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Licensed Unlicensed Requires Authentication Published by De Gruyter May 20, 2014

The braided Thompson's groups are of type F

Kai-Uwe Bux, Martin G. Fluch, Marco Marschler, Stefan Witzel and Matthew C. B. Zaremsky
An erratum for this article can be found here: https://doi.org/10.1515/crelle-2021-0033

Abstract

We prove that the braided Thompson’s groups Vbr and Fbr are of type F, confirming a conjecture by John Meier. The proof involves showing that matching complexes of arcs on surfaces are highly connected.

In an appendix, Zaremsky uses these connectivity results to exhibit families of subgroups of the pure braid group that are highly generating, in the sense of Abels and Holz.

Funding statement: The project was carried out by the research group C8 of the SFB 701 in Bielefeld, and all five authors are grateful for the support of the SFB. The fourth and fifth named authors also gratefully acknowledge support of the SFB 878 in Münster.

Appendix: Higher generation for pure braid groups. By Matthew C. B. Zaremsky at Binghamton

In this appendix we use techniques and results from the main body of the paper to derive higher generation properties for families of subgroups of pure braid groups. The notion of a family of subgroups of a group being highly generating was introduced by Abels and Holz [1]. It is a very natural condition, with many strong consequences, but to date few examples have been explicitly constructed of highly generating subgroups for “interesting” groups. One prominent existing example, given by Abels and Holz, is standard parabolic subgroups of Coxeter groups, or standard parabolic subgroups of groups with a BN-pair. The relevant geometry is given by Coxeter complexes and buildings. Higher generation is also used in [29] as a tool to calculate the Bieri–Neumann–Strebel–Renz invariants of right-angled Artin groups.

As an addition to the collection of interesting examples, we produce two classes of families of subgroups of the n-string pure braid group 𝑃𝐵n that we show to be highly generating. In the first case the geometry is given by complexes of arcs on a surface, related to the complexes 𝒜(Ln) from Definition 7. In the second case the geometry is given by complexes of “dangling flat braiges”, related to the complexes 𝒫n analyzed in Section 4.

In Section A we recall some definitions and results from [1], and establish a criterion for detecting coset complexes in Proposition 5. In Section B.1 we define the restricted arc complex on a surface, and in Section B.2 we define the complex of dangling flat pure braiges. The relevant families of subgroups of 𝑃𝐵n are defined in the paragraphs before Lemma 3 and Corollary 8, and in Definition 9. Finally in Section C we calculate the connectivity of these complexes and deduce that the families of subgroups are highly generating. See Propositions 6 and 13 for the exact bounds.

A Higher generation

Higher generation is defined using nerves of coverings of groups by cosets. The relevant definitions are as follows.

Definition 1 (Nerve).

Let X be a set and let 𝒰 be a collection of subsets covering X. The nerve of the cover 𝒰, denoted 𝒩(𝒰), is a simplicial complex with vertex set 𝒰 such that pairwise distinct vertices U0,,Uk span a k-simplex if and only if U0Uk.

The type of nerve we are interested in is the following coset complex.

Definition 2 (Coset complex and higher generation).

Let G be a group and let be a family of subgroups. Let 𝒰:=HG/H be the covering of G by cosets of subgroups in . We call 𝒩(𝒰) the coset complex of G with respect to , and denote it CC(G,). We say that n-generatesG if CC(G,) is (n-1)-connected, and -generatesG if CC(G,) is contractible.

The following theorem indicates some ways higher generation can be used. The first part says that 1-generation equals generation, and the second part says that a 2-generating family yields a decomposition of G as an amalgamated product.

Theorem 3 ([1, Theorem 2.4]).

Let F={HααΛ} be a family of subgroups of G. The following statements hold.

  1. is 1 -generating if and only if Hα generates G.

  2. is 2 -generating if and only if the natural map HαG is an isomorphism.

Here by Hα we mean the amalgamated product of the Hα over their intersections. We remark that another equivalent condition in part (1) is that the map HαG be surjective.

An important observation about coset complexes is that the action of the group on the complex has a very nice fundamental domain.

Observation 4 (Fundamental domain).

With the above notation, assume is finite. Since HH, we see that itself is the vertex set of a maximal simplex in CC(G,). This maximal simplex, which we call C, is a fundamental domain for the action of G on the complex CC(G,) by left multiplication.

Proof.

For any simplex σ in CC(G,), there exist H0,,Hk and gG such that the vertices of σ are the cosets gHi for 0ik. Then g-1σ is a face of C. This shows that every G-orbit intersects C, and indeed intersects C uniquely since if gHi=Hj, then gHi=Hj. ∎

A sort of converse of this observation is the following proposition, which allows us to detect highly generating families of subgroups as stabilizers of “nice” actions.

Proposition 5 (Detecting coset complexes).

Let G be a group acting by simplicial automorphisms on a simplicial complex X, with a single maximal simplex C as fundamental domain. Let

:={StabG(v)v is a vertex of C}.

Then CC(G,F) is isomorphic to X as a simplicial G-complex.

Proof.

Define a map φ:CC(G,)X by sending the coset gStabG(v) to the vertex gv of X. This is a G-invariant map between the 0-skeleta, and it induces a simplicial map since the vertices of a simplex in CC(G,) can be represented as cosets with a common left representative. Since C is a fundamental domain, φ is bijective. ∎

A good first example is when X is a tree, on which a group G acts edge transitively and without inversion. Then Theorem 3 and Proposition 5 imply that G decomposes as an amalgamated product. Namely, if e is a fundamental domain with endpoints v and w, then G=GvGeGw (this is standard Bass–Serre theory). Indeed, the vertex stabilizers are not just 2-generating, but -generating.

This example is generalized by looking at groups acting on buildings.

Example 6 (Buildings).

Let G be a group acting chamber transitively on a building Δ, by type preserving automorphisms. See [2] for the relevant background. Let C be the fundamental chamber, and let

:={StabG(v)v is a vertex of C}.

Then CC(G,)Δ, and so is highly generating for G. More precisely, if Δ is spherical of dimension n, then is n-generating, and if Δ is not spherical, then is -generating. If the action is not just chamber transitive, but is even Weyl transitive, as in [2, Chapter 6], then the stabilizers StabG(v) are precisely the maximal standard parabolic subgroups. An even stronger condition is that the action is strongly transitive, in which case G has a BN-pair, and we recover the situation in [1, Section 3.2].

We also have examples from the world of Artin groups.

Example 7 (Deligne complexes).

Background for this example can be found in [17]. Let (A,S) be an Artin system with associated Coxeter system (W,S). For TS let AT (respectively WT) be the subgroup generated by T. Let

:={ATTS with |WT|<}.

The coset complex CC(A,) is, up to homotopy equivalence, the Deligne complex of A. The connectivity of this complex, and hence the higher generation properties of this family of subgroups, is tied to the K(π,1) Conjecture described in [17]. Namely, is conjecturally -generating; see [17], in particular the conjecture, the subsequent discussion, and the definition on page 5. This is known to hold for many Artin groups, including for braid groups.

B Some variations on arc complexes and braige complexes

In this section we define and analyze some complexes on which the braid group and pure braid group act. In the first subsection we look at the restricted arc complex on a surface, and in the second subsection we look at the complex of flat dangling (pure) braiges. The restricted arc complex here will provide a coset complex for 𝑃𝐵n using arc stabilizers as subgroups. The flat dangling pure braige complex will provide a coset complex for 𝑃𝐵n using subgroups obtained via the “strand cloning maps”. These subgroups are smaller than the arc stabilizers, and more visualizable when using strand pictures for braids. We will save the connectivity calculations for Section C, after which we will conclude that these families of subgroups are highly generating.

B.1 Arc complexes

We maintain the definitions and notation from Section 3.1. Consider 𝒜(Γ) for Γ a subgraph of Kn with the same vertex set.

Terminological convention

Throughout this appendix, a subgraphΓ′′ of a graph Γ always has the same vertex set as Γ.

Given an arc system σ={α0,,αk} in 𝒜(Γ), denote by Γσ the following subgraph of Γ. Every vertex of Γ is a vertex of Γσ, and an edge e of Γ is in Γσ if and only if the endpoints of e are the endpoints of some αi. Call Γσfaithful if it has precisely (k+1) edges. Since we only consider simplicial graphs, i.e., there are no loops or multiple edges, this condition is equivalent to saying that no distinct αi, αj share both endpoints (they may share one).

The complex we are presently interested in is a complex 𝒜(Γ), which we will call the restricted arc complex.

Definition 1 (Restricted arc complex).

The restricted arc complex𝒜(Γ) on (S,P) corresponding to Γ is the subcomplex of 𝒜(Γ) consisting of arc systems σ for which Γσ is faithful. We may also write 𝒜(S,P,Γ).

We could equivalently require that the subspace of S given by the union of the arcs is a simplicial graph, i.e., has no multiple edges. In this way we can view 𝒜(Γ) as the complex of embeddings of subgraphs of Γ into S that send vertices in a prescribed way to the points of P.

Notational convention

Throughout this appendix, Ln denotes not the linear graph with n edges, but rather the linear graph with n vertices, and hence n-1 edges.

The Γ=Ln case is especially nice, since all of Ln can be embedded into any connected surface. In fact, every simplex of 𝒜(Ln) is a face of a maximal simplex of dimension n-2. See Figure 18 for some examples of arc systems.

Figure 18

From top to bottom, an arc system in 𝒜(L8)𝒜(L8), one in 𝒜(L8)𝒜(L8) and one in 𝒜(L8).

Remark 2.

Embedding graphs into surfaces is an interesting enterprise in its own right, so the complex 𝒜(Γ) may be of further general interest. For instance, the dimension of 𝒜(S,P,Γ) is one less than the number of edges in a maximal subgraph of Γ embeddable into (S,P).

Recall that Bn acts on 𝒜(Kn), and this action stabilizes 𝒜(Kn) and 𝒜(Kn). For general Γ, Bn will not necessarily stabilize 𝒜(Γ), since general braids may not stabilize P pointwise. However, pure braids do stabilize P pointwise, and so 𝑃𝐵n stabilizes 𝒜(Γ), 𝒜(Γ) and 𝒜(Γ) for any Γ.

Denote by [m] the set {1,,m} for m. Let S be the unit disk, and fix an embedding LnS of the linear graph with n vertices into S. Let P be the image of the vertex set, so P is a set of n points in S, labeled 1 through n. Under this embedding, the edges of Ln yield a maximal simplex of 𝒜(Ln), which we will denote C. For each J[n-1] define σJ to be the face of C consisting only of those arcs with endpoints j,j+1 for jJ. In particular, σJ is a (|J|-1)-simplex in 𝒜(Ln).

For each J[n-1] define

𝑃𝐵nJ:=Stab𝑃𝐵n(σJ)

and set

𝒜n:={𝑃𝐵nJJ[n-1] with |J|=1}.

Lemma 3.

As simplicial 𝑃𝐵n-complexes, the coset complex CC(𝑃𝐵n,AFn) and the restricted arc complex RA(Ln) are isomorphic.

Proof.

It suffices by Proposition 5 to show that C is a fundamental domain for the action of 𝑃𝐵n on 𝒜(Ln). A maximal simplex of 𝒜(Ln) is an embedding of Ln into S such that the vertex labeled i maps to the point in P labeled i, for each 1in. Any such simplex is in the 𝑃𝐵n-orbit of C. Moreover, if pσJ=σK for p𝑃𝐵n and σJ,σK are faces of C, then since p is pure we know that J=K. We conclude that C is a fundamental domain. ∎

In Section C we will calculate the connectivity of 𝒜(Ln), and deduce that 𝒜n is highly generating for 𝑃𝐵n. Before doing that, we describe another complex with a nice 𝑃𝐵n action.

B.2 Flat braige complexes

Definition 4 (Flat braiges).

A flat braige on n strands is a pair (b,Γ), consisting of a braid bBn and a subgraph Γ of Ln. If the edges of Γ are disjoint, we call (b,Γ)elementary. If the braid is pure, then the braige is a (flat) pure braige. See Figure 19 for some examples.

Figure 19

A flat braige on six strands and an elementary pure braige on six strands.

Note the fundamental difference between flat braiges here and “braiges”, as in Section 1.2. For flat braiges, a “merge” amounts to just choosing some pairs of adjacent strands that should be stuck together at the bottom with edges. With braiges however, the merging is more subtle; strands merge two at a time, not in a square shape but in more of a triangle, and a new strand continues down out of the merge. This new strand may merge further with other strands, but one must keep track of the order of merging. However, the notions of elementary braiges are the same here and as before, since it does not matter in which order the merges occur. The spraige in Figure 8 is a good example of how, before, we kept track of the order of merging, but with flat braiges as in Figure 19, we do not, and so the bottom of the picture is flattened out.

Let n(Ln) be the set of all flat braiges on n strands. There is a left action of Bn on n(Ln), via b(c,Γ):=(bc,Γ). We can think of n(Ln) as a simplicial complex, where (b,Γ) is a face of (b,Γ) if b=b and Γ is a subgraph of Γ. Restricting to pure braids, we get the set 𝒫n(Ln) of flat pure braiges, with an action of 𝑃𝐵n. A nice feature of this is that (id,Ln) is a fundamental domain for the action of Bn on n(Ln), or 𝑃𝐵n on 𝒫n(Ln). However, it is easy to see that n(Ln) and 𝒫n(Ln) stand little chance of being connected, since we can only “move” by changing the merges, and not the braid. To get a highly connected complex, we consider an equivalence relation on these complexes via the notion of dangling, as in Section 1.3. First we need to define what it means for a strand in a braid to be a clone.

Definition 5 (Clones).

Let bBn. Number the strands of b from left to right at their tops by 1,,n. Let ρb be the permutation induced by b under BnSn. Think of b as living in 3-space 3, with the top of the ith strand at the point (i,1,0) and the bottom at (ρb(i),0,0), for each i[n]. In particular all the tops and bottoms of the strands are in the xy-plane. Note that for any given strand, b has a representation wherein that strand is entirely contained in the xy-plane. Now suppose that for some i[n-1], b can be represented in such a way that the ith and (i+1)st strands are simultaneously in the xy-plane, and moreover, no strands of the braid other than those two intersect the closed region of the xy-plane bounded by the two strands and the line segments from (i,1,0) to (i+1,1,0) and from (ρb(i),0,0) to (ρb(i+1),0,0). In this case we will refer to the (i+1)st strand as a clone, specifically a clone of the ith strand. Note that necessarily ρb(i+1)=ρb(i)+1.

Our convention is to always consider the strand on the right to be the clone of the strand on the left, as opposed to the other way around. See Figure 20 for an example.

Figure 20

The sixth strand is a clone of the fifth.

For each i[n-1] there is a cloning mapκi:Bn-1Bn given by cloning the ith strand. This is not a homomorphism, but becomes one when restricted to κi:𝑃𝐵n-1𝑃𝐵n. For I={i1,,ir}[n-r], with i1<<ir, define the cloning map

κI:=κi1κir:Bn-rBn.

The restriction κI:𝑃𝐵n-r𝑃𝐵n is again a homomorphism. For J={j1,,jr}[n-1], with j1<<jr, let IJ[n-r] be the set {ji-(i-1)1ir}. The point is that a braid bBn is in the image of κIJ if and only if for each jJ, the (j+1)st strand is a clone of the jth strand. Denote the subset of such braids by Bn(J), and the subgroup of such pure braids by 𝑃𝐵n(J). (The parentheses distinguish 𝑃𝐵n(J) from the arc system stabilizer 𝑃𝐵nJ from the previous section.)

We can now define the equivalence relation between flat braiges, given by dangling. This is closely related to the notion of dangling in Section 1.3.

Definition 6 (Dangling flat braiges).

Let (b,Γ) be a flat braige on n strands, and number the vertices of Γ by 1,,n from left to right. Let JΓ[n-1] be the set of left endpoints of edges of Γ. Now consider any braid c from the set Bn(JΓ). For each jJΓ, we know that ρc(j+1)=ρc(j)+1, so there is a subgraph of Ln whose edges are precisely those connecting ρc(j) and ρc(j+1) for jJΓ. Call this graph Γc. The point is that, if we draw c below the braige, and “pull” the merges through c, we get the flat braige (bc,Γc). Now declare that (b,Γ) is equivalent to (bc,Γc) for each cBn(JΓ). One checks that this is an equivalence relation, called equivalence under dangling. Denote the equivalence class of (b,Γ) by [(b,Γ)], and call it a dangling flat braige. The idea is that the top of a braige is static, but the strands at the bottom are free to “dangle”, modulo the restriction that the merges remain rigid (and oriented) during the dangling. We analogously get the notion of a dangling flat pure braige, where we only consider c as above coming from 𝑃𝐵n(JΓ), so in particular Γc always equals Γ in the pure case. An example of dangling can be seen in Figure 21, and refer back to Figure 9 for comparison with the non-flat case.

Figure 21

The two (elementary) braiges on the top are equivalent under pure dangling, but are not equivalent to the third.

The key difference between dangling for flat braiges and dangling for braiges is a matter of which braid is considered to be the one acting. For braiges, the braid acting by dangling has as many strands as feet of the braige; for flat braiges, the braid acting is the image of this braid under a cloning map, so has as many strands as there are strands of the flat braige just above the merges.

Let n(Ln) be the set of equivalence classes under dangling of flat braiges in n(Ln). The simplicial structure of the latter induces a simplicial structure on the former, for example the faces of [(b,Γ)] are precisely of the form [(bc,Γ)], for cBn(JΓ) and Γ a subgraph of Γc. Also let 𝒫n(Ln) be the set of dangling flat pure braiges. The faces of a dangling flat pure braige [(p,Γ)] are the dangling pure braiges of the form [(pc,Γ)] for c𝑃𝐵n(JΓ) and Γ a subgraph of Γ. Heuristically, in n(Ln) we can move around not only by changing the merges, but now also by changing the braid in certain controlled ways, so n(Ln) and 𝒫n(Ln) stand a chance of being connected (for large enough n), and even highly connected. In the pure case we can also define 𝒫n(Γ) for any subgraph Γ of Ln, by only considering flat braiges from 𝒫n(Γ). We also have the subcomplexes of dangling elementary braiges or dangling elementary pure braiges, denoted n(Ln) and 𝒫n(Ln) respectively. In the pure case, note that 𝒫n(Ln) is identical to the complex 𝒫n analyzed in Section 4; in particular we already know its connectivity. Moreover in the pure case we can use any subgraph Γ of Ln, and get the complex 𝒫n(Γ). This will be an important subcomplex for proving that 𝒫n(Γ) is highly connected.

The left action of Bn on n(Ln) induces an action of Bn on n(Ln); for cBn we have c[(b,Γ)]:=[(cb,Γ)]. Similarly, 𝑃𝐵n acts from the left on 𝒫n(Ln), and indeed stabilizes 𝒫n(Γ) for any subgraph Γ of Ln. The action of 𝑃𝐵n on 𝒫n(Ln) is of particular interest, since there is a fundamental domain consisting of a single maximal simplex, namely [(id,Ln)]. This tells us that 𝒫n(Ln) is a coset complex, using the family of stabilizers of faces of [(id,Ln)].

Lemma 7 (Stabilizers of dangling braiges).

Let Γ be a subgraph of Ln. Then the stabilizer Stab𝑃𝐵n([(id,Γ)]) is precisely the subgroup 𝑃𝐵n(JΓ).

Proof.

First let p𝑃𝐵n(JΓ). Then we have p[(id,Γ)]=[(p,Γ)]=[(id,Γ)]. Now suppose p[(id,Γ)]=[(id,Γ)], so [(p,Γ)]=[(id,Γ)]. Then there exists an element c𝑃𝐵n(JΓ) such that (p,Γ)=(c,Γ). But this implies that p=c, so we are done. ∎

Let n:={𝑃𝐵n(JΓ)Γ is a subgraph of Ln with one edge}.

Corollary 8.

The coset complex CC(𝑃𝐵n,BFn) is isomorphic to PBn(Ln) as a simplicial 𝑃𝐵n-complex.

Proof.

This is immediate from Proposition 5 above, since [(id,Ln)] is a fundamental domain. ∎

In the next section we will calculate the connectivity of 𝒜(Ln) and 𝒫n(Ln), and hence of CC(𝑃𝐵n,𝒜n) and CC(𝑃𝐵n,n), from which we deduce higher generation.

We close this section by setting up a generalization of the complexes we have constructed. Note that in the definition of 𝒜n we require |J|=1, and in the definition of n we require Γ to have only one edge (this is the same as saying |JΓ|=1). The subgroups in these families consist of braids that, respectively, stabilize some arc, or feature at least one cloned strand. Of course, as n grows, it becomes increasingly “easy” for a braid to be very complicated while still featuring a cloned strand, or stabilizing an arc. Hence, higher generation becomes an even more interesting question if we consider requirements like, e.g., all but five strands are clones. (Observe that any of the standard generators of 𝑃𝐵n satisfy this very requirement.)

Definition 9 (More restrictive families).

Let s. Define

𝒜ns:={𝑃𝐵nJJ[n-1] with |J|=s}.

Also define

ns:={𝑃𝐵n(JΓ)Γ is a subgraph of Ln with s edges}.

Hence 𝒜n1=𝒜n and 𝒜nn-1={Z(𝑃𝐵n)}, also n1=n and nn-1={{1}}.

C Connectivity of the complexes

For define η():=-24. The main goal of this section is to prove that 𝒜(Ln) and 𝒫n(Ln) are (η(n)-1)-connected. Note that this is slightly different from the function η defined before Theorem 10; we do this because here the symbol Ln denotes a graph with n-1 edges and there it had n edges.

Theorem 1 (Restatement of Theorem 10 using current notation).

Let Γm be a subgraph of Ln with m edges. Then MA(Γm) is (η(m+1)-1)-connected.

In particular 𝒜(Ln) is (η(n)-1)-connected.

C.1 Connectivity of arc complexes

Our first goal is to deduce the connectivity of the complex 𝒜(Ln) from Theorem 1. We will use the notion of defect from Section 3.2. Let Γm be a subgraph of Ln with m edges. For a k-simplex σ={α0,,αk} in 𝒜(Γm), define r(σ) to be the number of points in P that are used as endpoints of arcs in σ. As in Section 3.2, define the defectd(σ) to be 2(k+1)-r(σ). Let h be the function on the barycentric subdivision 𝒜(Γm) of 𝒜(Γm) given by h(σ)=(d(σ),-dim(σ)), ordered lexicographically. Note that d(σ)=0 if and only if the arcs are all disjoint, even at their endpoints. Hence, thinking of h as a height function on the vertices of 𝒜(Γm), in the sense of [5], we observe that the sublevel set (𝒜(Γm))d=0 is precisely 𝒜(Γm). Hence we can compare the homotopy types of the two complexes using discrete Morse theory, with [5, Corollary 2.6] as the guide. The key is to inspect the descending links with respect to h. This is very similar to the procedure used before to deduce connectivity of 𝒜(Kn) from connectivity of 𝒜(Kn), but we will repeat many arguments for convenience.

Proposition 2.

The complex RA(Γm) is (η(m+1)-1)-connected.

Proof.

By Theorem 1, 𝒜(Γm) is (η(m+1)-1)-connected. We claim that the inclusion 𝒜(Γm)𝒜(Γm) induces a surjection in homotopy πk for kη(m+1)-1, from which the proposition follows. To prove the claim, it suffices by [5, Corollary 2.6] to prove that for σ𝒜(Γm)𝒜(Γm), i.e., h(σ)>0, the descending link lk(σ) is (η(m+1)-2)-connected. We suppose that σ is a k-simplex, with σ={α0,,αk}.

There are two types of arc systems in the descending link lk(σ). First, we could have σ<σ and h(σ)<h(σ). Then σ is obtained from σ by removing arcs and strictly decreasing the defect. Call the full subcomplex of lk(σ) spanned by these σ the down-link. Second, we could have σ~>σ and h(σ~)<h(σ). Here σ~ is obtained by adding new arcs to σ, so that the new arcs are all disjoint from each other and from any existing arcs, even at endpoints. Call the full subcomplex of lk(σ) spanned by such σ~ the up-link. Any simplex in the down-link is a face of every simplex in the up-link, so lk(σ) is the join of the down-link and up-link.

First consider the down-link. A face σ of σ fails to be in the down-link if and only if each arc in σσ is disjoint from every other arc of σ, since then and only then do σ and σ have the same defect. Let σ0 be the face of σ consisting precisely of all such arcs, if any exist. Since d(σ)>0, we know σ0σ. The boundary of σ is a (k-1)-sphere, and the complement in the boundary of the down-link is either empty, or is a cone with cone point σ0. Hence the down-link is either a (k-1)-sphere or is contractible, so in particular is (k-2)-connected. At this point we may assume without loss of generality that the down-link is a (k-1)-sphere, and so every arc in σ shares an endpoint with some other arc in σ. This means that every edge of Γσ shares an endpoint with some other edge of Γσ. In particular k1.

Now consider the up-link. The simplices in the up-link are given by adding arcs to σ that are all disjoint from each other and from the arcs in σ. Consider the connected surface S:=S{α0,,αk}, obtained by cutting out the arcs αi. If P:=SP, then we have |P|=n-r(σ). Also let Γm-2k-2 be the subgraph of Γm obtained by removing the edges of Γσ, and all edges sharing a vertex with any of these, so Γm-2k-2 has at most m-2k-2 edges (here we use the fact that every edge of Γσ shares an endpoint with some other edge of Γσ). The up-link of σ is isomorphic to the matching complex 𝒜(S,P,Γm-2k-2), which is (η(m-2k-1)-1)-connected. Since lk(σ) is the join of the down- and up-links, we conclude that lk(σ) is (η(m-2k-1)+k-1)-connected.

We have

η(m-2k-1)+k-1m-2k-34+k-2
η(m+1)+k2-52η(m+1)-2

since k1, and so we are done. ∎

The next corollary is immediate, keeping in mind that with our notation the graph Ln has n-1 edges.

Corollary 3.

The complex RA(Ln) is (η(n)-1)-connected.∎

Corollary 4.

The complex CC(𝑃𝐵n,AFn) is (η(n)-1)-connected, and hence AFn is η(n)-generating for 𝑃𝐵n.

Proof.

This is immediate from Lemma 3 and Corollary 3. ∎

We also want to show that the families 𝒜ns from Definition 9 are highly generating. For s>1, the coset complex CC(𝑃𝐵n,𝒜ns) is obtained up to homotopy equivalence from CC(𝑃𝐵n,𝒜ns-1) by removing the open stars of the vertices, i.e., the cosets pPBnJ for |J|=s-1. Hence the problem amounts to showing high connectivity of links. This is more or less the procedure done in the proof of [1, Theorem 3.3], in the context of buildings. It is a bit harder here though; links in buildings are themselves buildings, but links in restricted arc complexes are not themselves restricted arc complexes. Nonetheless, we can get the right connectivity without too much extra work.

Lemma 5 (Links in RA(Γm)).

Let σ={α0,,αk} be a k-simplex in RA(Γm) for Γm as above (with m edges). Then the link lkRA(Γm)(σ) is (η(m-k)-1)-connected.

To make precise the terminology, here by “link” we mean the subcomplex of simplices τ disjoint from σ for which there exists a simplex with τ and σ as faces.

Proof.

Set L:=lk𝒜(Γm)(σ). An arc system τ is in L if and only if each arc of τ is distinct from, but compatible with, every αi. For such an arc system τ, by retracting each arc αi to a point, τ maps to an arc system in 𝒜(Γm-(k+1)). Here Γm-(k+1) is a subgraph of Γm with m-(k+1) edges. More formally, for 0dk consider the homotopy equivalence of surfaces rd:SSd, obtained by collapsing αi to a point, for each 0id. Recall S=Dn, and here Sd is just our name for the copy of Dn-(d+1) obtained by collapsing these arcs. Here we do not think of Dn as a punctured disk, but rather as a disk with n distinguished points; hence rd is really a homotopy equivalence. Also let Pd be the image of P under rd. We have induced maps of complexes Rd:L𝒜(Γm-(d+1)). Note that these maps are surjective, but not injective; see Figure 22 for an example of the non-injectivity. Note however that the connectivity of 𝒜(Γm-(k+1)) is precisely the connectivity we are trying to verify for L.

Figure 22

Distinct arcs in the link of σ that map to the same arc under Rd.

The rd also induce epimorphisms

φd:Stab𝑃𝐵n(σ)𝑃𝐵n-(d+1),

with kernels Kd:=ker(φd). Also declare K-1 to be the trivial subgroup. Note that

K-1K0Kk.

Colloquially, the pure braids p in KdKd-1 are precisely those that do “twist” αd but do not twist any αi for i>d. For pKk, define

D(p):=min{d+1pKd}.

We will call D(p) the deviation of p; note that D(p)=0 if and only if p=id. Now fix a map sid:SkS with sidrk homotopic to the identity. This essentially amounts to fixing a choice of how to “blow up” each arc αi to get from Sk back to S. We get an induced map

ιid:𝒜(Γm-(k+1))L,

with Rkιid equal to the identity on 𝒜(Γm-(k+1)). For each pKk, set ιp:=pιid. These maps are all injective simplicial maps that can be thought of as different choices of how to blow up each αi, and we see that Rkιp is the identity for all p. Every arc system in L is the image of an arc system in 𝒜(Γm-(k+1)) under some ιp, so L=p𝑃𝐵nIm(ιp). Also, each Im(ιp) is isomorphic to 𝒜(Γm-(k+1)), and hence is an (η(m-k)-1)-connected subcomplex of L. We now need to glue these images Im(ιp) together in a clever order, always along (η(m-k)-2)-connected relative links, from which we will deduce that the link L is (η(m-k)-1)-connected.

The measurement D(p) provides such an order. For 0dk let

Ld:=D(p)dIm(ιp).

We claim that Ld is (η(m-k)-1)-connected for all d. The base case d=0 is clear. For a given d, the intersection Im(ιp)Im(ιq) with pq and D(p)=D(q)=d+1 is contained in Ld. This is because p and q must twist the arc αd differently, and so if β is an arc in Im(ιp)Im(ιq), then β cannot share endpoints with αd. For this reason, we can build up from Ld to Ld+1 by attaching the Im(ιp) with deviation d+1, in any order, and the relative links will always be in Ld. Now, for p with D(p)=d+1, we attach Im(ιp) to Ld along the intersection Im(ιp)Ld. This intersection consists precisely of those arc systems in Im(ιp) that do not use arcs sharing endpoints with αd. Applying Rk (so retracting each αi to a point), this gives us the subcomplex of 𝒜(Γm-(k+1)) whose arcs are disjoint from the endpoint obtained by collapsing αd. But this is just 𝒜(Γ) for Γ a subgraph of Γm-(k+1) with at most two fewer edges. This is (η(m-k)-2)-connected, and so we are done. ∎

Proposition 6.

For sN, CC(𝑃𝐵n,AFns) is (η(n-(s-1))-1)-connected, and hence AFns is (η(n-(s-1)))-generating for 𝑃𝐵n.

Proof.

It suffices to show that for |J|=s-1, the link of 𝑃𝐵nJ in CC(𝑃𝐵n,𝒜ns-1) is (η(n-(s-1))-1)-connected. Equivalently, we need the link of σJ in 𝒜(Ln) to be (η(n-(s-1))-1)-connected. Since σJ is a (|J|-1)-simplex, its link is (η(n-|J|)-1)-connected by Lemma 5 (since Ln has n-1 edges), and since |J|=s-1, we conclude that indeed the link is (η(n-(s-1))-1)-connected. ∎

C.2 Connectivity of flat braige complexes

Now we inspect CC(𝑃𝐵n,n) or more accurately 𝒫n(Ln). To pass from the world of arcs to the world of flat braiges, we will project the flat braiges onto arcs in the following way. For each J[n-1], let σJ be the simplex of 𝒜(Ln) defined before Lemma 3. Consider the action of 𝑃𝐵n on 𝒜(Ln) as a right action, and define a map

π:𝒫n(Ln)𝒜(Ln),
(σJΓ)p-1,

where JΓ is as in Definition 6. We will use π to also denote the restrictions

𝒫n(Ln)𝒜(Ln),
𝒫n(Γ)𝒜(Γ),
𝒫n(Γ)𝒜(Γ)

for Γ a subgraph of Ln. As in Section 4, think of π as the procedure of combing the braid straight and watching where the arcs get moved.

Proposition 7 (Flat braige connectivity from arc connectivity).

For Γm a subgraph of Ln with m edges, EPBn(Γm) is (η(m+1)-1)-connected.

When Γm=Ln, this is just Corollary 5. Indeed the proof here is more or less the same, but we will repeat it for convenience.

Proof.

By Theorem 1, 𝒜(Γm) is (η(m+1)-1)-connected. Let σ={α0,,αk} be a k-simplex in 𝒜(Γm). The link lk(σ) of σ in 𝒜(Γm) is isomorphic to 𝒜(Γ) for Γ a subgraph of Γm with at least m-3(k+1) edges, so lk(σ) is (η(m-3(k+1)+1)-1)-connected, and hence (η(m+1)-k-2)-connected. It now suffices by [31, Theorem 9.1] to prove that the fiber π-1(σ) is (k-1)-connected (here we treat a simplex as a closed cell). Indeed, we will prove that π-1(σ) is the join of the fibers π-1(αi) of the vertices αi of σ. See also Proposition 3.

Let 𝒥𝒱:=i=0kπ-1(αi) be the join of the vertex fibers. Clearly π-1(σ)𝒥𝒱. Also, the 0-skeleton of 𝒥𝒱 is contained in π-1(σ). Now suppose that the same is true of the r-skeleton for some r0. An (r+1)-simplex in 𝒥𝒱 is the join of a 0-simplex and an r-simplex, both of which are contained in π-1(σ). It now suffices to prove the following claim.

Claim.

Let [(p,E)] be a vertex in EPBn(Γm), where p𝑃𝐵n and E is a one-edge subgraph of Γm. Let [(q,Γ)] be a simplex in EPBn(Γm) such that π([(q,Γ)]) does not contain π([(p,E)]) but does share a simplex with π([(p,E)]) in MA(Γm). Then [(q,Γ)] shares a simplex with [(p,E)] in EPBn(Γm).

This hypothesis is rephrased in terms of arcs as: (Γ)q-1 shares a simplex with (E)p-1. By acting from the left with 𝑃𝐵n, we can assume without loss of generality that p=id, so we have π([(p,E)])=E. Let {β0,,β}:=(Γ)q-1, chosen so that E is disjoint from the βi, even at endpoints (remember we are in 𝒜(Γm), not just 𝒜(Γm)). This is possible by the hypothesis, and implies that the dangling equivalence class [(q,Γ)] contains a representative in which the (j+1)st strand is a clone of the jth strand, where j and j+1 are the endpoints of the edge of E. We can assume (q,Γ) itself is such a representative, in which case the dangling flat braige [(q,ΓE)] is a simplex of 𝒫n(Γm) containing [(q,Γ)] and [(p,E)], proving the claim. ∎

It might be possible to mimic this proof using π:𝒫n(Γ)𝒜(Γ) instead, and get the connectivity of 𝒫n(Ln) right away, but the downside is that the fibers are not joins of vertex fibers. Hence one would have to do extra work to show that fibers have the right connectivity.

To calculate the connectivity of the complex 𝒫n(Γm), we will use a similar procedure as for 𝒜(Γm). Namely, we will build up from 𝒫n(Γm) to 𝒫n(Γm) using discrete Morse theory. A k-simplex in 𝒫n(Γm) is a dangling equivalence class of a pair (p,Γ), for p𝑃𝐵n and Γ a subgraph of Γm with k+1 edges. Let r(Γ) be the number of vertices that are endpoints of an edge in Γ. Then define the defectd(p,Γ) to be 2(k+1)-r(Γ). Extend these definitions to the dangling equivalence classes, and observe that 𝒫n(Γm) is the d=0 sublevel set of 𝒫n(Γm). We now apply Morse theory, as before.

Proposition 8.

The complex PBn(Γm) is (η(m+1)-1)-connected.

Proof.

By Proposition 7, 𝒫n(Γm) is (η(m+1)-1)-connected. Mimicking the proof of Proposition 2, it suffices to prove that for σ𝒫n(Γm)𝒫n(Γm), the descending link lk(σ) is (η(m+1)-2)-connected. Let σ be such a k-simplex, say σ=[(p,Γ)]. The down-link is either Sk-1, or contractible if Γ has an isolated edge. Suppose there is no such isolated edge, so the down-link is Sk-1. Now, the up-link is obtained by dangling and then adding extra edges to the graph such that the new edges are disjoint from Γ and from each other. Since Γ has no isolated edges, there are at most 2(k+1) edges of Γm that share an endpoint with an edge of Γ. Hence the up-link of σ is isomorphic to 𝒫(Γm-2k-2) for some , which is (η(m-2k-1)-1)-connected. The calculation from the proof of Proposition 2 now tells us that lk(σ) is (η(m+1)-2)-connected. ∎

Corollary 9.

The complex PBn(Ln) is (η(n)-1)-connected.∎

Corollary 10.

The complex CC(𝑃𝐵n,BFn) is (η(n)-1)-connected, and hence BFn is η(n)-generating for 𝑃𝐵n. ∎

Example 11.

For n6, CC(𝑃𝐵n,n) is connected, so 𝑃𝐵n has a generating set in which each generator features at least one cloned strand. Indeed, the standard generating set from [27, Section 1.3.1] satisfies this property for n6, and fails for n<6. For n10, CC(𝑃𝐵n,n) is simply connected, so 𝑃𝐵n is 2-generated by n. Hence there exists a presentation for 𝑃𝐵n in which every generator features a cloned strand, and the relations all arise from relations in the subgroups of braids with a cloned strand. Again we note that the standard presentation works precisely in this range.

We conclude by showing that the families ns for s, defined in Definition 9, are highly generating as well. Just like in the arc case, for s>1 the coset complex CC(𝑃𝐵n,ns) is obtained up to homotopy equivalence from CC(𝑃𝐵n,ns-1) by removing the open stars of vertices, i.e., cosets pPBn(J) for |J|=s-1.

Figure 23

The map φ takes an element of lk𝒫5(L5)(σ) to an element of 𝒫4(L4). Here σ is [(id,E4)], for E4 the subgraph with a single edge indicated by the dashed line.

Lemma 12 (Links in PBn(Γm)).

Let σ be a k-simplex in PBn(Γm) for Γm as above (with m edges). Then the link lkPBn(Γm)(σ) is (η(m-k)-1)-connected.

Proof.

Links in the flat braige case are nicer than links in the arc case, since they are actually isomorphic to smaller dangling flat braige complexes. In the arc case, namely in the proof of Lemma 5, we related a given link to a smaller arc complex, via a map that was not an isomorphism. In the present case, we claim that lk𝒫n(Γm)(σ) is just isomorphic to 𝒫n-(k+1)(Γm-(k+1)), for Γm-(k+1) a graph with m-(k+1) edges, and then the connectivity result is immediate. Say σ=[(p,Γk+1)] for Γk+1 a subgraph of Γm with k+1 edges.

Let L:=lk𝒫n(Γm)(σ). The simplices in the link L are dangling flat braiges of the form τ=[(pq,Γ)], where q𝑃𝐵n(JΓk+1) and Γ is a subgraph of Γm having no edges in common with Γk+1. The first condition ensures that τ and σ share a simplex, namely [(pq,ΓΓk+1)], and the second condition ensures that τ and σ are disjoint. Acting from the left with 𝑃𝐵n, we can assume p=id. We have a map φ:L𝒫n-(k+1)(Γm-(k+1)), where Γm-(k+1) is the graph with n-(k+1) vertices that is obtained from Γm by retracting each edge of Γk+1 to a point. The map φ sends τ=[(q,Γ)] to [(q,Γ)], where Γ is the image of Γ under the retraction ΓmΓm-(k+1), and q is the preimage of q under the cloning map κJΓk+1. See Figure 23 for an example.

Since q is uniquely determined by q, we have an inverse φ-1, induced by the cloning map. (This is the essential difference from the arc case, that there is only one way to “blow up” a braige via cloning.) Since φ and φ-1 are of course simplicial maps, we conclude that φ is a simplicial isomorphism, and the result follows. ∎

Proposition 13.

For sN, CC(𝑃𝐵n,BFns) is (η(n-(s-1))-1)-connected, and hence BFns is η(n-(s-1))-generating for 𝑃𝐵n.

Proof.

As in the proof of Proposition 6, it suffices to prove that for Γ with s-1 edges, the link of the (s-2)-simplex [(id,Γ)] in 𝒫n(Ln) is (η(n-(s-1))-1)-connected. Since Ln has n-1 edges, this follows from Lemma 12. ∎

Example 14.

To generalize Example 11, we have that for any n6, nn-5 is 1-generating for 𝑃𝐵n. This means that 𝑃𝐵n has a set of generators such that in each generator, all but 5 strands are clones (indeed the standard generators have this property). Similarly for n10, nn-9 is 2-generating for 𝑃𝐵n, so 𝑃𝐵n has a presentation in which each relation can be realized by using only nine non-clone strands. Again, the standard presentation fits the bill.

Example 15.

In the situation of arcs, the swing presentation for 𝑃𝐵n, described in [28, Section 4], provides an explicit example of 𝒜nn-5 being 1-generating for n6 and 𝒜nn-9 being 2-generating for n10. In this presentation the generators are Dehn twists, each of which must stabilize at least one arc of the form σj, as soon as n6. Each relation in [28, Theorem 4.10] (specifically the second presentation) is a product of Dehn twists, and for n10 this product stabilizes at least one arc of the form σj. See Figure 24 for an example.

Figure 24

With six points, each generator must stabilize an arc. With ten points, each relation must stabilize an arc. The dashed lines indicate the arcs stabilized in the examples. The relation pictured here is a lantern relation, as in [28, Figure 12].

Acknowledgements

We are grateful to Andy Putman for suggesting a new strategy to handle the complexes 𝒜(Γ) in Section 3, and for referring us to his paper [‘Representation stability, congruence subgroups, and mapping class groups’, preprint 2013]. We also thank Matt Brin and John Meier for explaining the backstory of this problem to us, and the anonymous referee for corrections and a helpful comment.

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Received: 2013-04-08
Revised: 2014-02-05
Published Online: 2014-05-20
Published in Print: 2016-09-01

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