and α U (g) is an isometry. We, however, warn that in general, α U (gh) may not coincide with α U (g) • α U (h) (the composition is from left to right) as maps Nevertheless, we observe that α U : G ∞ → Isom(M U ) is 0-almost fragmentary action at y U because lim m→∞ ǫ m = 0. Therefore, it is a genuine action on L ′ = {y U · α U (g) : g ∈ G ∞ }. For every g 1 , g 2 ∈ G ∞ , define Because α(g 1 ) • α(g 2 ) • α(g 1 g 2 ) −1 is an isometry and we assume that M U is uniquely geodesic, each L g 1 ,g 2 is a closed convex subset of M U with L ′ ⊆ L g 1 ,g 2 . (Observe that every isometry sends geodesics to geodesics.) Finally, take Then L 0 = L 0 · α U (G ∞ ) holds, and α U gives rise to a genuine action on L 0 . We rewrite the restriction of α U on L 0 as α U : L 0 G ∞ ; it satisfies the required two conditions.
Remark 3.8. We may remove the assumption of the unique geodesic property on M U = E U if M = E consists only of Banach spaces. Indeed, if we assume that all α m are complex affine, then take L 0 as the closure of the algebraic complex affine span of L ′ ; this L 0 is a non-empty complex affine subspace of E U . Even if we do not assume it, the Mazur-Ulam theorem states that all α m are real affine. Then we can take a desired L 0 as a non-empty real affine subspace of E U .

From fibred coarse embeddings to equivariant embeddings of groups in the Cayley boundary
In this section, we prove item (i) of Theorem A. As mentioned in Introduction, our idea of the proof(s) is based on a trick of Gromov. We first demonstrate the proof of (i).(1) in Subsection 4.1. Then we proceed to the proof of (i).(2) in Subsection 4.2.
4.1. Proof for finite marked group sequences. We already know from Proposition 1.4 the way to recover (non-equivariant) coarse embeddings of groups in the Cayley boundary from local information from the fibred coarse embedding. The point in our proof is how to recover moreover equivariant coarse embeddings. The key tool here is Proposition 3.5.
Proof of (i).(1) of Theorem A. Similarly to the proof of Proposition 1.4, we may assume that (G m ) m∈N is a convergent sequence. Let G ∞ be the Cayley limit of it. Assume that m∈N Cay(G m ) admits a fibred coarse embedding into M, M ∈ M, with control pair (ρ, ω). Take as s, R ′ m , t g,R ′ m , t g 1 ,g 2 ,R ′ m as in Lemma 2.9. Fix q ∈ [1, ∞).
For each m ∈ N, set recall Definition 2.14. For We claim the following.
Then the orbit map of y m by the fragmentary action α m is an (equivariant) coarse embedding from B Gm (e Gm , R ′ m ) into M m,q with control pair (ρ, ω). Proof of Lemma 4.1. Since all t x,gx,R ′ m are isometries, α m (g) is an isometry for all g ∈ G m . Assume that g 1 , g 2 , g 1 g 2 ∈ B Gm (e Gm , R ′ m ). Then since for each x ∈ G m , Therefore, we have that by setting w x = t x,g 1 x,R ′ m (z g 1 x ), This proves (1). For (2), observe that for every g ∈ B Gm (x, R ′ m ) and every x ∈ G m , d M (y m,x , t x,gx,R ′ m (y m,gx )) = d M (t x,R ′ m (x)(s(x)), t x,R ′ m (gx)(s(gx)).
By assumption and by recalling Remark 2.17, we verify (2). By applying Proposition 3.5 with r m = R ′ m , we obtain from Lemma 4.1 an equivariant coarse embedding from G ∞ into lim U (M m,q , y m ) with equivariant control pair (ρ, ω). Since lim U (M m,q , y m ) ∈ F q (M), it proves the desired assertions.

4.2.
Non-linear version of Gromov's trick and proof for amenable group sequences. In order to extend the argument as in Subsection 4.1 to the case of amenable marked group sequences, we employ a Følner set of G m instead of G m itself and utilize Proposition 3.7. This idea dates back to Gromov, and well-known if M = Hilbert. We extend this framework to possibly non-linear settings.
For ǫ > 0 and for R ∈ N, an (ǫ, R)-Følner set F of a marked group G is a non-empty finite subset of G that satisfies Amenability of G is characterized by the existence of (ǫ, R)-Følner sets for all ǫ(> 0) and for all R (this property does not depend on the choices of markings of G).
Proof of (i).
(2) of Theorem A. We describe the modifications needed from the proof (i).

From equivariant equi-coarse embeddings of the Cayley boundary to fibred coarse embeddings
Here we prove (ii) of Theorem A. Unlike the proofs in Section 4, we do not need to impose conditions on G m , m ∈ N. First, we provide the proof where (G m ) m is a convergent sequence.
Proof of (ii) of Theorem A for the case where ♯(∂ Cay (G m ) m ) = 1. Let G ∞ be the Cayley limit of (G m ) m . For each m ∈ N, take R m and β Gm,G∞,Rm as in Lemma 2.1. Assume that there exist M ∈ M and an equivariant coarse embedding from G ∞ into M with equivariant control pair (ρ, ω), Let α : M G ∞ be an action by isometries and y ∈ M such that the orbit map G ∞ ∋ g ∞ → y ·α(g) ∈ M gives the (equivariant) coarse embedding above. Write as X = m∈N Cay(G m ).
We proceed to the proof of the general case; we here employ the class ℓ q (M). Recall the definition of an open neighborhood N(G, R) of G from Lemma 2.1. In particular, for every R ∈ N and for every 0

Proof of (ii) of Theorem
and an action α is an (equivariant) coarse embedding with equivariant control pair (ρ, ω). Fix q ∈ [1, ∞) and define Note that this is an (at most) countable ℓ q -product; hence M q ∈ ℓ q (M).
By Lemma 2.10, it suffices to construct a fibred coarse embedding in a generalized sense from X ′ into M q . Let (M q ) x = M q for all x ∈ X ′ and s : X ′ → x∈X ′ M q be s(x) = (y (r) j ) r,j . For each n R ≤ m ≤ n R+1 + R with i (R) m = i, consider the component Cay(G m ) in X ′ associated with these R and i. Set R ′ m = ⌊R/2⌋ and construct t g,R ′ m by (t g,R ′ m (x))((z) r,j ) = (w r,j ) r,j for x ∈ B Gm (g, R ′ m ) and for (z) r,j ∈ M q , where, Then in a similar argument to one in the previous proof for the case where ♯(∂ Cay (G m )) = 1, we may verify conditions (1)-(3) in Lemma 2.9; recall also Remark 2.8. Furthermore, we obtain that (ρ, ω) ∈ CP fib
Proof of Corollary B. For the case where M = Hilbert, B type r,C or CAT (0) ≤δ 0 , the assertions immediate follow from Theorem A; see also arguments in Examples 2.18 and 2.19. For M = L q , Naor and Y. Peres employed the classification of separable closed subspaces of L q -spaces and indicated a way to coming back to L q from F S q (L q ); see the last assertion of [NP11, Theorem 9.1].

Gadgets in the space of marked groups
In this section, we explain some gadgets to produce different markings of a group (or a sequence of finite groups) whose accumulation points in the space of marked groups have different nature.
6.1. Standard (restricted) wreath products and two conditions on group properties. First, we recall the definition of standard (restricted) wreath products; see also [MS13,Proposition 2.9]. For two groups G and H, G ≀ H is defined to be h∈H G ⋊H, where H acts by permutation of coordinates by right. For g ∈ G and h ∈ H, by gδ h we denote the element in h∈H G whose h-entry is g and all of the other entries are e G . We use e for the group unit of h∈H G. If G = (G; s 1 , . . . , s k ) and H = (H; t 1 , . . . , t l ) are two marked groups, then we endow G ≀ H with the standard (k + l)-marking as follows: ((s 1 δ e H , e H ), (s 2 δ e H , e H ), . . . , (s k δ e H , e H ), (e, t 1 ), (e, t 2 ), . . . , (e, t l )).
We write the marked group of G ≀ H with the standard marking above as G ≀ H. Then, for G m → G ∞ and H n → H ∞ (respectively in G(k) and G(l)) in the Cayley topology, we have that as min{m, n} → ∞, see §2.4. Theorem in [VG97]. This may be clear to the readers who are familiar with a relationship between wreath products and random walks.
We consider a group property P for finitely generated groups (or more generally, for countable discrete groups) that satisfies the following two conditions: (Conditions on the property P.) • If G has P, then so does G ≀ H for every finitely generated infinite amenable group H. • If an infinite G has P, then so does S <ℵ 0 (G) ⋊ G; recall our notation of symmetric groups from Introduction. Here G acts on S <ℵ 0 (G) by the permutations induced by right action G G.
Example 6.1. We exhibit some examples of P that satisfy the two conditions above.
(1) Amenability satisfies the two conditions above. Indeed, it is defined also for countable discrete group, and it is a classical fact that amenability is closed under taking group extensions. For the second condition, observe that S <ℵ 0 (G) above is locally finite (an increasing union of finite groups) and hence amenable.
(2) A-T-menability fulfills both of the two conditions. Indeed, it is also defined for countable discrete groups, and [CCJ + 01, Example 6.1.6] states that the extension of an a-T-menable group (normal subgroup side) by an amenable group (quotient side) is a-T-menable. The validity of the second condition is explicitly written in a celebrated paper of Cornulier-Stalder-Valette [CSV12, Example 5.4], as follows.
If H is a-T-menable, then so is S <ℵ 0 (H) ⋊ H.
To see this, more precisely, apply Corollary 5.3 to Example 3.4 with X = G in the concerning reference [CSV12]. In this work, they showed that a-Tmenability is closed under taking standard wreath products.
(3) Both of property A and the coarse embeddability into a Hilbert space, respectively, satisfy the two conditions: The proof for property A is similar to one in (1); see [ADR00] for the fact that property A is closed under taking group extensions. For the coarse embeddability, the second condition is fulfilled due to [CSV12, Theorem 5.10]; compare with (2). (4) The failure of property A (equivalently, the non-exactness in the context of C * -algebras; see [Oza00]) satisfies the two conditions. More generally, if a group property Q passes to (finitely generated) subgroups, then the property P, defined as the negation of Q, fulfills both of the conditions. For instance, we may set P as the failure of a-T-menability as well. (5) A non-example is Kazhdan's property (T); see also Remark 6.3 below. We refer the readers to [BdlHV08] for comprehensive study on this property.
Remark 6.3. The first condition on P is used in the context of the absorption lemma, Lemma 6.4. For applications, we may relax the condition to saying that G≀H remains to have P for certain infinite H, according to the way we utilize the absorption lemma. We note that even in this setting, property (T) does not fit in our framework. Indeed, if ♯(H) = ∞, then G ≀ H never has property (T) unless G is trivial; see [BdlHV08, Proposition 2.8.2] (note that notation of Γ ≀ H in [BdlHV08] corresponds to that of H ≀ Γ in the present paper).
6.2. Absorption Lemma. The following lemma, which may be called an absorption lemma, enables us to absorb a group into some abelian group by taking the wreath product by an infinite group. The original form in the paper of Bartholdi and Erschler [BE15] stated it in terms of permutational (restricted) wreath products; here we formulate it for a simpler case.
Lemma 6.4 (Special case of Lemma 6.13 in Bartholdi-Erschler [BE15]). Let G be a finitely generated group and fix (g 1 , . . . , g k ) a marking of G. For each j ∈ [k], let C j be the cyclic group of the same order as for g j . Then, for every infinite and finitely generated group P , there exists a system of marking (S m ) m∈N of G ≀ P with fixed size such that (G ≀ P ; S m ) Cay −→ (C 1 × C 2 × · · · × C k ) ≀ P, with a suitable marking of the Cayley limit group.
For the sake of completeness, we include (idea of) the proof.
Proof. Fix a marking T = (t 1 , . . . , t l ) of P . Since P is infinite, for every m ∈ N, there exists e P = x , are mutually disjoint. Now, define a system (S m ) m∈N of markings of G ≀ P by S m = ((g 1 δ e P , e P ), (g 2 δ x (m) 2 , e P ), . . . , (g k δ x (m) k , e P ), (e, t 1 ), . . . , (e, t l )), where e means the group unit of P G. Let (S m ) 1 be the set of the first k elements in the marking S m . Then the following holds true: For γ 1 , γ 2 elements in G ≀ P of the form τ −1 στ , σ ∈ (S m ) 1 , τ ∈ P , if γ 1 , γ 2 and γ 1 γ 2 are all contained in the ball B (G≀P ;Sm) (e G≀P , m) of radius m, then γ 1 and γ 2 commute. By a similar reasoning to one in the proof of [MS13, Lemma 5.1], we conclude that as m → ∞, with a suitable marking of the Cayley limit group.
Since the constant sequence of G ≀ P with a fixed standard marking converge to itself, Lemma 6.4 can be utilized as a source of producing two systems of different markings of a group that produce Cayley limit groups of quite different nature.
Note that the construction above will be moreover used to produce two systems of different markings of a sequence of finite groups with the same feature as one above, in the following way: Let G ∞ and P ∞ be (infinite) LEF marked groups and take (G m ) m∈N and (P m ) m∈N LEF approximations; recall Definition 2.2, respectively, of them. For m ∈ N ∪ {∞}, let G m and P m , respectively, be the underlying group of G m and P m . Then in the terminology as in Subsection 6.1, the standard marking G m ≀ P m of G m ≀ P m converges to G ∞ ≀ P ∞ in the Cayley topology. On the other hand, by Lemma 6.4, G ∞ ≀P ∞ admits a system of markings with respect to which the resulting marked groups converge to one with underlying group (C 1 × · · · × C k ) ≀ P ∞ . Set this sequence of marked group as ((G ∞ ≀ P ∞ ; S (∞) m )) m∈N . Then, for each m ∈ N, there exists a marking S m of G m ≀ P m such that (G m ≀ P m ; S m ) is sufficiently close to (G ∞ ≀ P ∞ ; S (∞) m ), more precisely, lim m→∞ R m = +∞ holds for R m that is taken for each m ∈ N as in (⋆) in Lemma 2.1. Then, with respect to this system of markings (S m ) m , (G m ≀ P m ) m∈N converge to a marked group whose underlying group is (C 1 ×· · ·×C k )≀P ∞ ; in this Cayley limit group, the original Cayley limit group G ∞ is "absorbed into" the abelian group C 1 × · · · × C k inside standard wreath products.
J. Brieussel and T. Zheng [BZ15] employed a construction in a similar spirit to one in the proof below and constructed several groups with remarkable properties in terms of speed of random walks and compression exponents into Banach spaces.
We describe the construction of these two systems of markings above more explicitly for the case where P ∞ = (Z; 1).
Lemma 6.5. Let G be a LEF group. Let (G m ) m∈N be a sequence of finite groups that is obtained from the underlying groups of a LEF approximation of G (with a fixed marking). Then, there exist two different systems of markings (S m ) m and (T m ) m of (G m ≀ (Z/mZ)) m∈N ≥3 such that • The sequence (G m ≀ (Z/mZ); S m ) m∈N ≥3 converges in the Cayley topology to a solvable marked group. • The sequence (G m ≀ (Z/mZ); T m ) m∈N ≥3 converges in the Cayley topology to G ≀ Z with a suitable marking.
Proof. Fix (g 1 , . . . , g k ) a marking of G. For every m ∈ N ≥3 , let (g k δ e Z/mZ , e Z/mZ ), (e, 1)), where e means the group unit of Z/mZ G m . (Hence T m is the standard marking of G m ≀ (Z/mZ).) Then, we have that where for every j ∈ [k], C j is the cyclic group of the same order as for g j .
Remark 6.6. In Lemma 6.5, if G above is generated by torsion elements, then we may have the Cayley limit group in the former assertion as a solvable group with asymptotic dimension 1.
For convenience of the readers, we summarize our arguments above.
Corollary 6.7. Let G be an infinite LEF group and P be an infinite amenable LEF group. Let (G m ) m∈N and (P m ) m∈N be sequences of underlying groups of LEF approximations, respectively, of G and P (with some markings). Let P be a group property that satisfies the first conditions of ones in Subsection 6.1. Assume that G satisfies P.
Then there exist two systems of markings (S m ) m∈N and (T m ) m∈N , • The sequence ((G m ≀ P m ; S m )) m∈N converges in the Cayley topology to a marked group whose underlying group is of the form (C 1 × · · · × C k ) ≀ P , where C j , j ∈ [k], are cyclic. In particular, this group is amenable. • The sequence ((G m ≀P m ; T m )) m∈N converges in the Cayley topology to a marked group whose underlying group is G ≀ P . In particular, this group has P. Moreover, for every m ∈ N and for each i ∈ i h i holds true. Proof. All but the last assertion follow from the arguments above. The last assertion is by our construction; compare with it for the case P = Z and P m = Z/mZ in Lemma 6.5.
Remark 6.8. K. W. Gruenberg [Gru57] showed that a wreath product G ≀ H with an infinite H is never RF unless G is abelian. Hence, our construction as in Corollary 6.7 may be available only after we extend our framework from RF approximations to LEF ones. In addition, if G is not abelian, then the Cayley convergence of (G m ≀ P m ; S m ) to the amenable marked group (C 1 × · · · × C k ) ≀ P above is a LEF approximation, but not an RF one. This is because C 1 × · · · × C k is abelian but G m for large m is not. 6.3. A gadget to encode information into symmetric groups. In this subsection, we recall our construction in [MS13,Remark 5.3] in the Part I paper, which allows us to encode all information into symmetric groups. Recall our notation of symmetric groups from Introduction. For simplicity, we only consider the case of LEF approximations: Suppose that (G m = (G m ; t . Without loss of generality, we may assume that t is a 2k-marking of S(G m ). Indeed, for every γ ∈ G m \ {e Gm }, the transposition on {e Gm , γ} may be written as some product of these 2k elements. Thus, we obtain a Cayley convergent sequence of groups in G(2k), Recall from Subsection 6.1 that G ∞ acts on S <ℵ 0 (G ∞ ) by permutations given by right multiplications.
This construction yields the following lemma.
Lemma 6.9. Let G be an infinite LEF group, and (G m ) m∈N be a sequence of underlying groups of a LEF approximation of G (with some markings). Let P be a group property. Assume that P satisfies the second condition of the two conditions as in Subsection 6.1.
Then, there exists a sequence (l m ) m∈N of integers at least 2 with lim m→∞ l m = ∞ and a sequence of markings (S m ) m∈N of (S([l m ])) m∈N such that (S([l m ]);S m ) m∈N converges in the Cayley topology to a marked group whose underlying group is isomorphic to S <ℵ 0 (G) ⋊ G. In particular, the Cayley limit group has P. ih i holds true. Proof. Take an infinite amenable LEF group P . Combine Corollary 6.7 and Lemma 6.9. 7. Examples 7.1. Special linear groups. Here we discuss coarse properties of X ′ , Y ′ , V ′ , W ′ as in Example 1.5. In our Part I paper [MS13,Remark 5.9], we observed that Remark 7.2. If M = E only consists of Banach spaces, then we assume α to be moreover affine. However, we do not go in details on this issue; recall Remark 2.7.
The following are showed by several researchers.
Indeed, for (2), exactly the same estimate of the first positive Laplace eigenvalue λ 1 for a link graph as one for a uniform lattices in SL(n, Q p ), which is given in [IN05, Section 6, Example 1], applies to that for a uniform lattices in SL(n, F p ((t −1 ))); the estimate gives that λ 1 = 1 − ( √ p/(p + 1)), where F p ((t −1 )) denotes the (local) field of formal Laurent series with indeterminate t −1 over F p . This is because local information is the same for buildings associated with PGL(n, Q p ) and for those associated with PGL(n, F p ((t −1 ))). For every prime p, the estimate above of λ 1 is strictly bigger than 1/2. Then, by [IN05, Theorem 1.1], every uniform lattice in SL(n, F p ((t −1 ))) has property (F CAT (0) ≤0 ). Even though SL(n, F p [t]) is a nonuniform lattice in SL(n, F p ((t −1 ))), we obtain the same conclusion as in (2)  Remark 7.5. Osajda pointed out to the authors that although it is implicit in his paper, the resulting group (RF but without property A) in [Osa16] is furthermore a-T-menable. To see this, he used a method developed in [Osa14] to transfer wall structures on the finite presented graphical small cancellation groups in his construction at all finitary stages to that on the infinitely presented limit group. See also [AO14]. We employ this a-T-menability in our proof of Theorem C.
Here we state the precise assertion of (i) of Theorem C.
Proposition 7.6. There exist a sequence of finite groups (G n ) n∈N and d ∈ N such that the following holds true: For every prime p and for every sequence (l n ) n∈N of integers at least 2 such that lim n→∞ l n = ∞, there exist three systems (S n ) n , (T n ) n and (U n ) n of d-markings of (H n,p (= H n,p,(ln)n ) = G n ≀ SL(2n + 3, F p ln )) n∈N , such that the following hold true: • For every n ∈ N and for every i ∈ [d], there exist h i = h n,p,i ∈ H n,p and k i = k n,p,i ∈ H n,p such that i . • The sequence ((H n,p ; S n )) n∈N converges in the Cayley topology to an amenable group. • The sequence ((H n,p ; T n )) n∈N converges in the Cayley topology to a group without property A, but the Cayley limit group is a-T-menable.
• The sequence ((H n,p ; U n )) n∈N converges in the Cayley topology to a group without property A. Moreover, the Cayley limit group is not a-M-menable for M = B type>1 or M = CAT (0) <δ(p) .
An outline of our construction in Proposition 7.6 goes as follows: We combine the construction in Subsection 6.1 (Corollary 6.7) with our examples as in Subsection 7.1 (Example 1.5), with respect to suitable markings of the Cayley limit groups. Here recall that the former limit group N > (Z, F p [t]) ⋊ Z is amenable, whereas the latter SL(Z, F p [t]) ⋊ Z contains a copy of SL(3, F p [t]), which has property (F M ) for M = B type>1 and M = CAT (0) <δ(p) .
For the sake of completeness, we exhibit our proof in a more detailed way; compare with arguments in Lemma 6.5 and Corollary 6.7.
Proof of Proposition 7.6. Let G be the (finitely generated) RF group without property A constructed in [Osa16], and S = (g 1 , . . . , g k ) be a k-marking of G. Take (G n ) n∈N an RF approximation of (G; S) (in fact, what we need here in principle are a LEF group without property A and a LEF approximation of it; see [AO14]). For every n ∈ N, write G n = (G n ; g (n) 1 , . . . , g (n) k ). Recall from Remark 7.5 that this G is a-T-menable.
Secondly, we transfer this construction to the framework of symmetric groups. The precise statement of (ii) of Theorem C is as follows.
Proposition 7.7. There exists d ′ ∈ N with the following condition. For every prime p, there exist a sequence (h n,p ) n∈N of integers at least 2 such that lim n→∞ h n,p = ∞ that satisfies the following: There exist three systems (S n ) n , (T n ) n and (Ũ n ) n of d ′ -markings of (S([h n,p ])) n∈N such that the following hold true: • For every n ∈ N and for every i ∈ [d ′ ], there existh i =h n,p,i ∈ S([h n,p ]) and k i =k n,p,i ∈ H n,p such that i . • The sequence ((S([h n,p ]);S n )) n∈N converges in the Cayley topology to an amenable group. • The sequence ((S([h n,p ]);T n )) n∈N converges in the Cayley topology to a group without property A, but the Cayley limit group is a-T-menable. Proof. Our contruction is built upon that in the proof of Proposition 7.6. Set d ′ = 2d, and take l n = n for n ∈ N in that proof. For every n ∈ N, set h n,p = ♯(H n,p ) and identify S([h n,p ]) with S(H n,p ). Transfer all constructions of (S n ) n , (T n ) n and (U n ) n into three systems (S n ) n , (T n ) n and (Ũ n ) n of d ′ -markings of (S([h n,p ])) n in the way indicated in Subsection 6.3. This together with arguments in Example 6.1 will complete our proof. Indeed, employ Theorem 6.2 for the proof of a-T-menability of the Cayley limit group of ((S([h n,p ]);T n )) n . More precisely, the concerning limit group is of the form and G is as in the proof of Proposition 7.6.
The proofs of Propisitions 7.6 and 7.7 complete the proof of Theorem C.
7.3. Embedded Banach expanders. In this subsection, we give a definition of embedded Banach expanders.
Definition 7.8. Let E be a non-empty class of Banach spaces and fix q ∈ [1, ∞). A sequence of finite connected graphs (Γ m ) m∈N of uniformly bounded degree is said to admit embedded Banach (E, q)-expanders if there exist a subsequence (m n ) n∈N of (m) m and a sequence of finite connected graphs (Λ mn ) n∈N such that all of the following hold true: • There exists D > 0 such that for each n ∈ N, there exists an injective map ι mn : V (Λ mn ) → V (Γ mn ) between the vertex sets such that the map ι mn : (V (Λ mn ), d Λm n ) → (V (Γ mn ), d Γm n ) is D-Lipschitz. • There exists d ∈ N ≥2 such that for every n, each vertex of Λ mn has degree at most d. • The number ♯(V (Λ mn )) tends to ∞ as n → ∞. • (Poincaré-type inequality) For every E ∈ E, there exists C E > 0 such that the following holds true: For every n ∈ N and for every map f mn : V (Λ mn ) → E, it holds that where m(f mn ) n denotes the mean of f mn : The sum on the right-hand side of the inequality above runs over all edges e ∈ E(Λ mn ) in Λ mn , and for each e ∈ E(Λ mn ), by writing e = (v, w) we express that e connects the vertices v and w. We say that (Γ m ) m∈N is a family of Banach (E, q)-expanders if we can take m n = m and Λ mn = Γ m (that also means that ι m = id V (Γm) ) for every n ∈ N.
The concept of ordinary expanders is one with (E, q) = (Hilbert, 2). It is known from work of Q. Cheng [Che16] that the condition of being Banach (E, q)-expanders does not depend on the choice of the exponent q ∈ [1, ∞).
The following is a variant of the well-known fact asserting that expanders do not admit a coarse embedding into a Hilbert space. For the sake of completeness, we provide a proof; compare with the proof of [NY12, Theorem 5.6.5].
Proposition 7.9. Let E be a non-empty class of Banach spaces and let q ∈ [1, ∞). If a sequence of finite connected graphs (Γ m ) m∈N of uniformly bounded degree admits embedded Banach (E, q)-expanders (Λ mn ) n∈N , then for the disjoint union m∈N (Γ m , d Γm ) does not admit a coarse embedding into E.
In particular, if (Γ m ) m∈N admits embedded (ordinary) expanders, then m∈N (Γ m , d Γm ) does not admit a coarse embedding into a Hilbert space.
Proof. Suppose that there exists a coarse embedding f : m∈N (Γ m , d Γm ) → E with control pair (ρ, ω). Then for every n ∈ N and for every v, w ∈ V (Λ mn ) adjacent in Λ mn , it holds that f (ι(v)) − f (ι(w)) ≤ ω(D). By the Poincaré-type inequality in the conditions above, we therefore have that 1 Since the right-hand side of the inequality above is independent of n, the images f (ι mn (V (Λ mn ))) must be concentrated around its mean m((f • ι mn )| V (Λm n ) ). It contradicts the properness of ρ as n → ∞, because ι mn is injective, ♯(V (Λ mn )) → ∞, and (Γ m ) m is of uniformly bounded degree.
The proof above works for a more general setting of graphs that admit weakly embedded expanders; see [AT15].
Proposition 7.10. Let F be a Banach space and let q ∈ [1, ∞). Let E be a Banach space that is sphere equivalent to F , namely, there exists a bijection Φ : S(F ) → S(E) between unit spheres such that Φ and Φ −1 are both uniformly continuous. If a sequence of finite connected graphs (Γ m ) m∈N admits embedded Banach (F, q)expanders, then it admits embedded Banach (E, q)-expanders.
There exists a notion of expanders with target in non-linear metric spaces; see [MN15] and [Nao14]. By combining this with [IN05, Proposition 6.3], we have the following.
Proposition 7.11. If a sequence of finite connected graphs of uniformly bounded degree (Γ m ) m∈N admits embedded expanders, then m∈N Γ m does not admit a coarse embedding into CAT (0) <1 .
Mendel and Naor [MN15] constructed a complete CAT(0) space M and a sequence of graphs (Γ m ) m such that (Γ m ) m forms an expander family with respect to M, but that expanders coming from random graphs are not expanders with respect to M. This M must have the Izeki-Nayatani invariant 1. 7.4. Uniformity is not automatic for a-M-menability. For a non-empty class of metric spaces, we say that a non-empty set K ⊆ G(k) is pointwise a-M-menable if every G ∈ K is a-M-menable. Concerning amenability and property (T), uniformity is automatic for Cayley-compact subsets, namely, the pointwise property automatically implies the uniform one; see [MS13,Proposition 3.4] and [MOSS15, Proposition 5.1]. In contrast, concerning a-M-menablity, uniformity is not automatic, as the example below indicates.
For each odd prime p, consider mod p reduction. Then G 0 maps onto SL(2, F p ) and (SL(2, F p ); 1 2 0 1 mod p, 1 0 2 1 mod p) as p → ∞. We write the marked group in the left-hand side as G p . Then K = {G p : p odd prime.} ∪ {F 2 } is a compact subset in G(2). This set K is pointwise a-T-menable, but not uniformly a-T-menable. Indeed, for the latter assertion, by work of A. Selberg [Sel65], it follows that (Cay(G p )) p forms an expander family; see also [Lub10]. By Proposition 7.9, there does not exist a common pair (ρ, ω) that serves as a control pair of all of the G p , p odd primes.
In this example, the obstruction to uniformity is the coarse non-embeddability, not the equivariant one of the sequence. Hence, we are able to utilize this observation to prove Corollary 1.8, as follows.
Proof of Corollary 1.8. By the way of contradiction. Assume that m∈N Γ m does not admit a coarse embedding into M. Choose an element Λ ∞ in the rooted graph boundary ∂ r (Λ n ) n∈N ; recall the definition from Remark 3.2. Then, for every m ∈ N, the subsequence (Γ m ×Λ n ) n∈N , with changing roots, has Γ m ×Λ ∞ as an accumulation point in the space of rooted graphs. Hence by Proposition 3.3, in particular, (Γ m × Λ ∞ ) m∈N , must admit equi-coarse embeddings into M. This contradicts the coarse non-embeddability of m∈N Γ m into M. 7.5. Upper triangular products. We saw in the previous subsection that by taking the disjoint union m,n∈N (Γ m × Λ n ), we can embed a copy of each Γ m and Λ n (as an isometrically embedded subgraph) in the rooted graph boundary. In what follows, we slightly modify this construction and call the resulting object the upper triangular product. We exhibit it in the context of the space of marked groups.
We write the sequence (G m ×H n ) (m,n)∈N 2 , m≤n in G(k 1 +k 2 ), with the enumeration with respect to the order above, identified with that by l ∈ N, as (G m ) m ▽ (H n ) n .
Note that for the upper triangular product, In this way, we can embed (isomorphic and isometric copies of) (G m ) m in the Cayley boundary (as subgroups of respectively suitable Cayley boundary groups).
Proof of (i) of Theorem D. Take the sequence of marked groups (G p ) p over odd primes p as in Example 7.12, and construct the upper triangular product (H l ) l∈N = (G p ) p ▽ (G p ) p . Set Γ l as Cay(H l ). Since (Cay(G p )) p forms an expander family, so does (Γ l ) l∈N . The Cayley boundary of that sequence contains an isometric copy of (Cay(G p )) p ; hence by Proposition 1.4 together with Propositions 7.9, 7.10 and 7.11, we confirm the second assertion. To see the third assertion, G. A. Margulis showed that there exists c > 0 such that for all odd prime p, girth(Cay(G p )) ≥ c · diam(Cay(G p )) holds, where the girth of a connected graph is the length of shortest cycle; see [DSV03,Appendix A]. For such sequence of finite graphs (Cay(G p )) p , T. Kondo [Kon12] constructed a complete CAT(0) space M 0 = M 0 ((Cay(G p )) p ) such that the disjoint union p Cay(G p ) embeds biLipschitzly into M 0 . Therefore, the disjoint union of (Γ l ) l admits a biLipschitz embedding into M = (M 0 × M 0 ) ℓ 2 . 7.6. Embedded expanders from fixed point property, and exotic examples from symmetric groups. Here we prove (ii) of Theorem D. First we prove the following proposition, which may be of its own interest. It may be regarded as a generalization of [MOSS15, Corollary 1.2] of our Part III paper.
Proposition 7.14. Let (G m = (G m ; s (m) 1 , . . . , s (m) k )) m∈N be a Cayley convergent sequence consisting of finite marked groups and G = (G; s 1 , . . . , s k ) be the limit. Let E be a non-empty class of Banach spaces that satisfies both of the following two conditions: (1) There exists q ∈ [1, ∞) such that for every E ∈ E, it holds that ℓ q (N, E) ∈ E.
(2) The class E can be written as a union of subclasses E = λ E λ such that each such subclass E λ satisfies the following: For every (E m ) m∈N with E m ∈ E λ for every m, there exists a non-principal ultrafilter U over N such that lim U (E m , 0) ∈ E λ . equivalent to the celebrated property (T) of D. Kazhdan; see [BdlHV08] on property (T), including this equivalence (the Delorme-Guichardet theorem).
Note that for each l ∈ N, the underlying groups of I l and J l are the same; we write it as K l . The marking of I l is of the form (b 3 , c (l) ). Here b 1 , b ′ 1 , b 2 , b 3 are associated, respectively, with σ, σ ′ , υ, τ , and c corresponds to the generator 1 of H n .
Finally, we deal with (3). In a similar argument to one above, we see that the Cayley boundary ∂ Cay (J l ) l contains an isomorphic and isometric copies of ((G m ; T m )) m∈2N+1 ≥3 (as subgroups of respectively suitable Cayley boundary groups). Now recall that (G m ; T m ) Cay −→ SL(Z, F p [t]) ⋊ Z with respect to a suitable marking of the limit, and that the Cayley limit group contains SL(3, F p [t]), which has property (F B type>1 ). Note that the class B type>1 fulfills the two conditions in Proposition 7.14. Indeed, to see (2), decompose B type>1 = r∈(1,2], C>0 B type r,C . Hence by Proposition 7.14, we conclude that (Cay(G m ; T m )) m∈2N+1 ≥3 admits embedded Banach (B type>1 , 2)-expanders. This with Propositions 7.9, 7.10 and 7.11 imply that ∂ Cay (J l ) l does not admit equi-coarse embeddings into M, where M is either of the two classes as in the assertion of (3). Thus by Proposition 1.4 we complete the proof. Here for every l ∈ N, we set k l = ♯(K l ) and identify S([k l ]) with S(K l ).
Remark 7.18. In this specific example above, we do not need to appeal to Proposition 7.17 to obtain a finitely presented lift with property (F B type>1 ). Indeed, it follows from work of H. Behr [Beh98] that SL(n, F p r [t]) is finitely presented for every prime p and for every r ∈ N ≥1 , provided that n ≥ 4. Thus the Cayley limit group SL(Z, F p [t]) ⋊ Z of our concern in the example above contains a copy of a finitely presented group SL(4, F p [t]) with property (F B type>1 ) as a subgroup.
We make a final remark, which is similar to one in the Part I paper [MS13]: The construction above is "semi -explicit" because in general, there is an issue to have an explicit generator of F × p nm . To obtain a fully explicit construction, replace coefficient rings (F p nm ) m with explicit other quotient rings of F p [t]; for instance take (F p [t]/(t nm − t)) m , and replace (t nm ∈ F p nm ) m with (t ∈ F p [t]/(t nm − t)) m .