Proof.

Since T has isotropy groups in $\mathcal{ℱ}$, we have that $G_t\in \mathcal{ℱ}$ and hence the sequence

$\mathbb{Z}[T^{G_t}]\stackrel{𝑝}{\to }\mathbb{Z}[V^{G_t}]\stackrel{\mathit{ϵ}}{\to }\mathbb{Z}\to 0$

is exact. Therefore the fixed-point set $V^{G_t}$ is non-empty. Choose ${\star }_t\in V^{G_t}$. Consider the expression

$p(t)=\sum _{v\in V}{n}_{t,v}v$

where each $n_{t,v}$ is an integer. Define $W_t=\{v\in V:n_{t,v}\ne 0\}$. Define

$V_t=W_t\cup \{{\star }_t\},$

and observe that $V_t$ is a finite $G_t$-invariant subset of V. The edge set $E_t$ is defined as the following subset of $V_t\times \mathbb{N}$,

$E_t=\{v\times j:v\in V_t\text{ and }1\le j\le |n_{t,v}|\}.$

For $e=v\times j$ define $e_{+}=v$ and $e_{-}={\star }_t$ if $n_{t,v}>0$, and $e_{+}={\star }_t$ and $e_{-}=v$ if $n_{t,v}<0$. Since each edge of $E_t$ has ${\star }_t$ as one of its endpoints, ${\mathrm{\Gamma }}_t=(V_t,E_t)$ is connected (when considered as a 1-complex). The $G_t$-action on $V_t$ induces an action on $E_t$ by

$g.(v\times j)=g.v\times j$

respecting the adjacency relation. Since $ϵ\circ p(t)=0$ and ϵ is the augmentation map, we have that ${\sum }_{v\in V_t}{n}_{t,v}=0$. Therefore,

$\sum _{e\in E_t}(e_{+}-e_{-})=\sum _{v\in V_t}{n}_{t,v}(v-{\star }_t)=p(t).\mathit{∎}$

Proof.

Observe that E is isomorphic to the K-space ${\coprod }_{i\in I}K/K_i$ where I is a complete set of representatives of K-orbits of E and $K/K_i$ denotes the K-space of left cosets of $K_i$ in K. Define $\widehat{E}$ as ${\coprod }_{i\in I}G/K_i$ and observe that the assignment $K_i\mapsto K_i$ induces an injective K-map $E\hookrightarrow \widehat{E}$. The three statements of the lemma are observations. ∎

Proof.

Let Γ be the directed G-graph whose vertex set is the G-set V, and edge G-set E is defined as follows.

Let $R_T$ be a complete set of representatives of G-orbits of T. For $t\in R_T$, let ${\mathrm{\Gamma }}_t=(V_t,E_t)$ be the directed $G_t$-graph of Step 1. Let ${\widehat{E}}_t$ be the G-set provided by Step 2 for the $G_t$-set $E_t$. The $G_t$-maps ${(-)}_t:E_t\to V$ and ${(+)}_t:E_t\to V$ defining initial and terminal vertices extend to G-maps ${(-)}_t:{\widehat{E}}_t\to V$ and ${(+)}_t:{\widehat{E}}_t\to V$. Let E be the disjoint union of G-sets $E={\coprod }_{t\in R_T}{\widehat{E}}_t$ and let $(+):E\to V$ and $(-):E\to V$ be the induced maps. Note that E has finitely many G-orbits.

By construction, the group G acts on the directed graph $\mathrm{\Gamma }=(V,E)$ with finitely many orbits of vertices and edges. Moreover, the G-action on Γ as a complex has no inversions (if an edge is fixed setwise, the it is fixed pointwise). The stabilizer $G_e$ of an edge $e\in E$ is the intersection of two subgroups in $\mathcal{ℱ}$, hence $G_e\in \mathcal{ℱ}$. It remains to show that the fixed-point sets ${\mathrm{\Gamma }}^K$ are non-empty and connected for $K\in \mathcal{ℱ}$.

Let $h:\mathbb{Z}[T]\to \mathbb{Z}[E]$ be the morphism of $\mathbb{Z}G$-modules given by $t\mapsto {\sum }_{e\in E_t}e$ for $t\in R_T$. Note that h is well defined since the subgroup $G_t$ preserves $E_t$ setwise. Since

$\sum _{e\in E_t}(e_{+}-e_{-})=p(t)$

by the definition of $E_t$, it follows that for each $K\in \mathcal{ℱ}$ the following diagram commutes:

where $\partial$ is the standard boundary map from 1-chains to 0-chains of Γ. Since the top row is an exact sequence, it follows that $H_0({\mathrm{\Gamma }}^K,\mathbb{Z})\cong \mathbb{Z}$ and hence ${\mathrm{\Gamma }}^K$ is non-empty and connected. ∎

Proof of Corollary 3.5.

Consider the G-set $V={\coprod }_{P\in \mathcal{𝒫}}G/P$. Observe that $V/G$ is finite, every isotropy group of V is in $\mathcal{ℱ}$, and for every $K\in \mathcal{ℱ}$ the fixed-point set $V^K$ is non-empty. The first two properties imply that $\mathbb{Z}[\cdot ,V]$ is a finitely generated free ${\mathcal{𝒪}}_{\mathcal{ℱ}}(G)$-module, and the third property implies that the augmentation map

$\mathbb{Z}[\cdot ,V]\stackrel{\mathit{ϵ}}{\to }{\mathbb{Z}}_{G,\mathcal{ℱ}}$

is surjective. Since G is ${\mathrm{FP}}_{1,\mathcal{ℱ}}$, there is a partial ${\mathrm{FP}}_{1,\mathcal{ℱ}}$-resolution

$\mathbb{Z}[\cdot ,T]\stackrel{𝑝}{\to }\mathbb{Z}[\cdot ,V]\stackrel{\mathit{ϵ}}{\to }{\mathbb{Z}}_{G,\mathcal{ℱ}}\to 0.$

Then the proof of Proposition 3.3 provides an $F_{1,\mathcal{ℱ}}$-graph Γ with vertex set V. Since $\mathcal{𝒫}$ is an almost malnormal collection, the intersection of the G-stabilizers of two distinct vertices is finite, and hence stabilizers of edges of Γ are finite. ∎

Proof.

Let T be the coned-off Cayley graph of $F=({\ast }_{P\in \mathcal{𝒫}}P)\ast F(S)$ with respect to $\mathcal{𝒫}$ and S. Observe that T is simply-connected since it is a tree. Let $\widehat{\mathrm{\Gamma }}$ be the coned-off Cayley graph of G with respect to $\mathcal{𝒫}$ and S. Observe that $\widehat{\mathrm{\Gamma }}$ is recovered as the quotient $T/N$. The main observation is that $T\to \widehat{\mathrm{\Gamma }}$ is a normal covering space with N as a group of deck transformations. Indeed, the edge stabilizers of T with respect to N are trivial by definition; the vertex stabilizers of T with respect to N are trivial since each factor P of F embeds in G an therefore $N\cap P$ is trivial for every $P\in \mathcal{𝒫}$. It follows that the 1-skeleton of X has fundamental group isomorphic to N, and since N is the normal closure of $\{r_1,\mathrm{\dots },r_m\}$ in F the definition of X implies that ${\pi }_1(X)$ is trivial. ∎

Proof.

The space X is simply-connected by Lemma 3.9, and since the relative presentation is finite, the G-action on X is cocompact. Let $K\in \mathcal{ℱ}$ be a non-trivial subgroup. Since the G-stabilizers of 1-cells and 2-cells of X are trivial, the fixed-point set $X^K$ consists only of 0-cells. Since $\mathcal{𝒫}$ is a malnormal collection and K fixes at least one cone-vertex, it follows that $X^K$ is a single cone-vertex of X. Therefore X is an $F_{2,\mathcal{ℱ}}$-space with trivial 1-cell stabilizers. ∎

Proof.

Consider the commutative diagram

constructed as follows. Let A and B be finitely generated and based free $\mathbb{Z}G$-modules, and let $A\to P$ and $B\to Q$ be surjective morphisms inducing the filling norms $\parallel \cdot {\parallel }_P$ and $\parallel \cdot {\parallel }_Q$. Since A is free and $B\to Q$ is surjective, there is a lifting $\tilde{\phi }:A\to B$ of φ. Since $\tilde{\phi }$ is morphism between finitely generated, free, based $\mathbb{Z}G$-modules, by [19, Lemma 2.7], there is a constant C such that

${\parallel \tilde{\phi }(a)\parallel }_1\le C{\parallel a\parallel }_1$

for every $a\in A$. Let $p\in P$ and let $a\in A$ such that $\rho (a)=p$. It follows that

${\parallel \phi (p)\parallel }_Q\le {\parallel \tilde{\phi }(a)\parallel }_1\le C{\parallel a\parallel }_1.$

Since the above inequality holds for any $a\in A$ with $\rho (a)=p$, it follows that

$\begin{array}{cc}\hfill {\parallel \phi (p)\parallel }_Q& \le C\cdot \underset{\rho (a)=p}{\min }\{{\parallel a\parallel }_1\}=C{\parallel p\parallel }_P.\mathit{∎}\hfill \end{array}$

Proof of Theorem 4.4.

Let $P_{\ast }$ and $Q_{\ast }$ be two resolutions of ${\mathcal{𝒪}}_{\mathcal{ℱ}}(G)$-modules for ${\mathbb{Z}}_{G,\mathcal{ℱ}}$ of type ${\mathrm{FP}}_{2,\mathcal{ℱ}}$. For $i=1,2$, let $\parallel \cdot {\parallel }_{P_i}$ and $\parallel \cdot {\parallel }_{Q_i}$ be filling norms of the components $P_i[G/1]$ and $Q_i[G/1]$, respectively. Denote by ${\mathrm{FV}}_{G,\mathcal{ℱ}}^P$ and ${\mathrm{FV}}_{G,\mathcal{ℱ}}^Q$ the homological filling functions induced by $P_{\ast }$ and $Q_{\ast }$, respectively. By symmetry, it is enough to prove that there is $C\ge 0$ such that for every $k\in \mathbb{N}$,

${\mathrm{FV}}_{G,\mathcal{ℱ}}^Q(k)\le C\cdot {\mathrm{FV}}_{G,\mathcal{ℱ}}^P(Ck+C)+Ck+C.$(4.1)

Define C as follows. Since the category of ${\mathcal{𝒪}}_{\mathcal{ℱ}}(G)$-modules is an abelian category with enough projectives, any two projective resolutions of an ${\mathcal{𝒪}}_{\mathcal{ℱ}}(G)$-module are chain homotopy equivalent. Therefore, there are chain maps $f_i:Q_i\to P_i$, $g_i:P_i\to Q_i$, and a map $h_i:Q_i\to Q_{i+1}$ such that

${\partial }_{i+1}\circ h_i+h_{i-1}\circ {\partial }_i=g_i\circ f_i-\mathrm{Id}.$

From here on, we only consider the components at $G/1$ of these resolutions and morphisms, so we are in the category of $\mathbb{Z}G$-modules. We have two exact sequences of finitely generated $\mathbb{Z}G$-modules

$P_2\underset{}{\overset{{\delta }_2}{\to }}{P}_1\underset{}{\overset{{\delta }_1}{\to }}{P}_0\stackrel{}{\to }\mathbb{Z}\stackrel{}{\to }0,Q_2\stackrel{{\partial }_2}{\to }{Q}_1\stackrel{{\partial }_1}{\to }{Q}_0\stackrel{}{\to }\mathbb{Z}\stackrel{}{\to }0$

and the morphisms $f_i:Q_i\to P_i$, $g_i:P_i\to Q_i$, and $h_i:Q_i\to Q_{i+1}$. Let C be the maximum of the constants given by Lemma 4.6 for $g_2$, $h_1$, and $f_1$ and the filling-norms provided.

Inequality (4.1) is proved exactly as in [19, Proof of Theorem 3.5] by using Lemma 4.6 instead of [19, Lemma 2.8]. We include the short argument for completeness.

Fix k. Let $\alpha \in \mathrm{𝖪𝖾𝗋}({\partial }_1)$ be such that ${\parallel \alpha \parallel }_{Q_1}\le k$. Choose $\beta \in P_2$ such that ${\delta }_2(\beta )=f_1(\alpha )$ and ${\parallel f_1(\alpha )\parallel }_{{\delta }_2}={\parallel \beta \parallel }_{P_2}$. Then

$\begin{array}{cc}\hfill \alpha & =g_1\circ f_1(\alpha )-{\partial }_2\circ h_1(\alpha )+h_0\circ {\partial }_1(\alpha )\hfill \\ & =g_1\circ {\delta }_2(\beta )-{\partial }_2\circ h_1(\alpha )\hfill \\ & ={\partial }_2\circ g_2(\beta )-{\partial }_2\circ h_1(\alpha )\hfill \\ & ={\partial }_2(g_2(\beta )-h_1(\alpha )).\hfill \end{array}$

It follows that

${\parallel \alpha \parallel }_{{\partial }_2}\le {\parallel g_2(\beta )-h_1(\alpha )\parallel }_{Q_2}$$\le {\parallel g_2(\beta )\parallel }_{Q_2}+{\parallel h_1(\alpha )\parallel }_{Q_2}$$\le C\cdot {\parallel \beta \parallel }_{P_2}+C\cdot {\parallel \alpha \parallel }_{Q_1}$$=C\cdot {\parallel f_1(\alpha )\parallel }_{{\delta }_2}+C\cdot {\parallel \alpha \parallel }_{Q_1}$$\le C\cdot {\mathrm{FV}}_{G,\mathcal{ℱ}}^P({\parallel f_1(\alpha )\parallel }_{P_1})+C{\parallel \alpha \parallel }_{Q_1}$$\le C\cdot {\mathrm{FV}}_{G,\mathcal{ℱ}}^P(C{\parallel \alpha \parallel }_{Q_1})+C{\parallel \alpha \parallel }_{Q_1}$$\le C\cdot {\mathrm{FV}}_{G,\mathcal{ℱ}}^P(Ck)+Ck.$