Homological growth of Artin kernels in positive characteristic

Abstract

We prove an analogue of the Lück Approximation Theorem in positive characteristic for certain residually finite rationally soluble (RFRS) groups including right-angled Artin groups and Bestvina–Brady groups. Specifically, we prove that the mod p homology growth equals the dimension of the group homology with coefficients in a certain universal division ring and this is independent of the choice of residual chain. For general RFRS groups we obtain an inequality between the invariants. We also consider a number of applications to fibring, amenable category, and minimal volume entropy.

1 Introduction

A celebrated theorem of Lück relates the rational homology growth in degree m through finite covers to the mth $\ell ^2$-Betti number of a residually finite group. More precisely:

Theorem 1.1

(Lück [26]) Let G be a residually finite group of type ${\textsf{F}}_{m+1}$ and let $(G_n)_{n\in {\mathbb {N}}}$ be a residual chain of finite index normal subgroups. Then

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{b_m(G_n; {\mathbb {Q}})}{[G:G_n]} = b_m^{(2)}(G), \end{aligned}$$

where $b_m^{(2)}(G)$ denotes the mth $\ell ^2$-Betti number of G.

An immediate consequence of Lück’s approximation theorem is that the left-hand limit always exists and is independent of the chosen residual chain. We remind the reader that a residual chain is a sequence $G=G_0\geqslant G_1\geqslant \cdots \geqslant G_n\geqslant \cdots$ such that each $G_n$ is a finite index normal subgroup of G and $\bigcap _{n\in {\mathbb {N}}} G_n=\{1\}$. A related invariant is the mth ${\mathbb {F}}_p$-homology gradient for a finite field ${\mathbb {F}}_p$. It is defined by

$$\begin{aligned} b^{(2)}_{m}(G, (G_n);{\mathbb {F}}_p):= \limsup _{n\rightarrow \infty }\frac{b_m(G_n;{\mathbb {F}}_p)}{[G:G_n]} \end{aligned}$$

for any G of type ${\textsf{F}}_{m+1}$ and residual chain $(G_n)_{n\in {\mathbb {N}}}$. We will recall the definition of various finiteness properties in Sect. 5.

In [27, Conjecture 3.4] Lück conjectured that $b^{(2)}_{m}(G;{\mathbb {F}}_p)$ should equal $b^{(2)}_{m}(G)$ (and hence be independent of the residual chain). This was disproved by Avramidi–Okun–Schreve [1] (using a result of Davis–Leary [13]) where they showed that

$$\begin{aligned} b^{(2)}_{3}(A_{{\mathbb {R}}{\textbf{P}}^2})=0 \quad \text {but} \quad b^{(2)}_{3}(A_{{\mathbb {R}}{\textbf{P}}^2};{\mathbb {F}}_2)=1 \end{aligned}$$

independently of the choice of residual chain. Here, $A_{{\mathbb {R}}{\textbf{P}}^2}$ is the right-angled Artin group (RAAG) on (the 1-skeleton of) any flag triangulation of the real projective plane.

For torsion-free groups satisfying the Atiyah conjecture, the $\ell ^2$-Betti numbers may be computed via the dimensions of group homology with coefficients in a certain skew field $\mathcal D_{{\mathbb {Q}}G}$, known as the Linnell skew field of G. In [19], Jaikin-Zapirain introduces analogues of the Linnell skew field with ground ring any skew field ${\mathbb {F}}$, denoted $\mathcal D_{{\mathbb {F}}G}$, and called the Hughes-free division ring of ${\mathbb {F}} G$. He proves that ${\mathcal {D}}_{{\mathbb {F}} G}$ exists and is unique up to ${\mathbb {F}}G$-isomorphism for large classes of groups, including residually finite rationally soluble (RFRS) groups (in particular compact special groups) and conjectures they should exist for all locally indicable groups.

One may compute group homology with coefficients in $\mathcal D_{{\mathbb {F}}G}$ and take $\mathcal D_{{\mathbb {F}}G}$-dimensions to obtain $\mathcal D_{{\mathbb {F}}G}$-Betti numbers, denoted $b_m^{\mathcal D_{{\mathbb {F}}G}}(G)$. We emphasise that when G is RFRS and ${\mathbb {F}} = {\mathbb {Q}}$, the Linnell skew field and the Hughes free division ring coincide [19, Appendix], and therefore that $b_m^{{\mathcal {D}}_{{\mathbb {Q}}G}}(G) = b_m^{(2)}(G)$. The ${\mathcal {D}}_{{\mathbb {F}}{G}}$-Betti numbers share a number of properties with $\ell ^2$-Betti numbers (see for example [18, Theorem 3.9] and [15, Lemmas 6.3 and 6.4]). In light of this we raise the following conjecture:

Conjecture A

Let ${\mathbb {F}}$ be a skew field. Let G be a torsion-free residually finite group of type $\textsf{FP}_{n+1}({\mathbb {F}})$ such that $\mathcal D_{{\mathbb {F}}G}$ exists. Let $(G_i)_{i\in {\mathbb {N}}}$ be a residual chain of finite index normal subgroups. Then,

$$\begin{aligned} b^{(2)}_{m}(G,(G_i);{\mathbb {F}}) = b^{{\mathcal {D}}_{{\mathbb {F}}{G}}}_m(G) \end{aligned}$$

for all $m \leqslant n$. In particular, the limit supremum in the definition of $b^{(2)}_{m}$ is a genuine limit and is independent of the choice of residual chain.

Our main result verifies this conjecture for various families of groups—notably for RAAGs, which provided the counterexamples to Lück’s original conjecture.

Theorem B

Let ${\mathbb {F}}$ be a skew field and G be a group commensurable with any of

• (1) a residually finite Artin group satisfying the $K(\pi ,1)$ Conjecture, such as a right-angled Artin group or RAAG.

• (2) an Artin kernel, i.e. the kernel of a homomorphism from a RAAG to ${\mathbb {Z}}$ (these include Bestvina–Brady groups);

• (3) a graph product of amenable RFRS groups.

If G is type $\textsf{FP}_n({\mathbb {F}})$, then $b^{(2)}_{m}(G, (G_n);{\mathbb {F}})=b^{{\mathcal {D}}_{{\mathbb {F}}{G}}}_m(G)$ for $m \leqslant n$. In particular, the limit supremum in the definition of $b^{(2)}_{m}$ is a genuine limit independent of the choice of residual chain $(G_n)_{n\in {\mathbb {N}}}$.

In the case when G is not torsion-free but contains a finite-index torsion-free subgroup H, each of the Betti numbers $b_i(G)$ for G appearing in the statement is defined to be $b_i(H)/[G:H]$. This extension to the definition is clearly consistent for $b^{(2)}_{i}(G;{\mathbb {F}})$ and is consistent for $b^{{\mathcal {D}}_{{\mathbb {F}}{G}}}_i(G)$ because of [15, Lemma 6.3].

In each case we are able to compute $b^{\mathcal D_{{\mathbb {F}}G}}_m(G)$ explicitly; in cases (1) and (3) we find that it is equal to the ${\mathbb {F}}$-homology gradient previously computed in [1] and [30]. In case (2) we compute both the ${\mathcal {D}}_{{\mathbb {F}}{G}}$-Betti numbers and the ${\mathbb {F}}$-homology gradients and show they are equal. We highlight the computation for (2), which we expect will be of independent interest. Note that this generalises the computation of Davis and Okun for the $\ell ^2$-Betti numbers of Bestvina–Brady groups [14].

Theorem C

Let ${\mathbb {F}}$ be a skew field, let $\varphi :A_L \rightarrow {\mathbb {Z}}$ be an epimorphism and let $BB_L^\varphi$ denote $\text {ker} \varphi$. If $BB_L^\varphi$ is of type $\textsf{FP}_n({\mathbb {F}})$ then

$$\begin{aligned} b_m^{{\mathcal {D}}_{{\mathbb {F}}{BB_L^\varphi }}}(BB_L^\varphi ) = b_m^{(2)}(BB_L^\varphi ; {\mathbb {F}}) = \sum _{v \in L^{(0)}} \left| \varphi (v)\right| \cdot {\widetilde{b}}_{m-1}({{\,\textrm{lk}\,}}(v); {\mathbb {F}}). \end{aligned}$$

for all $m \leqslant n$.

One may extend Conjecture A to G-spaces with finite $(n+1)$-skeleton. In this more general setting we are able to verify the conjecture for certain polyhedral product spaces (Theorem 3.15) and certain hyperplane arrangements (Theorem 3.21). We remark that [25] effectively proves the conjecture for torsion-free amenable groups.

For groups where ${\mathcal {D}}_{{\mathbb {F}}{G}}$ exists and is universal (see Sect. 4 for a definition), we obtain that the agrarian Betti-numbers give a lower bound for the homology gradients as an easy consequence of a result of Jaikin-Zapirain [19, Corollary 1.6]. Note that the following theorem applies to all residually (amenable and locally indicable) groups by [19, Corollary 1.3], and in particular to RFRS groups.

Theorem D

Let ${\mathbb {F}}$ be a skew-field and let G be a residually finite group of type $\textsf{FP}_{n+1}({\mathbb {F}})$ such that ${\mathcal {D}}_{{\mathbb {F}}{G}}$ exists and is the universal division ring of fractions of $\mathbb {F}G$. Then

$$\begin{aligned} b_m^{{\mathcal {D}}_{{\mathbb {F}}{G}}}(G) \leqslant b_m^{(2)}(G, (G_i); {\mathbb {F}}) \end{aligned}$$

for all $m \leqslant n$, where $(G_i)_{i \in {\mathbb {N}}}$ is any residual chain of finite-index subgroups of G.

We also mention the work of Bergeron–Linnell–Lück–Sauer [4], which we believe provides some more evidence for Conjecture A. Let $\Gamma$ be the fundamental group of a finite CW complex X with a homomorphism $\varphi :\Gamma \rightarrow \text {GL} _n({\mathbb {Z}}_p)$. Let G be the closure of $\varphi (\Gamma )$ and let $G_i = \text {ker} (G \rightarrow \text {GL} _n({\mathbb {Z}}/p^i{\mathbb {Z}}))$. Recall that the Iwasawa algebra of G over ${\mathbb {F}}_p$ is ${\mathbb {F}}_p\llbracket G \rrbracket = \varprojlim {\mathbb {F}}_p[G/G_i]$. If G is torsion-free, then ${\mathbb {F}}_p\llbracket G \rrbracket$ has no zero divisors and is Ore with respect to its nonzero elements. Letting ${\overline{\Gamma }} = \Gamma / \text {ker} \varphi$, we have ring homomorphisms

$$\begin{aligned} {\mathbb {F}} {{\overline{\Gamma }}} \rightarrow {\mathbb {F}} G \rightarrow {\mathbb {F}}_p\llbracket G \rrbracket \rightarrow {\mathcal {D}}, \end{aligned}$$

where ${\mathcal {D}}$ is the Ore localisation of ${\mathbb {F}}_p\llbracket G \rrbracket$. If M is an ${\mathbb {F}}_p\llbracket G \rrbracket$-module and G is torsion-free, then the dimension of M is

$$\begin{aligned} \dim _{{\mathbb {F}}_p\llbracket G \rrbracket } M:= \dim _{\mathcal {D}} ({\mathcal {D}} \otimes _{{\mathbb {F}}_p\llbracket G \rrbracket } M). \end{aligned}$$

If G is not torsion-free, then we can pass to a uniform finite-index subgroup $G_0 \leqslant G$ and then define $\dim _{{\mathbb {F}}_p\llbracket G \rrbracket } M = [G:G_0]^{-1}\dim _{{\mathbb {F}}_p\llbracket G \rrbracket } M$. There is then a natural mod p analogue of $\ell ^2$-Betti numbers given by

$$\begin{aligned} \beta _k({\overline{X}}, {\overline{\Gamma }}; {\mathbb {F}}_p) = \dim _{\mathbb {F}_p\llbracket G \rrbracket } H_k( {\mathbb {F}}_p\llbracket G \rrbracket \otimes _{{\mathbb {F}}_p{\overline{\Gamma }}} C_\bullet ({\overline{X}}; {\mathbb {F}}_p)) \end{aligned}$$

where ${{\overline{X}}}$ is the cover of X corresponding to $\text {ker} \varphi$. With this setup, Bergeron–Linnell–Lück–Sauer prove the following mod-p Lück approximation style theorem.

Theorem 1.2

With the notation above, let $\Gamma _i = \varphi ^{-1}(G_i)$ and let $X_i$ be the corresponding cover of X. Then

$$\begin{aligned} b_k(X_i; {\mathbb {F}}_p) = [\Gamma : \Gamma _i] \cdot \beta _k({{\overline{X}}}, {\overline{\Gamma }}; {\mathbb {F}}_p) + O\left( [\Gamma :\Gamma _i]^{1 - \frac{1}{\dim G}} \right) \end{aligned}$$

for all k.

1.1 Outline of the paper

In Sect. 2 we give the relevant background on group rings and the computational tools we will need. In fact in many computations we are able to work in the more general setting of agrarian invariants as defined in [18] and so summarise the relevant theory. The remaining three sections of the paper are described in the sequel.

1.1.1 Computations

In Sect. 3 we introduce the notion of a confident complex; roughly this is a CW complex admitting a finite cover by subcomplexes with amenable fundamental group such that the nerve of the cover has good properties. We then compute the ${\mathcal {D}}_{{\mathbb {F}}{G}}$-Betti numbers and ${\mathbb {F}}_p$-homology gradients of these complexes showing they are related to ${\mathbb {F}}_p$-Betti numbers of the nerve. The computation of the first invariant a uses a spectral sequence collapsing result of Davis–Okun [14] building on work of Davis–Leary [13]. The computation of the second invariant is similar in spirit to the work of Avramidi–Okun–Schreve [1] and Okun–Schreve [30] and again relies on a spectral sequence argument.

The remainder of the section involves computations of the homological invariants for various spaces and groups with the goal of showing they satisfy Conjecture A. The computations are summarised as follows. In Theorem 3.12 we show that Artin kernels of type $\textsf{FP}_n$ are fundamental groups of confident spaces with an n-acyclic covering space, and use this to prove Theorem C. In Theorem 3.15 we compute the invariants for graph products of amenable RFRS groups (including RAAGs) and polyhedral products of classifying spaces of amenable RFRS groups. In Theorem 3.19 we compute the invariants for the Artin groups alluded to earlier. In fact the method applies to RFRS groups admitting a strict fundamental domain with certain stabilisers (Remark  3.20). Finally, inspired by [12], we compute the invariants for hyperplane arrangements whenever the invariants are defined (Theorem 3.21).

1.1.2 A lower bound for homology gradients

In Sect. 4, we recall the notion of a universal division ring of fractions and prove Theorem D as a consequence of work of Jaikin-Zapirain in [19].

1.1.3 Applications to fibring

In Theorem 5.1 we prove that if a RFRS group of type $\textsf{FP}_{n+1}({\mathbb {F}})$ is not virtually $\textsf{FP}_n({\mathbb {F}})$-fibred, then there is some $m\leqslant n$ such that $b_m^{(2)}(G,(G_n);{\mathbb {F}})>0$ for every residual chain $(G_n)_{n\in {\mathbb {N}}}$.

In the remainder of Sect. 5 we apply the computations of the agrarian invariants of RAAGs and Artin kernels to obtain some results about fibring in RAAGs. In particular, we make progress towards the following question of Matthew Zaremsky, communicated to us by Robert Kropholler: If a RAAG virtually algebraically fibres with kernel of type ${\textsf{F}}_n$, then does it algebraically fibre with kernel of type ${\textsf{F}}_n$? We are able to answer this question if one replaces ${\textsf{F}}_n$ with $\textsf{FP}_n(R)$, where R is either a skew field, ${\mathbb {Z}}$, or ${\mathbb {Z}}/m$ for some integer $m > 1$ (Theorem 5.2). Since finitely presented groups of type $\textsf{FP}_n$ are of type ${\textsf{F}}_n$, this leaves ${\textsf{F}}_2$ as the main case of interest in Zaremsky’s question.

In [15], the first author showed that if G is RFRS and $\ell ^2$-acyclic in dimensions $\leqslant n$, then G virtually $\textsf{FP}_n({\mathbb {Q}})$-fibres, but left unanswered whether $\ell ^2$-Betti numbers control virtual $\textsf{FP}_n$-fibring. We resolve that here by showing that there are RAAGs that are $\ell ^2$-acyclic but do not virtually $\textsf{FP}_2$-fibre (Proposition 5.5). Finally, we use the explicit computation of the agrarian invariants of Artin kernels to show that if $A_L$ is a RAAG and ${\mathbb {F}}$ is a skew field, then either all of the $\textsf{FP}_n({\mathbb {F}})$-fibres of $A_L$ are themselves virtually $\textsf{FP}_n({\mathbb {F}})$-fibred or none of them are (Theorem 5.6, Corollary 5.8).

1.1.4 Applications to amenable category and minimal volume entropy

In Sect. 6 we relate the Agrarian invariants to the amenable category of [9] and the minimal volume entropy of Gromov [16]. The relevant background is described in the section.

We show via an argument of Sauer [31] that having a small amenable category implies vanishing of ${\mathcal {D}}_{{\mathbb {F}}{G}}$-Betti numbers for residually finite groups (Proposition 6.1). We also show that for residually finite groups with uniformly uniform exponential growth admitting a finite K(G, 1), having a non-zero ${\mathcal {D}}_{{\mathbb {F}}{G}}$-Betti number implies the minimal volume entropy of G is non-zero (Corollary 6.2). The analogous results for ${\mathbb {F}}_p$-homology gradients were established by Sauer [31] and Haulmark–Schreve [17] respectively. In some sense this provides more evidence towards Conjecture A. Finally, we give a condition for the minimal volume entropy of an Artin kernel admitting a finite K(G, 1) to be non-zero (Corollary 6.3) and conjecture a converse.

2 Preliminaries

Throughout all rings are assumed to be associative and unital.

2.1 Finiteness properties

Let R be a ring and G be a group. Then G is said to be of type $\textsf{FP}_n(R)$ if there is a projective resolution $P_\bullet \rightarrow R$ of the trivial RG-module R such that $P_i$ is finitely generated for all $i \leqslant n$. If G is of type $\textsf{FP}_n(R)$ and S is an R-algebra, then G is of type $\textsf{FP}_n(S)$. Thus, if G is of type $\textsf{FP}_n({\mathbb {Z}})$, then G is of type $\textsf{FP}_n(R)$ for any ring R; because of this, we write $\textsf{FP}_n$ to mean $\textsf{FP}_n({\mathbb {Z}})$. Note that finite generation is equivalent to $\textsf{FP}_1(R)$ for any ring R, though $\textsf{FP}_2$ is in general a stronger condition than $\textsf{FP}_2(R)$.

We also the mention the homotopical analogue of the $\textsf{FP}_n(R)$ condition: a group G is of type ${\textsf{F}}_n$ if G has a classifying space with finite n-skeleton. Note that ${\textsf{F}}_1$ is equivalent to $\textsf{FP}_1(R)$ for any ring R, but that ${\textsf{F}}_n$ is in general strictly stronger than $\textsf{FP}_n(R)$ for $n \geqslant 2$.

2.2 Agrarian invariants

Let G be a group and let ${\mathbb {F}}$ be a skew field. The group ring ${\mathbb {F}} G$ is the set of formal sums $\sum _{g \in G} \lambda _g g$, where $\lambda _g \in {\mathbb {F}}$ is zero for all but finitely many $g \in G$, equipped with the obvious addition and multiplication operations. Let ${\mathcal {D}}$ be a skew field. Then an agrarian embedding is a ring monomorphism $\alpha :{\mathbb {F}} G \hookrightarrow {\mathcal {D}}$. Agrarian embeddings were first studied by [28, 29], who proved that group rings of bi-orderable groups have agrarian embeddings. Note that the existence of an agrarian embedding implies that G is torsion-free and that the Kaplansky zero divisor and idempotent conjectures hold for ${\mathbb {F}} G$. There is no known example of a torsion-free group G and a skew field ${\mathbb {F}}$ such that ${\mathbb {F}} G$ does not have an agrarian embedding.

Let X be a CW-complex with a cellular G action such that for every $g \in G$ and every open cell e of X, if $g \cdot e \cap e \ne \varnothing$ then g fixes e pointwise. The cellular chain complex $C_\bullet (X; {\mathbb {F}})$ is naturally an $\mathbb {F}G$-module. In this situation, X is called a G-CW complex. If $\alpha :{\mathbb {F}} G \rightarrow {\mathcal {D}}$ is an agrarian map, then we can define the ${\mathcal {D}}$-homology and ${\mathcal {D}}$-Betti numbers of X by

$$\begin{aligned} H_p^{\mathcal {D}} (X):= H_p({\mathcal {D}} \otimes _{\mathbb {F}G} C_\bullet (X)) \qquad \text {and} \qquad b_p^{\mathcal {D}}(X) = \dim _{\mathcal {D}} H_p^{\mathcal {D}}(X) \end{aligned}$$

where $\dim _{\mathcal {D}}$ denotes the dimension as a $\mathcal D$-module, which is well-defined since ${\mathcal {D}}$ is a skew field. Taking X to be a classifying space of G, we obtain the $\mathcal D$-homology and ${\mathcal {D}}$-Betti numbers of G.

The following theorem gives a central example of an agrarian embedding.

Theorem 2.1

(Linnell [24]) If G is a torsion-free group satisfying the strong Atiyah conjecture, then there is a skew field $\mathcal D_{{\mathbb {Q}}G}$, known as the Linnell skew field of G, and an agrarian embedding ${\mathbb {Q}} G \hookrightarrow {\mathcal {D}}_{{\mathbb {Q}}G}$ such that $b_p^{(2)}(G) = b^{{\mathcal {D}}_{{\mathbb {Q}}G}}(G)$.

The strong Atiyah conjecture asserts that if X is a free G-CW complex of finite type and G has finite subgroups of bounded order, then ${\text {lcm}}(G) \cdot b_p^{(2)}(X) \in {\mathbb {Z}}$ for all $p \in {\mathbb {N}}$, where ${\text {lcm}}(G)$ is the least common multiple of the orders of finite subgroups of G. The strong Atiyah conjecture is known for many groups, in particular for residually (torsion-free solvable groups) [32], for cocompact special groups [33], and for locally indicable groups [20]. Importantly for us, the Atiyah conjecture holds for all RFRS groups, and in particular all subgroups of RAAGs. The following theorem of Jaikin-Zapirain provides many examples of agrarian embeddings in positive characteristic.

Theorem 2.2

(Jaikin-Zapirain [19, Theorem 1.1]) Let ${\mathbb {F}}$ be a skew field and let G be either locally indicable amenable, residually (torsion-free nilpotent), or free-by-cyclic. Then there exists a division ring $\mathcal D_{{\mathbb {F}} G}$, known as the Hughes-free division ring of $\mathbb {F}G$, and an agrarian embedding ${\mathbb {F}} G \hookrightarrow {\mathcal {D}}_{{\mathbb {F}} G}$.

2.3 A Mayer–Vietoris type spectral sequence

The following construction of a Mayer–Vietoris type spectral sequence is due to Davis and Okun [14]. We will state the homological version of the spectral sequence with arbitrary coefficients. Let ${\textsf{P}}$ be a poset. We define ${{\,\textrm{Flag}\,}}({\textsf{P}})$ to be the simplicial realisation of ${\textsf{P}}$, i.e. ${{\,\textrm{Flag}\,}}({\textsf{P}})$ is the simplicial complex whose simplices are the totally ordered, finite, nonempty subsets of ${\textsf{P}}$. Hence, every simplex $\sigma \in {{\,\textrm{Flag}\,}}({\textsf{P}})$ has a well-defined minimum vertex, denoted $\min (\sigma )$.

If Y is a CW complex, then a poset of spaces in Y over ${\textsf{P}}$ is a cover ${\mathcal {Y}} = \{Y_a\}_{a \in {\textsf{P}}}$ of Y with each $Y_a$ a subcomplex such that

• (1) $a < b$ implies $Y_a \subseteq Y_b$;

• (2) ${\mathcal {Y}}$ is closed under finite, nonempty intersections.

Let R be a ring and let $\textsf{Mod}_R$ denote the category R-modules. A poset of coefficients for ${\textsf{P}}$ is a contravariant functor ${\mathcal {A}} :{\textsf{P}} \rightarrow \textsf{Mod}_R$. The functor ${\mathcal {A}}$ induces a system of coefficients on ${{\,\textrm{Flag}\,}}({\textsf{P}})$ via $\sigma \mapsto {\mathcal {A}}_{\min (\sigma )}$, which gives chain complex

$$\begin{aligned} C_j({{\,\textrm{Flag}\,}}({\textsf{P}}); {\mathcal {A}}):= \bigoplus _{\sigma \in {{\,\textrm{Flag}\,}}({\textsf{P}})^{(j)}} {\mathcal {A}}_{\min (\sigma )}, \end{aligned}$$

where ${{\,\textrm{Flag}\,}}({\textsf{P}})^{(j)}$ is the set of j-simplices in ${{\,\textrm{Flag}\,}}({\textsf{P}})$.

Lemma 2.3

([14, Lemmas 2.1 and 2.2]) Let M be an R-module and suppose ${\mathcal {Y}} = \{Y_a\}_{a \in {\textsf{P}}}$ is a poset of spaces over ${\textsf{P}}$ in a CW complex Y. There is a Mayer–Vietoris type spectral sequence

$$\begin{aligned} E_{p,q}^2 = H_p({{\,\textrm{Flag}\,}}({\textsf{P}}); {\mathcal {H}}_q({\mathcal {Y}}; M)) \rightarrow H_{p+q}(Y;M), \end{aligned}$$

where ${\mathcal {H}}_\bullet ({\mathcal {Y}}; M)$ is a system of coefficients given by

$$\begin{aligned} {\mathcal {H}}_\bullet ({\mathcal {Y}}; M)(\sigma ) = H_\bullet (Y_{\min (\sigma )}; M). \end{aligned}$$

Moreover, if the induced homomorphism $H_\bullet (Y_a; M) \rightarrow H_\bullet (Y_b; M)$ is zero whenever $a < b$ in ${\textsf{P}}$, then

$$\begin{aligned} E_{p,q}^2 = \bigoplus _{a \in {\textsf{P}}} H_p({{\,\textrm{Flag}\,}}({\textsf{P}}_{\geqslant a}), {{\,\textrm{Flag}\,}}({\textsf{P}}_{> a}); H_q(Y_a; M)). \end{aligned}$$

3 Computations

3.1 Approximation for spaces with confident covers

Fix a skew field ${\mathbb {F}}$ and let X be a compact CW complex with a finite poset of spaces ${\mathcal {X}} = \{X_\alpha \}_{\alpha \in {\textsf{P}}}$ over ${\textsf{P}}$. We call ${\mathcal {X}}$ confident if it satisfies the following conditions: for each $\alpha \in {\textsf{P}}$

• (i) each $X_\alpha$ has finitely many components;

• (ii) either $X_\alpha \subseteq X^{(0)}$ or each connected component of $X_\alpha$ is a classifying space with torsion-free amenable fundamental group such that ${\mathbb {F}} G$ has no zero-divisors;

• (iii) if $C \subseteq X_\alpha$ is a component, then the inclusion $C \subseteq X$ induces an injection $\pi _1(C) \rightarrow \pi _1(X)$;

• (iv) if $X_\alpha$ is a collection of points, then $\alpha$ is minimal in ${\textsf{P}}$; equivalently, $X_\alpha \cap X_\beta = \varnothing$ whenever $X_\alpha , X_\beta \subseteq X^{(0)}$.

Remark 3.1

If G is a torsion-free elementary amenable group, then ${\mathbb {F}} G$ has no zero-divisors.

We will show that spaces with confident covers satisfy Lück’s approximation theorem in arbitrary characteristic, following Avramidi, Okun, and Schreve who prove the same result for the Salvetti complex of a RAAG. We will then show that the result agrees with the agrarian Betti numbers of the space.

Before beginning, we fix some notation. For the rest of the section, X will be a compact CW complex with a confident cover ${\mathcal {X}} = \{X_\alpha \}_{\alpha \in {\textsf{P}}}$ and we assume that $G:= \pi _1(X)$ is residually finite with residual chain $(G_n)_{n \in {\mathbb {N}}}$. Let $X_n$ be the covering space of X corresponding to . Fix some $n \in {\mathbb {N}}$ and let $V = \mathbb {F}[G/G_n]$. Let K be the nerve of ${\mathcal {X}}$ and let $X_\sigma = X_{\min (\sigma )}$ for any simplex $\sigma \in K$. By an abuse of notation, we will use $\alpha$ to denote both an element of $\mathsf P$ and the corresponding vertex of K.

Recall the Mayer–Vietoris type spectral sequence

$$\begin{aligned} E_{p,q}^1 = C_p(K; H_q(X_\sigma ; V)) \Rightarrow H_{p+q}(X; V) \cong H_{p+q}(X_n; {\mathbb {F}}) \end{aligned}$$

(see, e.g., [7, VII.4]).

Lemma 3.2

We have that

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{\dim _{\mathbb {F}} H_q (X_\sigma ; V)}{[G: G_n]} = {\left\{ \begin{array}{ll} n_\sigma &{} \quad \text {if} \ q = 0 \ \text {and} \ X_\sigma \subseteq X^{(0)}, \\ 0 &{} \quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

Proof

The proof is similar to that of [1, Lemma 8]. The claim is clear when $X_\sigma$ consists of 0-cells. In the other case, since the homology growth of amenable groups satisfying (ii) is sublinear [25, Theorem 0.2], the only way $\dim _{\mathbb {F}} H_q (X_\sigma ; V)$ can grow linearly is if the number of components of the preimage of $X_\sigma$ in $X_n$ grows linearly with the index. But this does not occur since the sequence $\Gamma _n$ is residual and normal and the inclusions $X_\sigma \subseteq X$ induce $\pi _1$-injections of infinite groups on each component of $X_\alpha$. $\square$

The spectral sequence is therefore concentrated on the $E_{p,0}^1$ line, up to an error sublinear in the index $[G:G_n]$. This implies that

$$\begin{aligned} \limsup _{n \in {\mathbb {N}}} \frac{\dim _{\mathbb {F}} E_{p,0}^2}{[G:G_n]} = \limsup _{n \in {\mathbb {N}}} \frac{\dim _{\mathbb {F}} H_p(X;V)}{[G:G_n]}. \end{aligned}$$

(1)

Define a poset of coefficients on the vertices of K by

$$\begin{aligned} {\mathcal {A}}_\alpha = {\left\{ \begin{array}{ll} V^{n_\sigma } &{}\quad \text {if} \ X_\sigma \subseteq X^{(0)}, \\ 0 &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

Let $K^{(p)}$ denote the set of p-simplices of K. There is a chain projection

$$\begin{aligned} E_{p,0}^1 = \bigoplus _{\sigma \in K^{(p)}} V^{n_\sigma } \rightarrow C_p(K; {\mathcal {A}}_\alpha ) = \bigoplus _{\sigma \in K^{(p)}: X_{\min (\sigma )} \subseteq X^{(0)}} V^{n_\sigma }. \end{aligned}$$

By Lemma 3.2, the kernel of this projection has dimension sublinear in the index $[G:G_n]$ and therefore

$$\begin{aligned} \limsup _{n \in {\mathbb {N}}} \frac{\dim _{\mathbb {F}} E_{p,0}^2}{[G:G_n]} = \limsup _{n \in {\mathbb {N}}} \frac{\dim _{\mathbb {F}} H_p(X;\mathcal A_\alpha )}{[G:G_n]}. \end{aligned}$$

(2)

The proof of the following proposition is similar to that of [1, Lemma 9], except that in their case the nerve is contractible. Though K is not necessarily contractible, it does decompose nicely into contractible pieces centred at the vertices $\alpha \in K^{(0)}$ such that $X_\alpha \subseteq X^{(0)}$.

Proposition 3.3

Let $S = \{\alpha \in K^{(0)}: X_\alpha \subseteq X^{(0)}\}$. Then

$$\begin{aligned} \dim _{\mathbb {F}} H_p(K; {\mathcal {A}}_\sigma ) = [G:G_n] \cdot \sum _{\alpha \in S} n_\alpha {\widetilde{b}}_{p-1}({{\,\textrm{lk}\,}}(\alpha ); {\mathbb {F}}) \end{aligned}$$

In particular, $\lim _{n \rightarrow \infty } \frac{b_p(X_n; \mathbb {F})}{[G:G_n]}$ exists and is independent of the residual chain $(G_n)$.

Proof

By (iv), if $\alpha \in K^{(0)}$ is a vertex such that ${\mathcal {A}}_\alpha \ne 0$, then every vertex $\beta \in K^{(0)}$ adjacent to $\alpha$ has ${\mathcal {A}}_\beta = 0$. Therefore the chain complex $C_\bullet (K; {\mathcal {A}}_\sigma )$ decomposes as a direct sum of chain complexes $\bigoplus _{\alpha \in S} C_\bullet ({{\,\textrm{st}\,}}(\alpha ); {\mathcal {A}}_\sigma )$, where the coefficient system ${\mathcal {A}}_\sigma$ is restricted to each ${{\,\textrm{st}\,}}(\alpha ) \subseteq K$.

For each $\alpha \in S$, there is a short exact sequence of chain complexes

$$\begin{aligned} 0 \rightarrow C_\bullet ({{\,\textrm{lk}\,}}(\alpha ); {\mathbb {F}}) \otimes V^{n_\alpha } \rightarrow C_\bullet ({{\,\textrm{st}\,}}(\alpha ); {\mathbb {F}}) \otimes V^{n_\alpha } \rightarrow C_\bullet ({{\,\textrm{st}\,}}(\alpha ); {\mathcal {A}}_\sigma ) \rightarrow 0. \end{aligned}$$

Because ${{\,\textrm{st}\,}}(\alpha )$ is contractible, the middle term is acyclic and therefore

$$\begin{aligned} H_\bullet ({{\,\textrm{st}\,}}(\alpha ); {\mathcal {A}}_\sigma ) \cong H_{\bullet -1}({{\,\textrm{lk}\,}}(\alpha ); {\mathbb {F}}) \otimes V^{n_\alpha }. \end{aligned}$$

The formula in the statement of the proposition follows, since $V^{n_\alpha }$ is a vector space of dimension $[G:G_n] \cdot n_\alpha$.

This formula together with (1) and (2) show that $\limsup _n \frac{b_p(X_n; {\mathbb {F}})}{[G:G_n]}$ is independent of the residual chain $(G_n)$. Moreover, this implies that the $\limsup$ is a genuine limit. $\square$

Remark 3.4

In [1], the computation of Proposition 3.3 is carried out in the case that $X_L$ is the Salvetti complex of the RAAG determined by L. If ${\mathcal {X}}$ is the cover of $X_L$ by standard tori, then there is a single vertex v in the cover and the corresponding vertex $\alpha$ in the nerve has link isomorphic to L. Thus, we recover the formula $b_p^{(2)}(A_L; {\mathbb {F}}) = {\widetilde{b}}_{p-1}(L;{\mathbb {F}})$.

Remark 3.5

Condition (ii) can be weakened as follows: One only requires that first, $\pi _1X$ is residually finite; and second, that each $X_\sigma$ is (homotopy equivalent to) a compact CW complex with vanishing ${\mathbb {F}}$-$\ell ^2$-Betti numbers independent of the chain, or is a 0-cell. In this case the conclusion of Proposition 3.3 still holds.

Corollary 3.6

Let X be a confident CW complex with $\pi _1(X)$ residually finite. If $\pi :{\widehat{X}} \rightarrow X$ is a degree d cover, then $b_p^{(2)}(H; {\mathbb {F}}) = d \cdot b_p^{(2)}(G; {\mathbb {F}})$.

Proof

Let $\{X_\alpha \}$ be a confident cover of X. Then $\{ \pi ^{-1}(X_\alpha ) \}$ is a confident cover of ${\widehat{X}}$ and $\left| \pi ^{-1}(X_\alpha )\right| = d \cdot \left| X_\alpha \right|$ whenever $X_\alpha \subseteq X^{(0)}$. $\square$

3.2 Agrarian homology of spaces with confident covers

We continue with the same set-up as the previous subsection: X is a CW complex with a confident cover ${\mathcal {X}} = \{X_\alpha \}_{\alpha \in {\textsf{P}}}$ and let K be the covering. Notice that $K \cong {\text {Flag}}({\textsf{P}})$. Additionally, we will assume that there exists a skew field ${\mathcal {D}}$ and a fixed agrarian embedding $\mathbb {F}G \rightarrow {\mathcal {D}}$, where $G = \pi _1(X)$.

Proposition 3.7

Let $S = \{ \alpha \in K^{(0)}: X_\alpha \subseteq X^{(0)} \}$. Then

$$\begin{aligned} \dim _{\mathbb {F}} H_p^{\mathcal {D}}(X) = \sum _{\alpha \in S} n_\alpha {\widetilde{b}}_{p-1} ({{\,\textrm{lk}\,}}(\alpha ); {\mathbb {F}}). \end{aligned}$$

In particular, if $\pi _1(X)$ is residually finite then $b_p^{\mathcal {D}}(X) = b_p^{(2)} (X; {\mathbb {F}})$.

Proof

Suppose $X_\alpha$ does not consist of 0-cells. Then, each component of $X_\alpha$ is a classifying space for an infinite amenable group and therefore $H_p^{\mathcal {D}}(X_\alpha ) = 0$ by [18, Theorem 3.9(6)]. Since the elements $\alpha$ such that $X_\alpha \subseteq X^{(0)}$ are minimal in ${\textsf{P}}$, the spectral sequence of Lemma 2.3 collapses on the $E_{p,0}^2$ line. By Lemma 2.3,

$$\begin{aligned} H_p^{\mathcal {D}}(X) = \bigoplus _{\alpha \in S} H_p({{\,\textrm{Flag}\,}}(\mathsf P_{\geqslant \alpha }), {{\,\textrm{Flag}\,}}({\textsf{P}}_{>\alpha }); \mathcal D^{n_\alpha }), \end{aligned}$$

whence the stated formula follows. $\square$

Corollary 3.8

If X is confident, $G = \pi _1(X)$ is residually finite, and G has a Hughes-free division ring ${\mathcal {D}}_{{\mathbb {F}}{G}}$, then $b_p^{{\mathcal {D}}_{{\mathbb {F}}{H}}} = b_p^{(2)}({{\widehat{X}}}; {\mathbb {F}})$ for every finite index subgroup $H \leqslant G$ and corresponding finite covering space ${{\widehat{X}}} \rightarrow X$.

Proof

This follows from Corollary 3.6 and the fact that Hughes-free Betti numbers scale when passing to a finite index subgroup [15, Lemma 6.3]. $\square$

Remark 3.9

Condition (ii) can be weakened as follows: One only requires that first $\pi _1X$ admits an agrarian map $\pi _1 X\rightarrow \mathcal D$; and second that each $X_\sigma$ is (homotopy equivalent to) a compact CW complex with with vanishing $\mathcal D$-Betti numbers or is a 0-cell. In this case the conclusion of Proposition 3.7 still holds.

3.3 Artin kernels

An Artin kernel is simply the kernel of a non-zero homomorphism $\varphi :A_L \rightarrow {\mathbb {Z}}$, where L is a flag complex and $A_L$ is the RAAG it determines. We now apply the results of the previous two subsections to Artin kernels and obtain an explicit formula for their agrarian Betti numbers in terms of L and $\varphi$.

We fix a skew field ${\mathbb {F}}$, a flag complex L, and a surjective homomorphism $\varphi :A_L \rightarrow {\mathbb {Z}}$, where $A_L$ is the RAAG on L. Moreover, we fix a standard generating set for $A_L$, identified with the vertex set ${\text {Vert}}(L)$. We denote the Artin kernel by $BB_L^\varphi := \text {ker} \varphi$. If $\varphi$ is the map sending each of the generators to $1 \in {\mathbb {Z}}$, then $BB_L^\varphi = BB_L$ is the usual Bestvina–Brady group.

Let $T_L$ be the Salvetti complex on L and let $X_L$ be its universal cover. There is an affine map $T_L \rightarrow S^1$ inducing $\varphi$ constructed as follows. For each $v \in {\text {Vert}}(L)$, let $S^1_v:= {\mathbb {R}}/{\mathbb {Z}}$ be the corresponding circle in $T_L$. Let $\sigma = \{v_1, \dots , v_k\} \in L$ be a simplex and let $T_\sigma = S_{v_1}^1 \times \cdots \times S_{v_k}^1$ be the associated subtorus of $T_L$. There is a map

$$\begin{aligned} T_\sigma \rightarrow S^1 = {\mathbb {R}}/{\mathbb {Z}}, \quad (x_1, \dots , x_k) \longmapsto \varphi (v_1) x_1 + \cdots + \varphi (v_k) x_k + {\mathbb {Z}}. \end{aligned}$$

The maps on each of the subtori extend to a well-defined map $f :T_L \rightarrow S^1$ inducing $\varphi$ on the level of fundamental groups. Moreover, f induces a cube-wise affine $A_L$-equivariant height function $h :X_L \rightarrow {\mathbb {R}}$ making the diagram

commute—here the vertical arrows are the universal covering maps.

We borrow the following definition and terminology from [8].

Definition 3.10

A vertex v of L is living [resp., dead] if $\varphi (v) \ne 0$ [resp., $\varphi (v) = 0$]. Denote the full subcomplex of L spanned by the living [resp., dead] vertices by $L^{\textsf{a}}$ [resp., $L^{\textsf{d}}$].

Let $Z = h^{-1}(\{p\})$ for some $p \notin {\mathbb {Z}}$. The level set Z has a natural CW complex structure and $BB_L^\varphi$ acts cocompactly on Z; we denote the quotient $Z/BB_L^\varphi$ by Y. For each n-simplex $\sigma \in L$, the subtorus $T_\sigma \subseteq T_L$ lifts to $X_\sigma$, a collection of pairwise disjoint sheets in $X_L$. Each sheet is an isometrically embedded copy of $(n+1)$-dimensional Euclidean space.

Let ${\textsf{P}}$ be the poset of simplices of L that contain at least one vertex in $L^{\textsf{a}}$. Then Z is covered by the collection $\{ X_\sigma \cap Z \}_{\sigma \in {\textsf{P}}}$. Writing $Y_\sigma$ for the image of $X_\sigma \cap Z$ in Y, we obtain a poset of spaces ${\mathcal {Y}} = \{ Y_\sigma \}_{\sigma \in \mathsf P}$ of Y where each subcomplex $Y_\sigma$ is a disjoint union of tori or a set of vertices. Crucially, ${\mathcal {Y}}$ is a confident cover.

Lemma 3.11

If $\sigma = \{v\} \in L^{\textsf{a}}$ is a vertex, then $Y_\sigma$ is a collection of $\left| \varphi (v)\right|$ vertices.

Proof

We will show that there are exactly $\left| \varphi (v)\right|$ orbits of lines in $X_\sigma$ under the $BB_L^\varphi$-action on $X_L$. For each vertex v in $L^{\textsf{a}}$, evenly subdivide each edge of $X_v$ into $\left| \varphi (v)\right|$ segments. Note that the restriction ${\overline{X}}_L^{(1)} \rightarrow {\mathbb {R}}$ of h is cellular, where ${\overline{X}}_L^{(1)}$ is the subdivided 1-skeleton of $X_L$ and ${\mathbb {R}}$ is given the cell structure with vertex set ${\mathbb {Z}}$ and edge set $\{[n,n+1]: n \in {\mathbb {Z}}\}$.

Fix a vertex $\sigma = \{v\} \in L^{\textsf{a}}$ and let ${\overline{X}}_\sigma$ be the subdivision of $X_\sigma$. Let e be an edge of ${\overline{X}}_\sigma$ and let $e'$ be the unique edge of $X_\sigma$ such that $e \subseteq e'$. We say e is an edge of type i if it is the ith highest edge (under the height function h) contained in $e'$; the integer i can take values in $\{0, \dots , \left| \varphi (v)\right| -1 \}$.

Because the action of $BB_L^\varphi$ on $X_L$ is height preserving, it preserves the set of type i edges. Since $\varphi$ is surjective, $\gcd (( \varphi (v) )_{v \in L^{(0)}}) = 1$. Therefore, the generic level set Z intersects edges of type i for every $i \in \{0, \dots , \left| \varphi (v)\right| - 1\}$. Moreover, $BB_L^\varphi$ acts transitively on the set of edges of type i of the same height, which follows from the fact that $BB_L^\varphi$ acts transitively on the set of edges of $T_\sigma$ of the same height. Thus, we conclude that there are exactly $\left| \varphi (v)\right|$ orbits of lines in $X_\sigma$ under the $BB_L^\varphi$ action. $\square$

Theorem 3.12

Let $\varphi :A_L \rightarrow {\mathbb {Z}}$ be an epimorphism and let Y be a generic level set of the induced height function. If $\mathbb {F}BB_L^\varphi \rightarrow {\mathcal {D}}$ is an agrarian embedding, then

$$\begin{aligned} b_p^{\mathcal {D}}(Y) = b_p^{(2)}(Y; {\mathbb {F}}) = \sum _{v \in L^{(0)}} \left| \varphi (v)\right| \cdot {\widetilde{b}}_{p-1}({{\,\textrm{lk}\,}}(v); {\mathbb {F}}). \end{aligned}$$

Moreover, if $BB_L^\varphi$ is of type $\textsf{FP}_n({\mathbb {F}})$ then

$$\begin{aligned} b_p^{\mathcal {D}}(BB_L^\varphi ) = b_p^{(2)}(BB_L^\varphi ; {\mathbb {F}}) = \sum _{v \in L^{(0)}} \left| \varphi (v)\right| \cdot {\widetilde{b}}_{p-1}({{\,\textrm{lk}\,}}(v); {\mathbb {F}}) \end{aligned}$$

for all $p \leqslant n$. We also have $b_p^{\mathcal {D}}(H) = b_p^{(2)}(H; {\mathbb {F}})$ whenever H is a finite index subgroup of $BB_L^\varphi$ and ${\mathcal {D}}$ is the Hughes-free division ring of ${\mathbb {F}}H$.

Before beginning the proof, we remark that $BB_L^\varphi$ is a RFRS group, being a subgroup of a RAAG. Hence, by [19, Corollary 1.3], ${\mathcal {D}}_{{\mathbb {F}} BB_L^\varphi }$ exists. When ${\mathbb {F}} = {\mathbb {Q}}$, the Hughes free division ring and the Linnell skew field coincide, so in this case Theorem 3.12 computes the $\ell ^2$-Betti numbers of $BB_L^\varphi$. This generalises the computation of Davis and Okun in [14, Theorem 4.4]

Proof

The first statement is an immediate consequence of Propositions 3.3 and 3.7, Remark 3.9, Remark 3.5, Lemma 3.11, and the observation that ${\mathcal {Y}}$ is a confident cover. The second follows from the fact that if $BB_L^\varphi$ is of type $\textsf{FP}_n({\mathbb {F}})$ if and only if Z is n-acyclic with ${\mathbb {F}}$ coefficients [8] (Bux–Gonzalez consider only the case ${\mathbb {F}} = {\mathbb {Z}}$, but their result remains true over arbitrary coefficients). $\square$

Remark 3.13

The condition that $BB_L^\varphi$ is of type $\textsf{FP}_n(\mathbb {F})$ can be verified directly in the flag complex L. Recall that a topological space is n-acyclic if its reduced homology (with coefficients in ${\mathbb {F}}$ in our case) vanishes in degrees $\leqslant n$. Note that we use the convention that the reduced homology of the empty set is ${\mathbb {F}}$ in dimension $-1$, so if X is n-acyclic for $n \geqslant -1$, then X is nonempty.

Bux and Gonzalez show that $BB_L^\varphi$ is of type $\textsf{FP}_n({\mathbb {F}})$ if and only if $L^{\textsf{a}} \cap {{\,\textrm{lk}\,}}(\sigma )$ is $(n - \dim (\sigma ) - 1)$-acyclic (with coefficients in ${\mathbb {F}}$) for every simplex $\sigma \in L^{\textsf{d}}$, including the empty simplex which has dimension $-1$ and link L [8, Theorem 14]. Their result is stated in the case ${\mathbb {F}} = {\mathbb {Z}}$ but remains true, with the same proof, when stated over a general coefficient ring. We will use this characterisation of the finiteness properties of $BB_L^\varphi$ is Sect. 5.

It is known by work of Okun–Schreve [30] that for RAAGs the homology torsion growth $t_p^{(2)}(A_L)$ in degree p is equal to $|H_{p-1}(L;{\mathbb {Z}})_{\text {tors} }|$. We conjecture an analogous result for Artin kernels.

Conjecture 3.14

If $BB_L^\varphi$ is of type $\textsf{FP}_n$, then

$$\begin{aligned} t_p^{(2)}(BB_L^\varphi ) = \limsup _{n\rightarrow \infty }\frac{\log |H_p(G_n;{\mathbb {Z}})_{\text {tors} }|}{[BB_L^\varphi :G_n]} = \sum _{v \in L^{(0)}} \left| \varphi (v)\right| \cdot |H_{p-1}({{\,\textrm{lk}\,}}(v); {\mathbb {Z}})_{{\text {tors} }}| \end{aligned}$$

for $p \leqslant n$ and any residual chain $(G_n)$.

3.4 Graph products

Let K be a simplicial complex on the vertex set $[m]:= \{1, \dots , m\}$. Let $({\textbf{X}}, {\textbf{A}}) = \{(X_i, A_i): i \in [m] \}$ be a collection of CW-pairs. The polyhedral product of $({\textbf{X}}, {\textbf{A}})$ and K is the space

$$\begin{aligned} ({\textbf{X}}, {\textbf{A}})^K:= \bigcup _{\sigma \in K} \prod _{i = 1}^m Y_i^\sigma \quad \text {where} \quad Y_i^\sigma = {\left\{ \begin{array}{ll} X_i &{}\quad \text {if } i \in \sigma , \\ A_i &{}\quad \text {if } i \notin \sigma . \end{array}\right. } \end{aligned}$$

If $\varvec{\Gamma } = \{\Gamma _1, \dots , \Gamma _m\}$ is a finite set of groups, then the graph product of $\varvec{\Gamma }$ and K, denoted $\varvec{\Gamma }^K$, is the quotient of the free product $*_{i \in [m]} \Gamma _i$ by all the relations $[\gamma _i, \gamma _j] = 1$, where $\gamma _i \in \Gamma _i$, $\gamma _j \in \Gamma _j$ and i and j are adjacent vertices of K. Note that $\varvec{\Gamma }^K$ is the fundamental group of $X = (B\varvec{\Gamma }, *)^K$, where $B\varvec{\Gamma } = \{ B\Gamma _1, \dots , B\Gamma _m \}$ and $*$ is a set of one-point subcomplexes. Note that by [34, Theorem 1.1], if K is a flag complex and each $\Gamma _i$ is a discrete group, then $(B\varvec{\Gamma }, *)^K$ is a model for a $K(\varvec{\Gamma }^K,1)$.

Theorem 3.15

Let ${\mathbb {F}}$ and ${\mathcal {D}}$ be skew fields and let K be a finite simplicial flag complex. Let $G = \varvec{\Gamma }^K$ be a graph product of discrete groups such that there is an agrarian map $\mathbb {F}G \rightarrow {\mathcal {D}}$. If $H_\bullet (\Gamma ; {\mathcal {D}}) = 0$ for each $\Gamma \in \varvec{\Gamma }$, then

$$\begin{aligned} H_p^{\mathcal {D}}(\varvec{\Gamma }) \cong {\widetilde{H}}_{p-1}(K; {\mathcal {D}}). \end{aligned}$$

In particular, $b_p(G;{\mathcal {D}}) = b_{p-1}(K; {\mathbb {F}})$.

Proof

Let $X = (B\varvec{\Gamma }, *)^K$ and let ${\textsf{P}}$ be the poset whose elements are (possibly empty) simplices of K. Note that ${\textsf{P}}$ defines a poset of spaces $\{X_\sigma \}_{\sigma \in {\textsf{P}}}$, where $X_\varnothing$ is a single vertex. For each $\varnothing \ne \sigma \in {\textsf{P}}$, the group $\varvec{\Gamma }^J$ is a direct product of groups with vanishing ${\mathcal {D}}$-homology, and therefore $H^n(\varvec{\Gamma }^J; {\mathcal {D}}) = H^n(X_J; {\mathcal {D}}) = 0$ by the Künneth formula. Take the spectral sequence of Lemma 2.3 with the coefficient system $\sigma \mapsto H_q(X_{\min (\sigma )}; {\mathcal {D}})$. All of these coefficient system cohomology groups vanish except when $\sigma = \varnothing$ and $q = 0$ and therefore

$$\begin{aligned} H^p(X; {\mathcal {D}}) = H^p({{\,\textrm{Flag}\,}}({\textsf{P}}), {{\,\textrm{Flag}\,}}({\textsf{P}}>\varnothing ); {\mathcal {D}}) \cong {\widetilde{H}}^{p-1}(K; {\mathcal {D}}) \end{aligned}$$

by Lemma 2.3. The last isomorphism follows from the fact that ${\textsf{P}}$ is isomorphic to the cone on the barycentric subdivision of K, where the cone point corresponds to the empty simplex $\varnothing$. $\square$

Remark 3.16

There are certain situations in which one can easily deduce the fact that a graph product has an agrarian embedding. For example, if each group $\Gamma \in \varvec{\Gamma }^K$ is ordered, then so is $\varvec{\Gamma }^K$ by a result of Chiswell [10]. Thus, ${\mathbb {F}}\varvec{\Gamma }^K$ embeds into its Mal’cev–Neumann completion [28, 29].

Similarly, if each $\Gamma \in \varvec{\Gamma }^K$ is RFRS, then it is possible to show that $\varvec{\Gamma }^K$ is also RFRS and therefore has a Hughes-free division ring ${\mathcal {D}}_{{\mathbb {F}}\varvec{\Gamma }^K}$. We provide a sketch of an argument here, which proceeds by induction on the number of vertices in K. If K has one vertex, then the claim is trivial. Suppose now that K has more than one vertex. If the 1-skeleton of K is a complete graph, then the claim is again trivial since $\varvec{\Gamma }^K$ is a direct product of RFRS groups and therefore RFRS. If K is not a complete graph, then there is some vertex $v \in K$ such that ${{\,\textrm{st}\,}}(v) \ne K$ and we obtain a splitting $\varvec{\Gamma }^K \cong \varvec{\Gamma }^{{{\,\textrm{st}\,}}(v)} *_{\varvec{\Gamma }^{{{\,\textrm{lk}\,}}(v)}} \varvec{\Gamma }^{K \smallsetminus v}$. By induction, both $\varvec{\Gamma }^{{{\,\textrm{st}\,}}(v)}$ and $\varvec{\Gamma }^{K \smallsetminus v}$ are RFRS and by a result of Koberda and Suciu, the amalgam is also RFRS [22, Theorem 1.3]. Note that the Koberda–Suciu result is a combination theorem for a related class of groups called RFRp groups, but their proof is easily adapted to the RFRS case; we refer the interested reader to their paper for more details.

Thanks to the work of Okun and Schreve [30], we have the following corollary which holds, in particular, for RAAGs.

Corollary 3.17

Let ${\mathbb {F}}$ and ${\mathcal {D}}$ be skew fields and let K be a finite simplicial flag complex. Let $G = \varvec{\Gamma }^K$ be a graph product with an agrarian map $\mathbb {F}G \rightarrow {\mathcal {D}}$. If each $\Gamma \in \varvec{\Gamma }$ is residually finite, $\mathcal D$-acyclic, and ${\mathbb {F}}$-$\ell ^2$-acyclic, then

$$\begin{aligned} b_n^{\mathcal {D}}(G) = \lim _{i \rightarrow \infty } \frac{b_n(G_i; {\mathbb {F}}_p)}{[G: G_i]} \end{aligned}$$

for any residual chain of finite index normal subgroups.

Proof

Okun and Schreve [30, Theorem 5.1] showed that the right-hand side of the above equation is independent of the choice of residual chain and equal to ${\widetilde{b}}_{p-1}(K;\mathbb {F})$, which equals the left-hand side by Theorem 3.15. $\square$

Remark 3.18

Let K be a simplicial complex, $X=(B\varvec{\Gamma },*)^K$, and let ${\widetilde{X}}$ denote the universal cover. The above arguments apply equally well for computing $b^\mathcal D_p({\widetilde{X}})$ and $b^{(2)}_{p}(X;{\mathbb {F}})$. In both cases they are equal to ${\widetilde{b}}_{p-1}(K;{\mathbb {F}})$ whenever they are defined. In the case when K is not a flag complex, X is not aspherical.

3.5 Artin groups

Let A be a residually finite Artin group and suppose the K(A, 1) conjecture holds for A. Then there is a contractible complex $D_A$—the Deligne complex of A—with stabilisers the maximal parabolic subgroups of A admitting a strict fundamental domain $Q_A$. In [30, Section 4 and Theorem 5.2] the authors compute $b^{(2)}_p(A;{\mathbb {F}})={\tilde{b}}_{p-1}(\partial Q_A;{\mathbb {F}})$ independently of a choice of residual chain. Here $\partial Q$ is the subcomplex of Q with non-trivial stabilisers.

Theorem 3.19

Let ${\mathbb {F}}$ be a field. Let A be a residually finite Artin group. Suppose the K(A, 1) conjecture holds for A. If ${\mathbb {F}}A\rightarrow {\mathcal {D}}$ is an agrarian map, then

$$\begin{aligned} b^{(2)}_p(A;{\mathbb {F}})=b^{{\mathcal {D}}}_p(A;{\mathbb {F}})={\tilde{b}}_{p-1}(\partial Q_A;{\mathbb {F}}). \end{aligned}$$

Proof

In the action of A on $D_A$, the non-trivial stabilisers have a central ${\mathbb {Z}}$ subgroup and so they are $\mathcal D$-acyclic and have vanishing ${\mathbb {F}}$-homology growth. We take a poset of spaces $\mathcal X$ over the (barycentric subdivision of the) strict fundamental domain Q of A, where we assign a classifying space $BA_\sigma$ to each $\sigma \in Q$. Now, we apply Remark 3.9 and Remark 3.5. $\square$

Remark 3.20

The above argument applies to residually finite groups G acting on a contractible complex with strict fundamental domain and $\mathcal D$-acyclic stabilisers—whenever G admits an agrarian embedding ${\mathbb {F}}G\rightarrow \mathcal D$.

3.6 Complements of hyperplane arrangements

Let $\mathcal {A}$ be a collection of affine hyperplanes in ${\mathbb {C}}^k$ and let $\Sigma (\mathcal {A})$ denote their union. Let $M(\mathcal {A}):= {\mathbb {C}}^k \smallsetminus \Sigma (\mathcal {A})$. The rank of $M(\mathcal {A})$ is the maximum codimension n of any nonempty intersection of hyperplanes in $\mathcal {A}$. By [12, Proposition 2.1] the ordinary Betti numbers satisfy ${\widetilde{b}}_p(M(A);{\mathbb {F}})=0$ except possibly when $p=n$.

Theorem 3.21

Let ${\mathbb {F}}$ be a skew field, $\mathcal {A}$ be an affine hyperplane arrangement in ${\mathbb {C}}^k$ of rank n, and let $\Gamma :=\pi _1M(A)$.

• (1) If $\Gamma$ is residually finite, then $b^{(2)}_p(M(\mathcal {A});{\mathbb {F}})={\widetilde{b}}_p(M(\mathcal {A});{\mathbb {F}})$ which equals zero except possibly when $p=n$.

• (2) If $\alpha :{\mathbb {F}}\Gamma \rightarrow \mathcal D$ is an agrarian map, then $b^\mathcal D_p(M(\mathcal {A}))={\widetilde{b}}_p(M(\mathcal {A});{\mathbb {F}})$ which equals zero except possibly when $p=n$.

We will need a lemma.

Lemma 3.22

Let ${\mathbb {F}}$ be a skew field, $\mathcal {A}$ be a non-empty central affine hyperplane arrangement in ${\mathbb {C}}^k$ of rank n, and let $\Gamma :=\pi _1M(A)$.

• (1) If $\Gamma$ is residually finite, then $b^{(2)}_p(M(\mathcal {A});{\mathbb {F}})=0$ for all n independently of a choice of residual chain.

• (2) If $\alpha :{\mathbb {F}}\Gamma \rightarrow \mathcal D$ is an agrarian map, then $b^\mathcal D_p(M(\mathcal {A}))=0$ for all n.

Proof

In this case we have $M(\mathcal {A})=S^1\times B$ where $B=M(\mathcal {A})/S^1$ by [12, Proof of Lemma 5.2]. Both results are easy applications of the Künneth formula and vanishing of the relevant invariant for ${\mathbb {Z}}=\pi _1S^1$. We spell out the details in the first case to highlight the independence of the residual chain.

Observe that every finite cover $M_i$ of $M(\mathcal {A})$ can be written as $B_i\times S^1$ such that $S^1$ has m one cells. We have the index of the cover $|M:M_i|=m|B:B_i|$. Now, we compute via the Künneth formula that

$$\begin{aligned} b^{(2)}_p(M,(M_i);{\mathbb {F}})&=\lim _{i\rightarrow \infty }\frac{b_p(M_i)+b_{p-1}(M_i)}{|M:M_i|}\\&=\lim _{i,m\rightarrow \infty }\frac{b_p(M_i)+b_{p-1}(M_i)}{m|B:B_i|}\\&=0. \end{aligned}$$

$\square$

Proof of Theorem 3.21

By [12, Section 3] there is a cover $\mathcal U$ of $M(\mathcal {A})$ by central subarrangements $U_\sigma$ such that $\pi _1 U_\sigma \rightarrow \Gamma$ is injective and the nerve $N(\mathcal U)$ is contractible. There is also a cover of a deleted neighbourhood of $\Sigma (A)$, denoted $\mathcal U_{{\text {sing} }}$, such that $H_p(N(\mathcal U),N(\mathcal U_{{\text {sing} }}))$ is concentrated in degree n. It follows from Lemma 3.22 that either the homology gradients or the $\mathcal D$-Betti numbers vanish. In particular, by Remarks 3.5 and 3.9 the cover $\mathcal U$ is confident and the results follow. $\square$

4 A lower bound for homology gradients

Let R be a ring, let ${\mathcal {D}}$ be a skew-field, and let $\varphi :R \rightarrow {\mathcal {D}}$ be a ring homomorphism. There is a rank function ${{\,\textrm{rk}\,}}_{{\mathcal {D}}, \varphi } :{\text {Mat} }(R) \rightarrow {\mathbb {R}}_{\geqslant 0}$ defined by ${{\,\textrm{rk}\,}}_{{\mathcal {D}}, \varphi } A = {{\,\textrm{rk}\,}}\varphi _* A$, where $\varphi _* A$ is the matrix obtained by applying the homomorphism $\varphi$ to every entry of A and ${{\,\textrm{rk}\,}}\varphi _* A$ is the rank of $\varphi _* A$ as a matrix over ${\mathcal {D}}$. If $\varphi :R \hookrightarrow {\mathcal {D}}$ is an epic embedding (i.e. $\varphi (R)$ generates ${\mathcal {D}}$ as a skew-field) and ${{\,\textrm{rk}\,}}_{{\mathcal {D}}, \varphi } \geqslant {{\,\textrm{rk}\,}}_{{\mathcal {E}}, \psi }$ for every skew-field ${\mathcal {E}}$ and every ring homomorphism $\psi :R \rightarrow {\mathcal {E}}$, then ${\mathcal {D}}$ is said to be a universal division ring of fractions for R. If ${\mathcal {D}}$ is a universal division ring of fractions for R, it is then unique up to R-isomorphism [11, Theorem 4.4.1].

Theorem 4.1

(Jaikin-Zapirain [19, Corollary 1.3]) Let G be a residually (locally indicable and amenable) group and let ${\mathbb {F}}$ be a skew-field. Then the Hughes-free division ring ${\mathcal {D}}_{{\mathbb {F}}{G}}$ exists and is the universal division ring of fractions of $\mathbb {F}G$.

We note that Theorem 4.1 holds for RFRS groups, since they are residually poly-${\mathbb {Z}}$ (see, e.g., [19, Proposition 4.4]). The main theorem of this section will follow quickly from the observation that ${\mathcal {D}}_{{\mathbb {F}}{G}}$-Betti numbers scale under taking finite index subgroups and another result of Jaikin-Zapirain ( [19, Corollary 1.6]), which he states for $\ell ^2$-Betti numbers of CW complexes but also holds for agrarian homology. For the convenience of the reader, we reproduce a proof in the agrarian setting here.

Theorem 4.2

(Jaikin-Zapirain) Let ${\mathbb {F}}$ be a skew-field and suppose that G is a group of type $\textsf{FP}_{n+1}({\mathbb {F}})$ for some $n \in {\mathbb {N}}$ such that ${\mathcal {D}}_{{\mathbb {F}}{G}}$ exists and is the universal division ring of fractions of $\mathbb {F}G$. Then $b_m^{{\mathcal {D}}_{{\mathbb {F}}{G}}}(G) \leqslant b_m(G; {\mathbb {F}})$ for all $m \leqslant n$.

Proof

The embedding $\iota :\mathbb {F}G \hookrightarrow {\mathcal {D}}_{{\mathbb {F}}{G}}$ and the augmentation map $\alpha :\mathbb {F}G \rightarrow \mathbb {F}$ induce rank functions on ${\text {Mat} }(\mathbb {F}G)$ which we denote by ${{\,\textrm{rk}\,}}_G$ and ${{\,\textrm{rk}\,}}_{{\mathbb {F}}}$, respectively. By universality, ${{\,\textrm{rk}\,}}_G \geqslant {{\,\textrm{rk}\,}}_{\mathbb {F}}$.

Let $C_\bullet \rightarrow {\mathbb {F}}$ be a free-resolution of the trivial $\mathbb {F}G$-module ${\mathbb {F}}$ such that $C_m$ is finitely generated for all $m \leqslant n+1$. For all $m \leqslant n+1$, let $d_m$ be an integer such that $C_m \cong \mathbb {F}G^{d_m}$ and view the boundary maps $\partial _m :C_m \rightarrow C_{m-1}$ as matrices over $\mathbb {F}G$. The homologies we are interested in are $H_m({\mathcal {D}}_{{\mathbb {F}}{G}} \otimes _{\mathbb {F}G} C_\bullet )$ and $H_m(k \otimes _{\mathbb {F}G} C_\bullet )$ and note that the differentials ${\mathcal {D}}_{{\mathbb {F}}{G}} \otimes \partial _m$ and $k \otimes \partial _m$ correspond to the matrices $\iota _* \partial _m$ and $\alpha _* \partial _m$ under the identifications ${\mathcal {D}}_{{\mathbb {F}}{G}} \otimes _{\mathbb {F}G} C_m \cong {\mathcal {D}}_{{\mathbb {F}}{G}}^{d_m}$ and ${\mathbb {F}} \otimes _{\mathbb {F}G} C_m \cong \mathbb {F}^{d_m}$. Therefore

$$\begin{aligned} b_m^{{\mathcal {D}}_{{\mathbb {F}}{G}}}(G)&= d_m - {{\,\textrm{rk}\,}}_G \partial _m - {{\,\textrm{rk}\,}}_G \partial _{m+1} \\&\leqslant d_m - {{\,\textrm{rk}\,}}_{{\mathbb {F}}} \partial _m - {{\,\textrm{rk}\,}}_{{\mathbb {F}}} \partial _{m+1} \\&= b_m(G; {\mathbb {F}}) \end{aligned}$$

$\square$

As a consequence, we obtain that agrarian Betti numbers bound homology gradients from below.

Theorem 4.3

Let ${\mathbb {F}}$ be a skew-field and let G be a group of type $\textsf{FP}_{n+1}({\mathbb {F}})$ such that ${\mathcal {D}}_{{\mathbb {F}}{G}}$ exists and is the universal division ring of fractions of $\mathbb {F}G$. If $H \leqslant G$ is any subgroup of finite index, then

$$\begin{aligned} b_m^{{\mathcal {D}}_{{\mathbb {F}}{G}}} (G) \leqslant \frac{b_m(H; {\mathbb {F}})}{[G:H]} \end{aligned}$$

for all $m \leqslant n$. In particular, if G is residually finite and $(G_i)_{i \in {\mathbb {N}}}$ is a residual chain of finite-index subgroups, then $b_m^{{\mathcal {D}}_{{\mathbb {F}}{G}}} (G) \leqslant b_m^{(2)}(G, (G_i); {\mathbb {F}})$ for all $m \leqslant n$.

Proof

By [15, Lemma 6.3], $[G:H] \cdot b_m^{{\mathcal {D}}_{{\mathbb {F}}{G}}}(G) = b_m^{{\mathcal {D}}_{{\mathbb {F}}{H}}}(H)$, and by Theorem 4.2. The second claim is an immediate consequence of the first. $\square$

5 Applications to fibring

A group G is algebraically fibred (or, simply, fibred) if it admits a nontrivial homomorphism $G \rightarrow {\mathbb {Z}}$ with finitely generated kernel. More generally, if ${\mathcal {P}}$ is a finiteness property (e.g. type ${\textsf{F}}_n$ or $\textsf{FP}_n(R)$ for some ring R, see Sect. 2.1 for definitions), we say that G is ${\mathcal {P}}$-fibred if there is a nontrivial homomorphism $G \rightarrow {\mathbb {Z}}$ with kernel of type ${\mathcal {P}}$.

Theorem 5.1

Let ${\mathbb {F}}$ be a skew-field and let G be a RFRS group of type $\textsf{FP}_{n+1}({\mathbb {F}})$. If G is not virtually $\textsf{FP}_n({\mathbb {F}})$-fibred, then for every residual chain of finite-index subgroups $(G_i)_{i \in {\mathbb {N}}}$, we have $b_m^{(2)}(G, (G_i); \mathbb {F})>0$ for some $m \leqslant n$.

Proof

By [15] a RFRS group G is virtually $\textsf{FP}_n({\mathbb {F}})$-fibred if and only if $b_i^{{\mathcal {D}}_{{\mathbb {F}}G}}(G) = 0$ for every $i \leqslant n$. Since G is not virtually $\textsf{FP}_n({\mathbb {F}})$-fibred, we have $b_m^{{\mathcal {D}}_{{\mathbb {F}}{G}}}(G)>0$ for some $m\leqslant n$. The result now follows from Theorem 4.3. $\square$

The authors thank Robert Kropholler for communicating to us the following question due to Matthew Zaremsky: If a RAAG $A_L$ is virtually ${\textsf{F}}_n$-fibred, is it ${\textsf{F}}_n$-fibred? We are able to answer the analogous homological question over skew fields, ${\mathbb {Z}}$, and ${\mathbb {Z}}/m$ for $m \in {\mathbb {N}}_{>1}$.

Theorem 5.2

Let L be a finite flag complex and R be either a skew field ${\mathbb {F}}$, ${\mathbb {Z}}$, or ${\mathbb {Z}}/m$ for some $m \in {\mathbb {N}}_{>1}$. Then the RAAG $A_L$ is virtually $\textsf{FP}_n(R)$-fibred if and only if it is $\textsf{FP}_n(R)$-fibred.

Proof

We begin with the case $R = {\mathbb {F}}$. Let $A_L$ be virtually $\textsf{FP}_n({\mathbb {F}})$-fibred. By [15], a RFRS group G is virtually $\textsf{FP}_n({\mathbb {F}})$-fibred if and only if $b_i^{{\mathcal {D}}_{{\mathbb {F}}G}}(G) = 0$ for every $i \leqslant n$. In particular this applies in the case $G=A_L$. By Theorem 3.15, $b_i^{{\mathcal {D}}_{{\mathbb {F}}A_L}}(A_L)=0$ for every $i\leqslant n$ implies that ${\widetilde{b}}_i(L; {\mathbb {F}}) = 0$ for every $i \leqslant n-1$. By [5, Main Theorem], the Bestvina–Brady group is of type $\textsf{FP}_n({\mathbb {F}})$ and therefore $A_L$ is $\textsf{FP}_n(\mathbb {F})$-fibred.

Now suppose $R = {\mathbb {Z}}/m$ for some $m \in {\mathbb {N}}_{>1}$ and suppose that $A_L$ is virtually $\textsf{FP}_n({\mathbb {Z}}/m)$-fibred. If p is a prime factor of m, then there is a ring homomorphism ${\mathbb {Z}}/m \rightarrow {\mathbb {Z}}/p = {\mathbb {F}}_p$ and therefore ${\mathbb {F}}_p$ is a ${\mathbb {Z}}/m$-algebra. Thus, $A_L$ is virtually $\textsf{FP}_n({\mathbb {F}}_p)$-fibred. Therefore $b_i^{{\mathcal {D}}_{{\mathbb {F}}_p A_L}}(A_L) = 0$ for all $i \leqslant n$ and thus ${\widetilde{b}}_i(L; {\mathbb {F}}_p) = 0$ for all $i \leqslant n-1$ for all $i \leqslant n-1$ by Theorem 3.15. Then ${\widetilde{b}}_i(L; {\mathbb {Z}}/m) = 0$ for all $i \leqslant n-1$, so $BB_L$ is of type $\textsf{FP}_n({\mathbb {Z}}/m)$.

Finally, suppose $A_L$ is virtually $\textsf{FP}_n$-fibred. In particular, $A_L$ is virtually $\textsf{FP}_n({\mathbb {F}})$-fibred for every skew field ${\mathbb {F}}$, which implies that L is $(n-1)$-acyclic over every field by Theorem 3.15. Therefore L is $(n-1)$-acyclic over ${\mathbb {Z}}$, which implies that $BB_L$ is of type $\textsf{FP}_n$. $\square$

Remark 5.3

In the case ${\mathbb {F}} = {\mathbb {Q}}$, Theorem 5.2 could have been deduced from previous work since the $\ell ^2$-Betti numbers of RAAGs were computed by Davis and Leary in [13] and it is well known that a virtual $\textsf{FP}_n({\mathbb {Q}})$-fibring implies the vanishing of $\ell ^2$-Betti numbers in dimensions $\leqslant n$.

Since a finitely presented group of type $\mathsf {FP_n}$ is of type ${\textsf{F}}_n$ we can reduce Zaremsky’s question to one remaining case.

Question 5.4

(Zaremsky) Let L be a finite flag complex. If $A_L$ is virtually ${\textsf{F}}_2$-fibred, then is it ${\textsf{F}}_2$-fibred?

We can also give examples of RAAGs that show that [15, Theorem A] does not hold when ${\mathbb {Q}}$ is replaced by ${\mathbb {Z}}$. In other words, the vanishing of $\ell ^2$-Betti numbers of RFRS groups does not detect virtual $\textsf{FP}_n$-fibrations.

Proposition 5.5

Let p be a prime. There are RAAGs that are $\ell ^2$-acyclic but that do not virtually $\textsf{FP}_2({\mathbb {F}}_p)$-fibre. In particular, these RAAGs do not virtually $\textsf{FP}_2$-fibre.

Proof

Let L be a ${\mathbb {Q}}$-acyclic flag complex that has non-trivial $\mathbb {F}_p$-homology in dimension 1, e.g. we may take L to be a flag triangulation of M(1, p) the Moore space with homology ${\widetilde{H}}_n(M(1,p);{\mathbb {Z}})=0$ unless $n=1$, in which case it is isomorphic to ${\mathbb {Z}}/p$. Then $A_L$ is $\ell ^2$-acyclic by [13] (or Theorem 3.15) but it is not ${\mathcal {D}}_{{\mathbb {F}}_p A_L}$-acyclic by Theorem 3.15. By [15, Theorem 6.6], $A_L$ does not virtually $\textsf{FP}_2({\mathbb {F}}_p)$-fibre and in particular does not virtually $\textsf{FP}_n$-fibre. $\square$

In contrast to this result, Kielak showed that if G is RFRS, of cohomological dimension at most 2, and $\ell ^2$-acyclic, then G is virtually $\textsf{FP}_2$-fibred [21, Theorem 5.4].

The next application has to do with the following general question: If G fibres in two different ways, so that $G \cong K_1 \rtimes {\mathbb {Z}}\cong K_2 \rtimes {\mathbb {Z}}$ with $K_1$ and $K_2$ finitely generated, then what properties do $K_1$ and $K_2$ share? For example if G is a free-by-cyclic (resp. surface-by-cyclic) group and $G \cong K \rtimes {\mathbb {Z}}$, then K is necessarily a free (resp. surface) group. We thank Ismael Morales for bringing the following question to our attention: if $G \cong K_1 \rtimes {\mathbb {Z}}\cong K_2 \rtimes {\mathbb {Z}}$ with $K_1$ and $K_2$ finitely generated, then is $b_1^{(2)}(K_1) = 0$ if and only if $b_1^{(2)}(K_2) = 0$? We prove this is the case for RAAGs, and obtain a similar result for higher $\ell ^2$-Betti numbers and other agrarian invariants.

Theorem 5.6

Let $\varphi _0, \varphi _1 :A_L \rightarrow {\mathbb {Z}}$ be epimorphisms such that $BB_L^{\varphi _0}$ and $BB_L^{\varphi _1}$ are of type $\textsf{FP}_n({\mathbb {F}})$. If ${\mathbb {F}} BB_L^{\varphi _i} \hookrightarrow {\mathcal {D}}_i$ is an agrarian embedding for $i = 0,1$, then $BB_L^{\varphi _0}$ is ${\mathcal {D}}_0$-acyclic in dimensions $\leqslant n$ if and only if $BB_L^{\varphi _1}$ is ${\mathcal {D}}_1$-acyclic in dimensions $\leqslant n$.

Before proving Theorem 5.6, we need a technical lemma. First we fix some notation. If $\sigma _1 = [e_1, \dots , e_m]$ and $\sigma _2 = [f_1, \dots , f_m]$ are ordered simplices in a simplicial complex L such that $\sigma _1 \cup \sigma _2$ is a simplex (equivalently, if $\sigma _1 \in {{\,\textrm{lk}\,}}(\sigma _2)$), then $\sigma _1 \cup \sigma _2$ always denotes the ordered simplex $[e_1, \dots , e_m, f_1, \dots , f_n]$. Moreover, if $\tau = \alpha _1 \sigma _1 + \cdots + \alpha _n \sigma _n$ is a formal linear combination of simplices $\sigma _i$ (with coefficients $\alpha _i$ in some fixed skew field) such that $\sigma \cup \sigma _i \in L$ for every i, then $\sigma \cup \tau$ denotes the formal linear combination

$$\begin{aligned} \alpha _1 \sigma \cup \sigma _1 + \cdots + \alpha _n \sigma \cup \sigma _n. \end{aligned}$$

If $\varphi :A_L \rightarrow {\mathbb {Z}}$ is a homomorphism, recall that $L^{\textsf{a}}$ is the subcomplex of L spanned by the vertices $v \in L$ such that $\varphi (v) \ne 0$. We will write ${{\,\textrm{lk}\,}}_L(\sigma )$ (resp. ${{\,\textrm{lk}\,}}_{L^{\textsf{a}}}(\sigma )$) for the link of a simplex $\sigma$ in L (resp. $L^{\textsf{a}}$).

Lemma 5.7

Let $BB_L^\varphi$ be of type $\textsf{FP}_n({\mathbb {F}})$ and let v be a dead vertex of L. Then every simplicial $(n-1)$-cycle of ${\text {lk}}_{L}(v)$ is homologous to a cycle in ${\text {lk}}_{L^{\textsf{a}}}(v)$.

Proof

Let $\sigma = \alpha _1 \sigma _1 + \cdots + \alpha _k \sigma _k$ be a simplicial $(n-1)$-cycle in ${\text {lk}}_L(v)$, where each $\sigma _i$ is an ordered $(n-1)$-simplex of ${\text {lk}}_L(v)$ and $\alpha _i \in {\mathbb {F}}$ for each i. By induction on $m\geqslant 0$, we will show that the simplices $\sigma _i$ can be replaced with $(n-1)$-simplices having at least m living vertices such that the resulting chain is a cycle homologous to $\sigma$. The lemma follows from the $m = n$ case.

For the base case, suppose that $\sigma _i$ is a simplex with no living vertices. Then $\{v\} \cup \sigma _i$ is a dead n-simplex and therefore ${\text {lk}}_{L^{\textsf{a}}}(\{v\} \cup \sigma _i)$ is $(-1)$-connected (see Remark 3.13), i.e. it is nonempty. Thus, there is a living vertex u such that $\{u\} \cup \sigma _i \subseteq {\text {lk}}_L(v)$. Since

$$\begin{aligned} \partial ( \{u\} \cup \sigma _i ) = \sigma _i - \{u\} \cup \partial \sigma _i, \end{aligned}$$

where $\{u\} \cup \partial \sigma _i$ is a linear combination of $(n-1)$-simplices with one living vertex, we can replace $\sigma _i$ with $\{u\} \cup \partial \sigma _i$ in $\sigma$. Hence, we assume that the linear combination $\alpha _1 \sigma _1 + \cdots + \alpha _k \sigma _k$ only involves simplices with at least one living vertex.

Assume that $\alpha _1 \sigma _1 + \cdots + \alpha _k \sigma _k$ only involves simplices with at least $m \geqslant 1$ living vertices for some $m<n$ and let $\sigma _i$ be a simplex with exactly m living vertices. Let $\lambda \subseteq \sigma _i$ be the dead $(n-m-1)$-face of $\sigma _i$ and let $\sigma _i = \sigma _{i_1}, \dots , \sigma _{i_l}$ be the simplices among $\{\sigma _1, \dots , \sigma _k \}$ containing $\lambda$ as a face. For each $j \in \{1, \dots , l\}$, write $\sigma _{i_j} = \varepsilon _j \lambda \cup \tau _j$, where $\tau _j$ is a living $(m-1)$-simplex of ${\text {lk}}_L(v)$ and $\varepsilon _j \in \{\pm 1\}$. Then

$$\begin{aligned} \partial \left( \sum _{j=1}^l \alpha _{i_j} \sigma _{i_j} \right)&= \sum _{j=1}^l \alpha _{i_j} \varepsilon _j (\partial \lambda \cup \tau _j + (-1)^{n-m} \lambda \cup \partial \tau _j) \\&= \left( \sum _{j=1}^l \alpha _{i_j} \varepsilon _j \partial \lambda \cup \tau _j \right) + (-1)^{n-m} \lambda \cup \partial \left( \sum _{j=1}^l \alpha _{i_j} \varepsilon _j \tau _j \right) \\&= 0, \end{aligned}$$

since $\partial \sigma = 0$ and the simplices $\sigma _{i_j}$ are the only simplices among $\{\sigma _1, \dots , \sigma _k \}$ containing $\lambda$ as a face. Thus,

$$\begin{aligned} \lambda \cup \partial \left( \sum _{j=1}^l \alpha _{i_j} \varepsilon _j \tau _j \right) = 0, \end{aligned}$$

whence we conclude that $\sum _{j=1}^l \alpha _{i_j} \varepsilon _j \tau _j$ is an $(m-1)$-cycle in ${\text {lk}}_{L^\mathsf a}(\{v\} \cup \lambda )$. But $\{v\} \cup \lambda$ is a dead $(n-m)$-simplex and therefore ${\text {lk}}_{L^{\textsf{a}}}(\{v\} \cup \lambda )$ is $(m-1)$-connected. Hence, $\sum _{j=1}^l \alpha _{i_j} \varepsilon _j \tau _j = \partial \psi$ for some living m-chain $\psi$ in ${\text {lk}}_{L^{\textsf{a}}}(\{v\} \cup \lambda )$. Then

$$\begin{aligned} \partial (\lambda \cup \psi )&= \partial \lambda \cup \psi + (-1)^{n-m} \lambda \cup \left( \sum _{j=1}^l \alpha _{i_j} \varepsilon _j \tau _j \right) \\&= \partial \lambda \cup \psi + (-1)^{n-m} \sum _{j=1}^l \alpha _{i_j} \sigma _{i_j}. \end{aligned}$$

The chain $\partial \lambda \cup \psi$ is a linear combination of simplices with $m + 1$ living vertices. We can therefore replace $\sum _{j=1}^l \alpha _{i_j} \sigma _{i_j}$ with $\pm \partial \lambda \cup \psi$ and assume that $\sigma$ is a linear combination of simplices each with at least $m+1$ living vertices. $\square$

Proof of Theorem 5.6

Suppose that $b_p^{{\mathcal {D}}_0}(BB_L^{\varphi _0}) > 0$ for some $p \leqslant n$. By Theorem 3.12, there is a vertex v of L such that $\varphi _0(v) \ne 0$ and

$$\begin{aligned} {\widetilde{b}}_{p-1}({\text {lk}}(v); {\mathbb {F}}) > 0. \end{aligned}$$

Hence, there is a simplicial $(p-1)$-cycle $\sigma$ in ${\text {lk}}(v)$ that is not a boundary. If $\varphi _1(v) = 0$, then, by Lemma 5.7, $\sigma$ is homologous to a cycle in ${\text {lk}}_{L^{\textsf{a}}}(v)$ where $L^{\textsf{a}}$ denotes the living link with respect to $\varphi _1$. Thus $\widetilde{H}_{p-1}({\text {lk}}_{L^{\textsf{a}}}(v); {\mathbb {F}}) \ne 0$. But ${\text {lk}}_{L^{\textsf{a}}}(v)$ is $(n-1)$-connected over ${\mathbb {F}}$, so we must have $\varphi _1(v) \ne 0$, and therefore $b_p^{{\mathcal {D}}_1}(BB_L^{\varphi _1}) > 0$ by Theorem 3.12. $\square$

We highlight the following immediate corollary.

Corollary 5.8

Either all the $\textsf{FP}_n({\mathbb {F}})$-fibres of $A_L$ are virtually $\textsf{FP}_n({\mathbb {F}})$-fibred or none of them are. In particular, either all of $A_L$’s fibres virtually fibre or none of them do.

Proof

This follows from Theorem 5.6 and the fact that being ${\mathcal {D}}_{{\mathbb {F}} BB_L^{\varphi }}$-acyclic in dimensions $\leqslant n$ is equivalent to virtually fibring with kernel of type $\textsf{FP}_n({\mathbb {F}})$ [15, Theorem 6.6]. $\square$

6 Amenable category and minimal volume entropy

In this section we will relate ${\mathcal {D}}_{{\mathbb {F}}{G}}$-Betti numbers with amenable category and minimal volume entropy.

Let X be a path-connected space with fundamental group G. A (not necessarily path-connected) subset U of X is an amenable subspace if $\pi _1(U,x)\rightarrow \pi (X,x)$ has amenable image for all $x\in U$. The amenable category, denoted $\textsf{cat}_{\mathcal {AMN}}X$, is the minimal $n\in {\mathbb {N}}$ for which there exists an open cover of X by $n+1$ amenable subspaces. If no such cover exists we set $\textsf{cat}_{\mathcal {AMN}}X=\infty$. The definition of amenable category has been extracted from [9] and [23]. Note that we normalise the invariant as in the second paper. Also note that often in the literature the multiplicity of the cover is considered instead, however, the two definitions turn out to be equivalent for CW complexes [9, Remark 3.13]

Proposition 6.1

Let ${\mathbb {F}}$ be a skew field. Let G be residually finite of type ${\textsf{F}}$, and suppose ${\mathcal {D}}_{{\mathbb {F}}{G}}$ exists. If $\textsf{cat}_{\mathcal {AMN}}G=k$. Then, $b^{{\mathcal {D}}_{{\mathbb {F}}{G}}}_p(G)=b^{(2)}_{p}(G,(G_n);{\mathbb {F}})=0$ for $p\geqslant k-1$ and every residual chain $(G_n)$.

Proof

Let X be a finite model for a K(G, 1). As explained in [17, Theorem 3.2] we may adapt the proof of [31, Theorem 1.6] to apply to k-dimensional aspherical simplicial complexes. In particular, for a residual chain $(G_n)_{n\in {\mathbb {N}}}$ we obtain a sequence of covers $X_n\rightarrow X$, such that the number of p-cells in $X_n$ grows sublinearly in $[G:G_n]$. Since

$$\begin{aligned} b^{{\mathcal {D}}_{{\mathbb {F}}{G}}}_p(X_n)= [G:G_n] \cdot b^{{\mathcal {D}}_{{\mathbb {F}}{G}}}_p(G) \leqslant \mathcal |I_n(X_k)| \end{aligned}$$

where $I_p(X_k)$ is the set of p-cells of $X_n$. But now, as k tends to infinity, the left hand side of the equation grows linearly, and the right hand side of the equation grows sublinearly. This is only possible if $b^{{\mathcal {D}}_{{\mathbb {F}}{G}}}_n(G)=0$. The statement concerning $b^{(2)}_{p}(G,(G_n);{\mathbb {F}})$ is analogous. $\square$

Let X be a finite CW complex with a piecewise Riemannian metric g. Fix a basepoint $x_0$ in the universal cover ${{\widetilde{X}}}$ and let ${\widetilde{g}}$ be the pull-back metric. The volume entropy of (X, g) is

$$\begin{aligned} {\text {ent} }(X,g):=\lim _{t\rightarrow \infty }\frac{1}{t}{\text {Vol} }(B_{x_0}(t),{{\widetilde{g}}}). \end{aligned}$$

The minimal volume entropy of X is

$$\begin{aligned} \omega (X):= \inf _g {\text {ent} }(X,g){\text {Vol} }(X,g)^{1/\dim X} \end{aligned}$$

where g varies over all piecewise Riemannian metrics. The invariant was originally defined for Riemannian manifolds in [16].

Suppose G is a group admitting a finite K(G, 1). The minimal volume entropy of G is

$$\begin{aligned} \omega (G):= \inf (\omega (X)) \end{aligned}$$

where X ranges over all finite models of a K(G, 1) such that $\dim X=\text {gd} (G)$.

There are few calculations of minimal volume entropy of groups which are not fundamental groups of aspherical manifolds in literature. To date there is the work of Babenko–Sabourau [2] on which computations for free-by-cyclic groups [6] and RAAGs [17, 23] have been completed.

We say G has uniformly uniform exponential growth if each subgroup either has uniform exponential growth bounded below by some constant $\omega _0>1$ or is virtually abelian. Note that this property is sometimes called uniform uniform exponential growth or locally uniform exponential growth.

Corollary 6.2

Let ${\mathbb {F}}$ be a skew field. Let G be a residually finite group of type ${\textsf{F}}$, and suppose ${\mathcal {D}}_{{\mathbb {F}}{G}}$ exists. If G has uniformly uniform exponential growth and is not ${\mathcal {D}}_{{\mathbb {F}}{G}}$-acyclic, then $\omega (G)>0$.

Proof

This follows from [17, Paragraph after Theorem 3.3] swapping out their use of [17, Theorem 3.3] for our Proposition 6.1. $\square$

Corollary 6.3

Let ${\mathbb {F}}$ be a skew field and let $\varphi :A_L \rightarrow {\mathbb {Z}}$ be an epimorphism. Suppose $BB_L^\varphi$ is of type ${\textsf{F}}$. If $\bigoplus _{v \in L^{\textsf{a}}} {\widetilde{H}}_{p-1}({{\,\textrm{lk}\,}}(v); {\mathbb {Z}})\ne 0$, then $\omega (BBL^\varphi _L)>0$.

Proof

This follows from Corollary 6.2 and the fact that a right-angled Artin group has strongly uniform exponential growth by [3]. $\square$

We conjecture that the converse of the last corollary holds.

Conjecture 6.4

Let ${\mathbb {F}}$ be a skew field and let $\varphi :A_L \rightarrow {\mathbb {Z}}$ be an epimorphism. Suppose $BB_L^\varphi$ is of type ${\textsf{F}}$. If $\bigoplus _{v \in L^{\textsf{a}}} {\widetilde{H}}_{p-1}({{\,\textrm{lk}\,}}(v); {\mathbb {Z}})=0$, then $\omega (BBL^\varphi _L)=0$.