1. Introduction
Given a group $G$ , we say that a collection $\mathcal F$ of subgroups of $G$ is a family if it is nonempty and closed under conjugation and taking subgroups. We fix a group $G$ and a family $\mathcal F$ of subgroups of $G$ . We say that a $G$ CWcomplex $X$ is a model for the classifying space $E_{{\mathcal F}}G$ if all of its isotropy groups belong to $\mathcal F$ and if $Y$ is a $G$ CWcomplex with isotropy groups belonging to $\mathcal F$ , there is precisely one $G$ map $Y\to X$ up to $G$ homotopy. It can be shown that a model for the classifying space $E_{{\mathcal F}}G$ always exists and it is unique up to $G$ homotopy equivalence. We define the $\mathcal F$ geometric dimension of $G$ as
The $\mathcal F$ geometric dimension has its algebraic counterpart, the $\mathcal F$ cohomological dimension $\textrm{cd}_{{\mathcal F}}(G)$ , which can be defined in terms of Bredon cohomology (see Section 2). The $\mathcal F$ geometric dimension and the $\mathcal F$ cohomological dimension satisfy the following inequality (see [Reference Lück and Meintrup17, Theorem 0.1]):
It follows that if $\textrm{cd}_{{\mathcal F}}(G)\geq 3$ then $\textrm{cd}_{{\mathcal F}}(G)=\textrm{gd}_{{\mathcal F}}(G)$ . It is not generally true that $\textrm{cd}_{{\mathcal F}}(G)=\textrm{gd}_{{\mathcal F}}(G)$ . For example, for the family of finite subgroups ${\mathcal F}_0$ , in [Reference Brady, Leary and Nucinkis3] it was proved that there is a rightangled Coxeter group $W$ such that $\textrm{cd}_{{\mathcal F}_0}(W)=2$ and $\textrm{gd}_{{\mathcal F}_0}(W)=3$ . For other examples see [Reference Serre25].
Let $n\geq 0$ be an integer. A group is said to be virtually $\mathbb{Z}^n$ if it contains a subgroup of finite index isomorphic to $\mathbb{Z}^n$ . Define the family
The families ${\mathcal F}_0$ and ${\mathcal F}_1$ are relevant due to their connection with the Farrell–Jones and BaumConnes isomorphism conjectures; see for example [Reference Lück and Reich18]. The Farrell–Jones conjecture has been proved for braid groups in [Reference Aravinda, Farrell and Roushon1, Reference Flores and GonzálezMeneses12, Reference Álvarez and Saldaña15] and for some even Artin groups in [Reference Wu27].
For $n\geq 2$ , the families ${\mathcal F}_n$ have been recently studied by several people; see for example [Reference Huang and Prytuła13, Reference JuanPineda and Saldaña14, Reference Lück16, Reference Rolland, Álvarez and Saldaña23, Reference Serre25]. For a virtually $\mathbb{Z}^n$ group $G$ , it was proved in [Reference Rolland, Álvarez and Saldaña23] that $\textrm{gd}_{{\mathcal F}_k}(G)\leq n+k$ for all $0\leq k\lt n$ . For a free abelian group, this upper bound was also obtained by Corob Cook, Moreno, Nucinkis, and Pasini in [Reference Charney4], and they asked whether this upper bound was sharp:
Question 1 ([Reference Charney4], Question 2.7). For $0\leq k\lt n$ , is $\textrm{gd}_{{\mathcal F}_k}(\mathbb{Z}^n)=n+k$ ?
We answer this question affirmatively in Theorem 1.1. For $k=1$ , this was proved in [Reference Mislin and Valette20, Theorem 5.13] and for $k=2$ in [Reference Prytuła22, Proposition A.]. As an application, we provide lower bounds for the ${\mathcal F}_k$ geometric dimension of virtually abelian groups, braid groups, and rightangled Artin groups (RAAGs). Combining these lower bounds with previously known results in the literature, we show that they are sharp. We also prove that the ${\mathcal F}_k$ geometric dimension is equal to the ${\mathcal F}_k$ cohomological dimension in all these cases. On the other hand, inspired by [Reference Lück16], we use Bass–Serre theory to explicitly calculate, for all $k\geq 1$ , the ${\mathcal F}_k$ geometric dimension of graphs of groups whose vertex groups are infinite finitely generated virtually abelian groups.
There are few explicit calculations of the ${\mathcal F}_n$ geometric dimension for $n\geq 2$ . For example, the ${\mathcal F}_n$ geometric dimension for orientable $3$ manifold groups was explicitly calculated in [Reference Lück16] for all $n\geq 2$ . In [Reference Prytuła22, Proposition A.], it was shown that $\textrm{gd}_{{\mathcal F}_2}(\mathbb{Z}^k)=k+2$ for all $k\geq 3$ . With our results we add braid groups, RAAGs, and graphs of groups whose vertex groups are infinite finitely generated virtually abelian groups to this list. In what follows, we present more precisely these results.
The ${\mathcal F}_n$ dimension of virtually abelian groups
Let $G$ be a virtually $\mathbb{Z}^n$ group. In [Reference Rolland, Álvarez and Saldaña23, Proposition 1.3], it was proved that $\textrm{gd}_{{\mathcal F}_k}(G)\leq n+k$ for $0\leq k\lt n$ . For a free abelian group, this upper bound has also been proved in [Reference Charney4]. In this article, we prove that this upper bound is sharp.
Theorem 1.1. Let $k,n\in \mathbb{N}$ such that $0\leq k\lt n$ . Let $G$ be a virtually $\mathbb{Z}^n$ group. Then $\textrm{gd}_{{\mathcal F}_k}(G)=\textrm{cd}_{{\mathcal F}_k}(G)=n+k$ .
For $k=1$ , the Theorem 1.1 was proved in [Reference Mislin and Valette20, Theorem 5.13]. For $k=2$ , a particular case was proved in [Reference Prytuła22, Proposition A.], specifically $\textrm{gd}_{{\mathcal F}_2}(\mathbb{Z}^k)=k+2$ for all $k\geq 3$ . As a corollary of Theorem 1.1, we have
Corollary 1.2. Let $n\geq 1$ and let $G$ be a group that has a virtually $\mathbb{Z}^n$ subgroup. Then for $0\leq k\lt n$ we have $\textrm{gd}_{{\mathcal F}_k}(G)\geq n+k$ and $\textrm{cd}_{{\mathcal F}_k}(G)\geq n+k$ .
The ${\mathcal F}_n$ dimension of braid groups
There are various ways to define the (full) braid group $B_n$ on $n$ strands. For our purposes, the following definition is convenient. Let $D_n$ be the closed disc with $n$ punctures. We define the braid group $B_n$ on $n$ strands as the isotopy classes of orientation preserving diffeomorphisms of $D_n$ that restrict to the identity on the boundary $\partial D_n$ . In the literature, this group is known as the mapping class group of $D_n$ . It is well known that $\textrm{gd}_{{\mathcal F}_0}(B_n)=n1$ see for example [Reference Arnold2, Section 3]. In [Reference JuanPineda and Saldaña14, Theorem 1.4], it was proved that $\textrm{gd}_{{\mathcal F}_k}(B_n)\leq n+k1$ for all $k\in \mathbb{N}$ . Using Corollary 1.2 and [Reference Farrell and Roushon11, Proposition 3.7] we prove that this upper bound is sharp.
Theorem 1.3. Let $k,n \in \mathbb{N}$ such that $0\leq k\lt n1$ and $G$ be either the full braid group $B_n$ or the pure braid group $P_n$ . Then $\textrm{gd}_{{\mathcal F}_k}(G)=\textrm{cd}_{{\mathcal F}_k}(G)=\textrm{vcd}(G)+k= n+k1.$
The ${\mathcal F}_n$ dimension of rightangled Artin groups
Let $\Gamma$ be a finite simple graph, that is, a finite graph without loops or multiple edges between vertices. We define the rightangled Artin group (RAAG) $A_{\Gamma }$ as the group generated by the vertices of $\Gamma$ with all the relations of the form $vw=wv$ whenever $v$ and $w$ are joined by an edge.
Let $A_\Gamma$ be a RAAG. It is wellknown $A_\Gamma$ is a $\textrm{CAT}(0)$ group, in fact $A_\Gamma$ acts on the universal cover $\tilde{S}_\Gamma$ of its Salvetti CWcomplex $S_\Gamma$ , see Section 4.2. In [Reference Rolland, Álvarez and Saldaña23], it was proved that $\textrm{cd}_{{\mathcal F}_k}(A_\Gamma ) \leq \dim (S_{\Gamma })+k+1$ . Following the proof of [Reference Rolland, Álvarez and Saldaña23], Proof of Theorem 3.1] and using [Reference Huang and Prytuła13, Proposition 7.3], we can actually show that $\textrm{cd}_{{\mathcal F}_k}(A_\Gamma ) \leq \dim (S_{\Gamma })+k$ in Theorem 4.6. Moreover, by using Corollary 1.2 and Remark 4.2, we can prove that this upper bound is sharp.
Theorem 1.4. Let $A_\Gamma$ be a rightangled Artin group. Then for $0 \leq k \lt \textrm{cd}(A_\Gamma )$ we have $ \textrm{gd}_{{\mathcal F}_k}(A_\Gamma )= \textrm{cd}_{{\mathcal F}_k}(A_\Gamma )=\dim (S_{\Gamma })+k=\textrm{cd}(A_\Gamma )+k$ .
This calculation of the ${\mathcal F}_k$ geometric dimension of a RAAG $A_\Gamma$ is explicit because the dimension of the Salvetti CWcomplex $S_{\Gamma }$ is the maximum of all natural numbers $n$ such that there is a complete subgraph $\Gamma '$ of $\Gamma$ with $V(\Gamma ') = n$ (see Lemma 4.5).
Using Corollary 1.2, we can give a lower bound for the ${\mathcal F}_k$ geometric dimension of the outer automorphism group $\textrm{Out}(A_\Gamma )$ of some RAAGs $A_\Gamma$ .
Proposition 1.5. Let $n\geq 2$ . Let $F_n$ be the free group in $n$ generators. Then for all $0\leq k\lt 2n3$ we have
Proposition 1.6. Let $A_d$ be the rightangled Artin group given by a string of $d$ diamonds. Then $\textrm{gd}_{{\mathcal F}_k}(\textrm{Out}(A_d))\geq \textrm{vcd}(\textrm{Out}(A_d))+k\geq 4d +k1$ for all $0\leq k\lt 4d1$ .
Question 2. Given Theorems 1.3 and 4.7 , it is natural to ask whether it is true that in Proposition 1.5 we can have $\textrm{gd}_{{\mathcal F}_k}(\textrm{Out}(F_n)) \leq \textrm{vcd}(\textrm{Out}(F_n)) + k \leq 2n + k  3$ for all $0 \leq k \lt 2n  3$ . Similarly, if it is true that in Proposition 1.6 , we can have $\textrm{gd}_{{\mathcal F}_k}(\textrm{Out}(A_d)) \leq \textrm{vcd}(\textrm{Out}(A_d)) + k \leq 4d + k  1$ for all $0 \leq k \lt 4d  1$ .
The ${\mathcal F}_n$ geometric dimension for graphs of groups of finitely generated virtually abelian groups
Inspired by [Reference Lück16], we use Bass–Serre theory, Theorem 1.1 and Corollary 1.2 to compute the ${\mathcal F}_n$ geometric dimension of graphs of groups whose vertex groups are finitely generated virtually abelian groups.
Theorem 1.7. Let $Y$ be a finite graph of groups such that for each $v\in V(Y)$ the group $G_v$ is infinite finitely generated virtually abelian, with $\textrm{rank}(G_e)\lt \textrm{rank}(G_v)$ . Suppose that the splitting of $G=\pi _1(Y)$ is acylindrical. Let $m=\max \{\textrm{rank}(G_v) v\in V(Y) \}$ . Then for $1\leq k\lt m$ we have $\textrm{gd}_{{\mathcal F}_k}(G)=m+k.$
Corollary 1.8. Let $Y$ be a finite graph of groups such that for each $v\in V(Y)$ the group $G_v$ is infinite finitely generated virtually abelian and for each $e\in E(Y)$ the group $G_e$ is a finite group. Let $m=\max \{\textrm{rank}(G_v) v\in V(Y) \}$ . Then for $1\leq k\lt m$ we have $\textrm{gd}_{{\mathcal F}_k}(G)=m+k.$
Outline of the paper
In Section 2, we introduce the Lück–Weiermann construction, which enables us to build models inductively for the classifying space of $E_{{\mathcal F}_n\cap H}\mathbb{Z}^n$ . Later in the same section, we define Bredon cohomology and present the Mayer–Vietoris sequence, which is as a crucial tool in proving Theorem 3.6. In Section 3, we prove Theorem 1.1. In Section 4, we present some applications of Corollary 1.2, for instance, we explicitly calculate the ${\mathcal F}_k$ geometric dimension of braid groups and RAAGs. Furthermore, we provide a lower bound for the ${\mathcal F}_k$ geometric dimension of the outer automorphism group of certain RAAGs. Finally, we use Bass–Serre theory to prove Theorem 1.7.
2. Preliminaries
The Lück–Weiermann construction
In this subsection, we give a particular construction of Lück–Weiermann [Reference Mislin and Valette20, Theorem 2.3] that we will use later.
Definition 2.1. Let ${\mathcal F}\subset{\mathcal G}$ be two families of subgroups of $G$ . Let $\sim$ be an equivalence relation in ${\mathcal G}{\mathcal F}$ . We say that $\sim$ is strong if the following is satisfied

(a) If $H, K \in{\mathcal G}{\mathcal F}$ with $H\subseteq K$ , then $H\sim K$ ;

(b) If $H, K \in{\mathcal G}{\mathcal F}$ and $g\in G$ , then $H\sim K$ if and only if $gHg^{1} \sim gKg^{1}$ .
Definition 2.2. Let $G$ be a group and $L, K$ be subgroups of $G$ . We say that $L$ and $K$ are commensurable if $L\cap K$ has finite index in both $L$ and $K$ .
Definition 2.3. Let $G$ be a group and let $H$ be a subgroup of $G$ . We define the commensurator of $H$ in $G$ as
Definition 2.4. Let $G$ be a group, let $H$ be a subgroup of $G$ , and $\mathcal F$ a family of subgroups of $G$ . We define the family ${\mathcal F}\cap H$ of $H$ as all the subgroups of $H$ that belong to $\mathcal F$ . We can complete the family ${\mathcal F}\cap H$ in order to get a family $\overline{{\mathcal F}\cap H}$ of $G$ .
Remark 2.5. Following the notation of Definition 2.4 note that:

• If $H=G$ then $\overline{{\mathcal F}\cap H}={\mathcal F}$ .

• If $H$ is normal subgroup of $G$ , then $\overline{{\mathcal F}\cap H}={\mathcal F}\cap H$ .
Let $G$ be a group, $H$ a subgroup of $G$ and $n\geq 0$ . Consider the following nested families of $G$ , $\overline{{\mathcal F}_n\cap H}\subseteq \overline{{\mathcal F}_{n+1}\cap H}$ , let $\sim$ the equivalence relation in $\overline{{\mathcal F}_{n+1}\cap H}\overline{{\mathcal F}_n\cap H}$ given by commensurability. It is easy to check that this is a strong equivalence relation.
We introduce the following notation:

• We denote by $(\overline{{\mathcal F}_{n+1}\cap H}\overline{{\mathcal F}_n\cap H})/\sim$ the equivalence classes in $\overline{{\mathcal F}_{n+1}\cap H}\overline{{\mathcal F}_n\cap H}$ . Given $L\in (\overline{{\mathcal F}_{n+1}\cap H}\overline{{\mathcal F}_n\cap H})$ we denote by $[L]$ its equivalence class.

• Given $[L]\in (\overline{{\mathcal F}_{n+1}\cap H}\overline{{\mathcal F}_n\cap H})/\sim$ , we define the next family of subgroups of $N_{G}[L]$
\begin{equation*}(\overline {{\mathcal F}_{n+1}\cap H})[L]\;:\!=\;\{K\leq N_G[L] K\in (\overline {{\mathcal F}_{n+1}\cap H}\overline {{\mathcal F}_n\cap H}), [K]=[L]\}\cup (\overline {{\mathcal F}_n\cap H}\cap N_G[L]).\end{equation*}
Theorem 2.6 ([Reference Mislin and Valette20], Theorem 2.3). Let $G$ be a group, let $H$ be a subgroup of $G$ and $n\geq 0$ . Consider the following nested families of $G$ , $\overline{{\mathcal F}_n\cap H}\subseteq \overline{{\mathcal F}_{n+1}\cap H}$ , let $\sim$ be the equivalence relation given by commensurability in $\overline{{\mathcal F}_{n+1}\cap H}\overline{{\mathcal F}_n\cap H}$ . Let $I$ be a complete set of representatives of conjugation classes in $(\overline{{\mathcal F}_{n+1}\cap H}\overline{{\mathcal F}_n\cap H})/\sim$ . Choose arbitrary $N_G[L]$ CWmodels for $E_{(\overline{{\mathcal F}_n\cap H})\cap N_{G}[L]}N_{G}[L]$ and $E_{ (\overline{{\mathcal F}_{n+1}\cap H})[L]}N_{G}[L]$ and an arbitrary model for $E_{\overline{{\mathcal F}_n\cap H}}G$ . Consider the following $G$ pushout
such that $f_{[L]}$ is a cellular $G$ map for every $[L]\in I$ and either (1) $i$ is an inclusion of $G$ CWcomplexes or (2) such that every map $f_{[L]}$ is an inclusion of $G$ CWcomplexes for every $[L]\in I$ and $i$ is a cellular $G$ map. Then $X$ is a model for $E_{\overline{{\mathcal F}_{n+1}\cap H}}G$ .
Remark 2.7. The conditions in Theorem 2.6 are not restrictive. For instance, to satisfy the condition $(2)$ , we can use the equivariant cellular approximation theorem to assume that the maps $i$ and $f_{[L]}$ are cellular maps for all $[L]\in I$ and to make the function $f_{[L]}$ an inclusion for every $[L]\in I$ , we can replace the spaces by the mapping cylinders. See [ Reference Mislin and Valette20, Remark 2.5].
Following the notation from Theorem 2.6 we have
Corollary 2.8. $\textrm{gd}_{\overline{{\mathcal F}_{n+1}\cap H}}(G)\leq \max \{ \textrm{gd}_{ \overline{{\mathcal F}_{n}\cap H}}(G)+1, \textrm{gd}_{ (\overline{{\mathcal F}_{n+1}\cap H})[L]}(N_{G}[L]) L\in I\}.$
The pushout of a union of families
The following lemma will be also useful.
Lemma 2.9 ([Reference Day, Sale and Wade8], Lemma 4.4). Let $G$ be a group and ${\mathcal F},\,{\mathcal G}$ be two families of subgroups of $G$ . Choose arbitrary $G$  $CW$ models for $E_{{\mathcal F}} G$ , $E_{{\mathcal G}}G$ and $E_{{\mathcal F}\cap{\mathcal G}} G$ . Then, the $G$  $CW$ complex $X$ given by the cellular homotopy $G$ pushout
is a model for $E_{{\mathcal F}\cup{\mathcal G}}G$ .
With the notation Lemma 2.9 we have the following
Corollary 2.10. $\textrm{gd}_{{\mathcal G}\cup{\mathcal F}}(G)\leq \max \{\textrm{gd}_{{\mathcal F}}(G), \textrm{gd}_{{\mathcal G}}(G), \textrm{gd}_{{\mathcal G}\cap{\mathcal F}}(G)+1\}.$
Nested families
Given a group $G$ and two nested families ${\mathcal F}\subseteq{\mathcal G}$ of $G$ , we will use the following propositions to bound the geometric dimension $\textrm{gd}_{{\mathcal F}}(G)$ using the geometric dimension $\textrm{gd}_{{\mathcal G}}(G)$ .
Proposition 2.11 ([Reference Mislin and Valette20], Proposition 5.1 (i)). Let $G$ be a group and let $\mathcal F$ and $\mathcal G$ be two families of subgroups such that ${\mathcal F}\subseteq{\mathcal G}$ . Suppose for every $H\in{\mathcal G}$ we have $\textrm{gd}_{{\mathcal F}\cap H}(H)\leq d$ . Then $\textrm{gd}_{{\mathcal F}}(G)\leq \textrm{gd}_{{\mathcal G}}(G)+d.$
The proof of the following proposition is implicit in [Reference Lück and Weiermann19, Proof of Theorem 3.1] and [Reference Mislin and Valette20, Proposition 5.1].
Proposition 2.12. Let $G$ be a group. Let $\mathcal F$ and $\mathcal G$ be families of subgroups of $G$ such that ${\mathcal F}\subseteq{\mathcal G}$ . If $X$ is a model for $E_{\mathcal G} G$ , then
Bredon cohomology
In this subsection, we recall the definition of Bredon cohomology, the cohomological dimension for families and its connection with the geometric dimension for families. For further details see [Reference Onorio21].
Fix a group $G$ and $\mathcal F$ a family of subgroups of $G$ . The orbit category $\mathcal{O}_{{\mathcal F}}G$ is the category whose objects are $G$ homogeneous spaces $G/H$ with $H\in{\mathcal F}$ and morphisms are $G$ functions. The category of Bredon modules is the category whose objects are contravariant functors $M\colon \mathcal{O}_{{\mathcal F}}G \to Ab$ from the orbit category to the category of abelian groups, and morphisms are natural transformations $f\colon M\to N$ . This is an abelian category with enough projectives. The constant Bredon module $\underline{\mathbb{Z}}\colon \mathcal{O}_{{\mathcal F}}G \to Ab$ is defined in objects by $\underline{\mathbb{Z}}(G/H)=\mathbb{Z}$ and in morphisms by $\underline{\mathbb{Z}}(\varphi )=id_{\mathbb{Z}}$ . Let $P_{\bullet }$ be a projective resolution of the Bredon module $\underline{\mathbb{Z}}$ , and $M$ be a Bredon module. We define the Bredon cohomology of $G$ with coefficients in $M$ as
We define the $\mathcal F$ cohomological dimension of $G$ as
We have the following Eilenberg–Ganea type theorem that relates the $\mathcal F$ cohomological dimension and the $\mathcal F$ geometric dimension.
Theorem 2.13 ([Reference Lück and Meintrup17], Theorem 0.1). Let $G$ be a group and $\mathcal F$ be a family of subgroups of $G$ . Then
This Theorem 2.13 together with the following Mayer–Vietoris sequence will be used to give lower bounds for the $\mathcal F$ geometric dimension $\textrm{gd}_{{\mathcal F}}(G)$ .
Mayer–Vietoris sequence
Following the notation of Theorem 2.6, by [Reference Davis, Quinn and Reich7, Proposition 7.1] [Reference Mislin and Valette20] we have the next long exact sequence
Remark 2.14. The results presented in Corollaries 2.8 , 2.10 , and Proposition 2.11 have cohomological counterparts. Specifically, if we replace $\textrm{gd}_{{\mathcal F}}$ with $\textrm{cd}_{{\mathcal F}}$ , all the results hold true, see for instance [ Reference Rolland, Álvarez and Saldaña23, Remark 2.9].
3. The ${\mathcal F}_k$ dimension of a virtually $\mathbb{Z}^n$ group
The objective of this section is to prove Theorem 1.1. Let $G$ be a virtually $\mathbb{Z}^n$ group. By [Reference Rolland, Álvarez and Saldaña23, Proposition 1.3], Theorem 2.13 and since the $\mathcal F$ cohomological dimension is monotone, we have for all $0\leq k\lt n$ the following inequalities
Therefore, to prove Theorem 1.1, it is enough to show that $\textrm{cd}_{{\mathcal F}_k\cap \mathbb{Z}^n}(\mathbb{Z}^n)\geq n+k$ for $0\leq k\lt n$ . In Theorem 3.6, we prove this inequality. In order to prove Theorem 3.6 we need Lemma 3.1, Mayer–Vietoris sequence, Lemma 3.5, and Corollary 3.3.
Lemma 3.1. Let $k,t,n \in \mathbb{N}$ such that $0\leq k\lt t\leq n$ . Let $H$ be a subgroup of $\mathbb{Z}^n$ of $\textrm{rank}$ $t$ , then $\textrm{gd}_{{\mathcal F}_k\cap H}(\mathbb{Z}^n)\leq n+k$ .
Proof. The proof is by induction on $k$ . Let $G=\mathbb{Z}^n$ . For $k=0$ , we have $\textrm{gd}_{{\mathcal F}_0\cap H}(G)=\textrm{gd}_{}(G)=n.$ Suppose that the inequality is true for all $k\lt m$ . We prove that the inequality is true for $k=m$ . Let $\sim$ be the equivalence relation on ${\mathcal F}_m\cap H{\mathcal F}_{m1}\cap H$ defined by commensurability, and let $I$ a complete set of representatives classes in $({\mathcal F}_m\cap H{\mathcal F}_{m1}\cap H)/\sim$ . By Corollary 2.8 and Remark 2.5, we have
then to prove that $\textrm{gd}_{{\mathcal F}_m\cap H}(G)\leq n+m$ it is enough to prove that $\textrm{gd}_{({\mathcal F}_m\cap H)[L]}(G)\leq n+m$ for all $L\in I$ . Let $L\in I$ . We can write the family
as the union of two families $ ({\mathcal F}_m\cap H)[L]={\mathcal G} \cup ({\mathcal F}_{m1}\cap H)$ where $\mathcal G$ is the family generated by $\{ K\leq GK\in{\mathcal F}_m\cap H{\mathcal F}_{m1}\cap H, [K]=[L]\}$ . By Corollary 2.10, we have
We prove the following inequalities

(i) $\textrm{gd}_{{\mathcal G}}(G)\leq nm$ ,

(ii) $\textrm{gd}_{{\mathcal G} \cap ({\mathcal F}_{m1}\cap H)}(G)\leq n+m1$
and as a consequence we will have $\textrm{gd}_{{\mathcal F}_m\cap H[L]}(G)\leq n+m$ . First, we prove item (i). Note that a model for $E_{{\mathcal F}_0}(G/L)$ is a model for $E_{\mathcal G} G$ via the action given by the projection $G\to G/L$ . Since $ G/L$ is virtually $\mathbb{Z}^{nm}$ by [Reference Rolland, Álvarez and Saldaña23, Proposition 1.3] we have $\textrm{gd}_{{\mathcal F}_0}(G/L)\leq nm$ .
Now we prove item (ii). Applying Proposition 2.11 to the inclusion of families ${\mathcal G}\cap ({\mathcal F}_{m1}\cap H) \subset{\mathcal G}$ , we get
for some $d$ such that for any $K\in{\mathcal G}$ we have $\textrm{gd}_{{\mathcal G}\cap ({\mathcal F}_{m1}\cap H)\cap K}(K)\leq d$ . Since we already proved $\textrm{gd}_{{\mathcal G}}(G)\leq nm$ , our next task is to show that $d$ can be chosen to be equal to $2m1$ .
Recall that any $K\in{\mathcal G}$ is virtually $\mathbb{Z}^t$ for some $0\leq t \leq m$ . We split our proof into two cases. First assume that $K\in{\mathcal G}$ is virtually $\mathbb{Z}^t$ for some $0\leq t \leq m1$ . Hence, $K$ belongs to ${\mathcal F}_{m1}\cap H$ , it follows that $K$ belongs to ${\mathcal G} \cap ({\mathcal F}_{m1}\cap H)$ and we conclude $\textrm{gd}_{({\mathcal G} \cap ({\mathcal F}_{m1}\cap H))\cap K}(K)=0$ . Now assume $K\in{\mathcal G}$ is virtually $\mathbb{Z}^{m}$ . We claim that $({\mathcal G} \cap ({\mathcal F}_{m1}\cap H))\cap K={\mathcal F}_{m1}\cap K$ . The inclusion $({\mathcal G} \cap ({\mathcal F}_{m1}\cap H))\cap K \subset{\mathcal F}_{m1}\cap K$ is clear since ${\mathcal F}_{m1}\cap H\subset{\mathcal F}_{m1}$ . For the other inclusion let $M\in{\mathcal F}_{m1}\cap K$ . Since $K\leq H$ we get ${\mathcal F}_{m1}\cap K \subseteq{\mathcal F}_{m1}\cap H$ and as a consequence $M\in{\mathcal F}_{m1}\cap H$ , on the other hand $M\leq K\in{\mathcal G}$ , therefore $M\in ({\mathcal G} \cap ({\mathcal F}_{m1}\cap H))\cap K$ . This establishes the claim. We conclude that
where the inequality follows from [Reference Rolland, Álvarez and Saldaña23, Proposition 1.3].
The following proposition is a mild generalization of [Reference Charney4, Lemma 2.3].
Proposition 3.2. Let $H$ be a subgroup of $\mathbb{Z}^n$ that is maximal in ${\mathcal F}_t{\mathcal F}_{t1}$ . Then, for all $0\le k\le t$ , each $L\in ({\mathcal F}_k\cap H{\mathcal F}_{k1}\cap H)$ is contained in a unique maximal element $M\in ({\mathcal F}_k{\mathcal F}_{k1})$ and $M$ is a subgroup of $H$ .
Proof. We have two cases $\textrm{rank}(H)=n$ or $\textrm{rank}(H)\lt n$ . In the first case, by the maximality of $H$ we have that $H=\mathbb{Z}^n$ and ${\mathcal F}_{k}\cap H={\mathcal F}_k$ . Let $L\in ({\mathcal F}_k{\mathcal F}_{k1})$ , we consider the following short exact sequence:
Since $\operatorname{rank}(\mathbb{Z}^n) = \operatorname{rank}(L) + \operatorname{rank}(\mathbb{Z}^n/ L)$ and by the classification theorem of finitely generated abelian groups, we have that $\mathbb{Z}^n/ L$ is isomorphic to $\mathbb{Z}^{nk} \oplus F$ where $F$ is the torsion part. Therefore, it is clear that $p^{1}(F)$ is the unique maximal subgroup of $\mathbb{Z}^n$ of rank $k$ that contains $L$ .
Suppose that $\textrm{rank}(H)=t\lt n$ . Let $L\in{\mathcal F}_{k}\cap H{\mathcal F}_{k1}\cap H$ , in particular $L\in{\mathcal F}_k$ then by the first case $L$ is contained in a unique maximal $M\in{\mathcal F}_{k}{\mathcal F}_{k1}$ . We claim that $M\leq H$ . Note that $MH$ is virtually $\mathbb{Z}^t$ because
it follows that $MH\in{\mathcal F}_t$ , and then the maximality of $H$ implies $H=MH$ . This finishes the proof of claim. Now it is easy to see that $M\in{\mathcal F}_{k}\cap H{\mathcal F}_{k1}\cap H$ is the unique maximal in ${\mathcal F}_{k}\cap H{\mathcal F}_{k1}\cap H$ containing $L$ . In fact, suppose that there is another $N\in{\mathcal F}_{k}\cap H{\mathcal F}_{k1}\cap H$ that is maximal and contains $L$ . Then we have
which implies $NM\in{\mathcal F}_k\cap H$ . This contradicts the maximality of $N$ .
Corollary 3.3. Let $H$ be a subgroup of $\mathbb{Z}^n$ that is maximal in ${\mathcal F}_t{\mathcal F}_{t1}$ . Then, for all $0\le k\le t$ the following statements hold

(a) Each $L\in ({\mathcal F}_k\cap H{\mathcal F}_{k1}\cap H)$ is contained in a unique maximal element $M\in ({\mathcal F}_k\cap H{\mathcal F}_{k1}\cap H)$ .

(b) Let $S\in ({\mathcal F}_{k}\cap H{\mathcal F}_{k1}\cap H)$ be a maximal element, then $S$ is maximal in ${\mathcal F}_k{\mathcal F}_{k1}$ .
Lemma 3.4. Let $n, t\in \mathbb{N}$ such that $0\leq t\lt n$ . Let $L$ be a subgroup of $\mathbb{Z}^n$ that is maximal in ${\mathcal F}_t{\mathcal F}_{t1}$ . Let $SUB(L)$ be the family of all the subgroups of $L$ . Then $\textrm{gd}_{SUB(L)}(\mathbb{Z}^n)\leq nt$ .
Proof. A model for $E_{{\mathcal F}_0}(\mathbb{Z}^n/L)$ is a model for $E_{SUB(L)}\mathbb{Z}^n$ via the action given by the projection $\mathbb{Z}^n\to \mathbb{Z}^n/L$ . Since $\mathbb{Z}^n/L=\mathbb{Z}^{nt}$ , a model for $E_{{\mathcal F}_0}(\mathbb{Z}^n/L)$ is $\mathbb{R}^{nt}$ with the action given by translation.
Lemma 3.5. Let $p,t,n\in \mathbb{N}$ such that $0\leq k\leq p\lt t\leq n$ . Let $H$ be a subgroup of $\mathbb{Z}^n$ that is maximal in ${\mathcal F}_t{\mathcal F}_{t1}$ , and let $S$ be maximal in ${\mathcal F}_{p}\cap H{\mathcal F}_{p1}\cap H$ (note that $S$ is a subgroup of $H$ ). Then, we can choose a model $X$ of $E_{{\mathcal F}_k\cap S}\mathbb{Z}^n$ with $\dim (X)\leq n+k$ , and a model $Y$ of $E_{{\mathcal F}_k\cap H}\mathbb{Z}^n$ with $\dim (Y)\leq n+k$ such that we have an inclusion $X\hookrightarrow Y$ .
Proof. The proof is by induction on $k$ . Let $G=\mathbb{Z}^n$ . For $k=0$ , we have $E_{{\mathcal F}_0\cap S}G=EG$ and $E_{{\mathcal F}_0\cap H}G=EG$ . A model for $EG$ is $\mathbb{R}^n$ and the claim follows. Assuming the claim holds for all $k \lt m$ , we prove that it holds for $k = m$ , that is, we show that there is a model $X$ of $E_{{\mathcal F}_m\cap S}G$ with $\dim (X)\leq n+m$ , and a model $Y$ of $E_{{\mathcal F}_m\cap H}G$ with $\dim (Y)\leq n+m$ such that we have a inclusion $X\hookrightarrow Y$ . Let $\sim$ be the equivalence relation on ${\mathcal F}_m\cap H{\mathcal F}_{m1}\cap H$ defined by commensurability. Let $I_1$ be a complete set of representatives of classes of subgroups in $ ({\mathcal F}_m\cap H {\mathcal F}_{m1}\cap H)/\sim$ . By Corollary 3.3, these representatives can be chosen to be maximal within their class. Applying Theorem 2.6 and Remark 2.5, the following homotopy $G$ pushout gives us a model $X_1$ for $E_{{\mathcal F}_m\cap H}G$
For $L\in I_1$ , by maximality of $L$ in its commensuration class we can write the family
as the union of two families
where $SUB(L)$ is the family of all the subgroups of $L$ .
On the other hand, let $\sim$ be the equivalence relation on ${\mathcal F}_m\cap S{\mathcal F}_{m1}\cap S$ defined by commensurability. Let $I_2$ be a complete set of representatives of classes of subgroups in $ ({\mathcal F}_m\cap S {\mathcal F}_{m1}\cap S)/\sim$ . By Corollary 3.3, these representatives can be chosen to be maximal within their class. Applying Theorem 2.6, we obtain a homotopy $G$ pushout that gives us a model $X_2$ for $E_{{\mathcal F}_m\cap S}G$
Let $T \in I_2$ . We claim that a model for $E_{SUB(T) \cup ({\mathcal F}_{m1} \cap H)}G$ is also a model for $E_{SUB(L) \cup ({\mathcal F}_{m1} \cap H)}G$ for every $L \in I_1$ . Let $L \in I_1$ . Note that $T$ and $L$ are maximal subgroups of $H$ , thus $H = L \oplus N_1$ and $H = T \oplus N_2$ . We can construct an automorphism of $H$ , $\sigma \colon L \oplus N_1 \to T \oplus N_2$ , that maps $L$ to $T$ isomorphically. Since $H$ is maximal in $G$ , we can split $G$ as $G = H \oplus R$ . Therefore, we can extend the automorphism $\sigma$ to an automorphism of $G$ , $\hat{\sigma }\colon L \oplus N_1 \oplus R \to T \oplus N_2 \oplus R$ , that maps $L$ to $T$ isomorphically and preserves the subgroup $H$ . It follows that $E_{SUB(T)\cup ({\mathcal F}_{m1}\cap H)}G$ is a model for $E_{SUB(L)\cup ({\mathcal F}_{m1}\cap H)}G$ via the action given by the automorphism $\hat{\sigma }$ . From Corollary 3.3 it follows that $I_1=I_2\sqcup (I_1I_2)$ . Therefore, we can replace the homotopy $G$ pushouts in equations (3.1) and (3.2) with the following homotopy $G$ pushouts.
By induction hypothesis there is a model $X$ of $E_{{\mathcal F}_{m1}\cap S}G$ with $\dim (X)\leq n+m1$ , and a model $Y$ of $E_{{\mathcal F}_{m1}\cap H}G$ with $\dim (Y)\leq n+m1$ , such that we have a inclusion $X\hookrightarrow Y$ . By the $G$ pushouts in equations (3.3) and (3.4), to prove that there is a inclusion $E_{{\mathcal F}_{m}\cap S}\hookrightarrow E_{{\mathcal F}_{m}\cap H}$ it is enough to prove that there is a inclusion $E_{SUB(T)\cup ({\mathcal F}_{m1}\cap S) }G \hookrightarrow E_{SUB(T)\cup ({\mathcal F}_{m1}\cap H)}G$ . By Lemma 2.9, the following $G$ pushouts gives us a model for $E_{SUB(T)\cup ({\mathcal F}_{m1}\cap S) }G$ and $E_{SUB(T)\cup ({\mathcal F}_{m1}\cap H) }G$ , respectively.
Note that $SUB(T)\cap ({\mathcal F}_{m1}\cap S)={\mathcal F}_{m1}\cap T= SUB(T)\cap ({\mathcal F}_{m1}\cap H)$ . It follows from these $G$ pushouts that we have a inclusion $E_{SUB(T)\cup ({\mathcal F}_{m1}\cap S) }G \hookrightarrow E_{SUB(T)\cup ({\mathcal F}_{m1}\cap H)}G$ .
Finally, we prove that $\dim (X_1)\leq n+m$ and $\dim (X_2)\leq n+m$ . From equation (3.3) it follows
Then to prove that $\dim (X_1)\leq n+m$ it is enough to prove $\textrm{gd}_{SUB(T)\cup ({\mathcal F}_{m1}\cap H)}(G)\leq n+m$ . By equation (3.5) and since $SUB(T) \cap ({\mathcal F}_{m1}\cap H)={\mathcal F}_{m1}\cap T$ we have
Theorem 3.6 (The lower bound). Let $m,t,n\in \mathbb{N}$ such that $0\leq m\lt t\leq n$ . Let $H$ be a subgroup of $\mathbb{Z}^n$ that is maximal in ${\mathcal F}_t{\mathcal F}_{t1}$ , then $H_{{\mathcal F}_m\cap H}^{n+m}(\mathbb{Z}^n;\;\underline{\mathbb{Z}})\neq 0$ .
Proof. Let $G=\mathbb{Z}^n$ . The proof is by double induction on $(t,m)$ . The claim is true for all $(t,0)\in \mathbb{N}\times \{0\}$ . Let $H$ be a subgroup of $G$ that is maximal in ${\mathcal F}_t{\mathcal F}_{t1}$ , then
Suppose that the claim is true for all $(t,s)\in \mathbb{N}\times \{0,1,\dots, m1\}$ , we prove that the claim is true for $(t,m)$ , i.e. $H_{{\mathcal F}_m\cap H}^{n+m}(G;\;\underline{\mathbb{Z}})\neq 0$ .
Applying Mayer–Vietoris to the $G$ pushout in equation (3.3) and Lemma 3.1, we have the following long exact sequence
We now show that $\prod _{L\in{I}_1}H^{n+m}(E_{SUB(T)\cup ({\mathcal F}_{m1}\cap H)}G/G)=0$ . It is enough to show that $\textrm{gd}_{SUB(T)\cup ({\mathcal F}_{m1}\cap H)}(G)\leq n+m1$ . By Lemma 2.9 the following homotopy $G$ pushout gives us a model $Y$ for $E_{SUB(T)\cup ({\mathcal F}_{m1}\cap H)}G$ .
Note that $SUB(T)\cap ({\mathcal F}_{m1}\cap H)={\mathcal F}_{m1}\cap T$ . By Lemma 3.5, the map $g$ can be taken as an inclusion, then by [Reference Waner26, Theorem 1.1] the homotopy $G$ pushout can be taken as a $G$ pushout. It follows that
Then the sequence equation (3.6) reduce to
Then to prove that $H_{{\mathcal F}_k\cap H}^{n+m}(G;\;\underline{\mathbb{Z}})=H^{n+m}(X_1/G)\neq 0$ is enough to prove that $\varphi$ is not surjective. By equation (3.3) we have $\varphi = (\prod _{L\in I_1}f_{T}^{*})\Delta$ , where $\Delta$ is the diagonal embedding. First, we prove that $f_{T}^{*}$ is not surjective.
Applying Mayer–Vietoris to the $G$ pushout in equation (3.7) we have the following long exact sequence
Since $\textrm{gd}_{SUB(T)}(G)\leq nm$ and since there is precisely one $G$ map $E_{{\mathcal F}_{m1}\cap H}G\to E_{SUB(T)\cup ({\mathcal F}_{m1}\cap H)}G$ up to $G$ homotopy we can reduce the sequence to
By hypothesis $T$ is maximal in ${\mathcal F}_m\cap H{\mathcal F}_{m1}\cap H$ , then by Corollary 3.3 (b) we have that $T$ is maximal in ${\mathcal F}_m{\mathcal F}_{m1}$ , by induction hypothesis we have that $H^{n+m1}(E_{{\mathcal F}_{m1}\cap T}G/G)\neq 0$ , thus $f_T^{*}$ is not surjective.
Finally, we see that $\varphi$ is not surjective. In fact, let $b_K\notin \textrm{Im}(f_{T}^{*})$ , for some $K\in I_1$ , then $(0,0, \cdots, b_K,\cdots,0)\notin \textrm{Im}(\varphi )$ . Suppose that is not the case, that is, there is
such that $(0,0, \cdots, b_K,\cdots,0)=\varphi ((\prod _{L\in{I}_1}a_L,c))=\prod _{L\in{I}_1}f_{T}^{*}(a_L)\Delta (c)=(f_{T}^{*}(a_L)c)_{L\in I_1}.$ Then $f_{T}^{*}(a_L)=c$ for $L\neq K$ and $f_{T}^{*}(a_K)c=b_K$ , it follows that
then $b_K\in \textrm{Im}(f_{T}^{*})$ and this is a contradiction.
Proposition 3.7. Let $k,t,n\in \mathbb{N}$ such that $0\leq k\lt t\leq n$ . Let $H$ be a subgroup of $\mathbb{Z}^n$ that is maximal in ${\mathcal F}_t{\mathcal F}_{t1}$ . Let ${\mathcal F}_k\cap H$ be the family that consists of all the subgroups of $H$ that belong to ${\mathcal F}_k$ . Then $\textrm{cd}_{{\mathcal F}_k\cap H}(\mathbb{Z}^n)=\textrm{gd}_{{\mathcal F}_k\cap H}(\mathbb{Z}^n)=n+k$ .
4. Some applications of Corollary 1.2
4.1. The ${\mathcal F}_k$ dimension of braid groups
In this subsection, we compute the ${\mathcal F}_n$ dimension of full and pure braid groups. For our purposes, it is convenient to define the braid group as follows: let $D_n$ the closed disc with $n$ punctures, we define the braid group $B_n$ on $n$ strands, as the isotopy classes of orientation preserving diffeomorphisms of $D_n$ that restrict to the identity on the boundary $\partial D_n$ . We define the pure braid group, $P_n$ , as the finite index subgroup of $B_n$ consisting of elements that fixe pointwise the punctures.
Theorem 4.1. Let $k,n \in \mathbb{N}$ such that $0\leq k\lt n1$ and let $G$ be either the braid group $B_n$ or the pure braid group $P_n$ . Then $\textrm{gd}_{{\mathcal F}_k}(G)=\textrm{cd}_{{\mathcal F}_k}(G)=n+k1$ .
Proof. It is enough to prove the following inequalities
In [Reference JuanPineda and Saldaña14, Theorem 1.4] was proved that $\textrm{gd}_{{\mathcal F}_k}(B_n)\leq \textrm{vcd}(B_n)+k$ for all $0\leq k\lt n1$ . Since $P_n$ has finite index in $B_n$ also we have $\textrm{gd}_{{\mathcal F}_k}(P_n)\leq \textrm{vcd}(P_n)+k$ for all $0\leq k\lt n1$ . On the other hand, it is well known that $\textrm{vcd}(B_n)=n1$ see for example [Reference Arnold2, Section 3]. This proves the first inequality. The second inequality is by Theorem 2.13. In [Reference Farrell and Roushon11, Proposition 3.7] it is shown that $P_n$ has a subgroup isomorphic to $\mathbb{Z}^{n1}$ . Therefore, by monotonicity of the ${\mathcal F}_k$ geometric dimension and Corollary 1.2, we have $\textrm{cd}_{{\mathcal F}_k}(B_n)\geq \textrm{cd}_{{\mathcal F}_k}(P_n)\geq n+k1$ for all $0\leq k\lt n1$ . This proves the last inequality.
For $k=1$ , this theorem has been proved in [Reference Farrell and Roushon11].
4.2. The ${\mathcal F}_k$ dimension of RAAGs and their outer automorphism groups
In this subsection, we compute the ${\mathcal F}_n$ dimension of RAAGs and we give a lower bound for the ${\mathcal F}_n$ geometric dimension of the outer automorphism group of some RAAGs.
We recall some basic notions about RAAGs, for further details see for instance [Reference Cook, Moreno, Nucinkis and Pasini5]. Let $\Gamma$ be a finite simple graph, that is, a finite graph without loops or multiple edges between vertices. We define the rightangled Artin group (RAAG) $A_{\Gamma }$ as the group generated by the vertices of $\Gamma$ with all the relations of the form $vw=wv$ whenever $v$ and $w$ are joined by an edge.
The Salvetti complex
For the construction of the Salvetti complex we follow [Reference Cook, Moreno, Nucinkis and Pasini5, Subsection 3.6]. Let $A_\Gamma$ be a RAAG, its Salvetti complex $S_\Gamma$ is a CWcomplex that can be constructed as follows:

• The $S_\Gamma ^{(1)}$ skeleton is constructed as follows: we take a point $x_0$ , and for each $v\in V(\Gamma )$ , we attach a $1$ cell $I=[0,1]$ that identifies the endpoints of $I$ to $x_0$ . Then, the $S_\Gamma ^{(1)}$ skeleton is a wedge of circles.

• The $S_\Gamma ^{(2)}$ skeleton is constructed as follows. For each edge of $\Gamma$ we attach a $2$ cell $I\times I$ to $S_\Gamma ^{(1)}$ by the boundary $\partial (I\times I)$ as $s_vs_w s_{v}^{1}s_{w}^{1}$ .

• In general the $S_\Gamma ^{(n)}$ skeleton is constructed as follows. For each complete subgraph $\Gamma '$ of $\Gamma$ with $V(\Gamma ')=n$ , we attach a $n$ cell $I^n$ to the $S_\Gamma ^{(n1)}$ skeleton using the generators $V(\Gamma ')$ .
Remark 4.2. Note that, by the construction of the Salvetti complex $S_\Gamma$ , its fundamental group is $A_\Gamma$ . Additionally, $S_\Gamma$ has a $\dim (S_\Gamma )$ dimensional torus embedded in it, which follows from its construction. Therefore, the fundamental group $\pi _1(S_\Gamma,x_0)=A_\Gamma$ has a subgroup that is isomorphic to $\mathbb{Z}^{\dim (S_\Gamma )}$ .
Theorem 4.3 ([Reference Cook, Moreno, Nucinkis and Pasini5], Theorem 3.6). The universal cover of the Salvetti complex, $\tilde{S_\Gamma }$ , is a $\textrm{CAT}(0)$ cube complex. In particular, $S_\Gamma$ is a $K(A_\Gamma, 1)$ space.
Corollary 4.4. Let $G$ be a RAAG. Then $G$ is torsionfree.
Lemma 4.5. Let $A_\Gamma$ be a RAAG then $\textrm{gd}(A_\Gamma )=\textrm{cd}(A_\Gamma )=\dim (S_\Gamma )$ . Moreover,
Proof. It is enough to prove the following inequalities
The first inequality follows from Theorem 4.3. The second inequality follows from Theorem 2.13. By [Reference Cook, Moreno, Nucinkis and Pasini5, Subsection 3.7] $H^{\dim (S_\Gamma )}(S_\Gamma )=H^{\dim (S_\Gamma )}(A_\Gamma )$ is a free abelian generated by each $\dim (S_\Gamma )$ cell. The third inequality follows.
By construction of the Salvetti complex $S_\Gamma$ , we have that
Since $\textrm{cd}(A_\Gamma )= \dim (S_\Gamma )$ the claim follows.
Let $G$ be a rightangled Artin group. In [Reference Rolland, Álvarez and Saldaña23, Corollary 1.2], it was proved that $\textrm{cd}_{{\mathcal F}_k}(G)\leq \textrm{cd}(G)+k+1$ for all $0\leq k\lt \textrm{cd}(G)$ . However, by following their proof in [Reference Rolland, Álvarez and Saldaña23, Proof of Theorem 3.1] and using [Reference Huang and Prytuła13, Proposition 7.3], we can actually prove that $\textrm{cd}_{{\mathcal F}_k}(G)\leq \textrm{cd}(G)+k$ for all $0\leq k\lt \textrm{cd}(G)$ . In [Reference Rolland, Álvarez and Saldaña23] and [Reference Huang and Prytuła13, Proposition 7.3], they work with the ${\mathcal F}_k$ cohomological dimension instead of ${\mathcal F}_k$ geometric dimension, that is the reason the following Theorem 4.6 is stated in terms of ${\mathcal F}_k$ cohomological dimension.
Theorem 4.6. Let $G$ be a RAAG. Then $\textrm{cd}_{{\mathcal F}_k}(G)\leq \textrm{cd}(G)+k$ for $k\in \mathbb{N}$ .
Proof. The proof is by induction on $k$ . For $k=0$ it follows from Lemma 4.5. Suppose that the inequality is true for all $k\lt m$ . We prove the inequality for $k=m$ . Let $\sim$ be the equivalence relation on ${\mathcal F}_m{\mathcal F}_{m1}$ defined by commensurability, and let $I$ be a complete set of representatives of conjugacy classes in $({\mathcal F}_m{\mathcal F}_{m1})/\sim$ . Then by the cohomological version of Corollary 2.8 (see Remark 2.14) we have
Then to prove that $\textrm{cd}_{{\mathcal F}_{m}}(G)\leq \textrm{cd}(G)+m$ it is enough to prove that $\textrm{cd}_{{\mathcal F}_{m}[L]}(N_{G}[L])\leq \textrm{cd}(G)+m$ for all $L\in I$ . Let $L\in I$ , we can write the family
as the union of two families ${\mathcal F}_m[L]={\mathcal G} \cup ({\mathcal F}_{m1}\cap N_{G}[L])$ where $\mathcal G$ is the family generated by $\{ K\leq N_G[L]K\in{\mathcal F}_m{\mathcal F}_{m1}, K\sim L\}$ . By the cohomological version of Corollary 2.10 (see Remark 2.14) we have
We prove that

1. $\textrm{cd}_{{\mathcal G}}(N_{G}[L])\leq \textrm{cd}(G)m$

2. $\textrm{cd}_{{\mathcal G}\cap{\mathcal F}_{m1}}(N_{G}[L])\leq \textrm{cd}(G)+m1$
As a consequence we will have $\textrm{cd}_{{\mathcal F}_m[L]}(N_{G}[L])\leq \textrm{cd}(G)+m$ . First, we prove item $(1)$ . We define the family ${\mathcal F}=\{K\leq N_{G}[L] \mid [K\;:\;K\cap L]\lt \infty \}$ . We claim that ${\mathcal F} ={\mathcal G}$ . To show that ${\mathcal G} \subseteq{\mathcal F}$ , note that
since, by definition, $\mathcal G$ is the smallest family that contains $\{ K \leq N_G[L] \mid K \in{\mathcal F}_m {\mathcal F}_{m1}, K \sim L \}$ , it follows that ${\mathcal G} \subseteq{\mathcal F}$ . Now let’s prove the other inclusion ${\mathcal F} \subseteq{\mathcal G}$ . Let $S \in{\mathcal F}$ , then $[S \;:\; S \cap L] \lt \infty$ . Note that $[LS \;:\; L] = [S \;:\; S \cap L] \lt \infty$ , it follows that $LS$ is commensurable with $L$ , and as a consequence $S \leq LS \in{\mathcal G}$ , in particular it follows that $S \in{\mathcal G}$ . This proves the claim. Since ${\mathcal G}={\mathcal F}$ we have by [Reference Huang and Prytuła13, Proposition 7.3 and Definition 7.2] that $\textrm{cd}_{{\mathcal G}}(N_{G}[L])\leq \textrm{cd}(G)m$ .
We now prove the item $(2)$ . Applying the cohomological version of Proposition 2.11 (see Remark 2.14) to the inclusion of families $({\mathcal G}\cap{\mathcal F}_{m1}) \subset{\mathcal G}$ we get
for some $d$ such that for any $K\in{\mathcal G}$ we have $\textrm{cd}_{({\mathcal G}\cap{\mathcal F}_{m1})\cap K}(K)\leq d$ . Since we already proved $\textrm{cd}_{{\mathcal G}}(N_{G}[L])\leq \textrm{cd}(G)m$ , our next task is to show that $d$ can be chosen to be equal to $2m1$ .
Recall that any $K\in{\mathcal G}$ is virtually $\mathbb{Z}^t$ for some $0\leq t \leq m$ . We split our proof into two cases. First assume that $K\in{\mathcal G}$ is virtually $\mathbb{Z}^t$ for some $0\leq t \leq m1$ . Hence $K$ belongs to ${\mathcal F}_{m1}$ , it follows that $K$ belongs to ${\mathcal G} \cap{\mathcal F}_{m1}$ and we conclude $\textrm{cd}_{{\mathcal G} \cap{\mathcal F}_{m1}\cap K}(K)=0$ . Now assume $K\in{\mathcal G}$ is virtually $\mathbb{Z}^{m}$ . We claim that $({\mathcal G} \cap{\mathcal F}_{m1})\cap K={\mathcal F}_{m1}\cap K$ . The inclusion $({\mathcal G} \cap{\mathcal F}_{m1})\cap K \subset{\mathcal F}_{m1}\cap K$ is clear. For the other inclusion let $M\in{\mathcal F}_{m1}\cap K$ . Since $M\leq K\in{\mathcal G}$ , therefore $M\in ({\mathcal G} \cap{\mathcal F}_{m1})\cap K$ . This establishes the claim. We conclude that
where the inequality follows from [Reference Rolland, Álvarez and Saldaña23, Proposition 1.3].
Theorem 4.7. Let $G$ be a rightangled Artin group. Then for $0 \leq k \lt \textrm{cd}(G)$ we have $ \textrm{cd}_{{\mathcal F}_k}(G)=\textrm{cd}(G)+k$ .
Proof. By Theorem 4.6, we have $\textrm{cd}_{{\mathcal F}_k}(G)\leq \textrm{cd}(G)+k$ . On the other hand, by Lemma 4.5 $G$ has a subgroup isomorphic to $\mathbb{Z}^{\textrm{cd}(G)}$ , then the claim it follows from Corollary 1.2.
Theorem 4.8. Let $G$ be a rightangled Artin group. Then for $0 \leq k \lt \textrm{cd}(G)$ we have $ \textrm{gd}_{{\mathcal F}_k}(G)= \textrm{cd}_{{\mathcal F}_k}(G)$ .
Proof. If $k=0$ the claim follows from Lemma 4.5. Suppose that $k\geq 1$ , hence by hypothesis, $\textrm{cd}(G)\geq 2$ . By Theorem 4.7, we have $\textrm{cd}_{{\mathcal F}_k}(G)\geq 3$ , then by Theorem 2.13, $\textrm{gd}_{{\mathcal F}_k}(G)= \textrm{cd}_{{\mathcal F}_k}(G)$ .
Given a fixed rightangled Artin group $A_{\Gamma }$ , we denote by ${\textrm{Aut}}(A_\Gamma )$ the group of automorphisms of $A_\Gamma$ and by $\textrm{Inn}(A_\Gamma )$ the subgroup consisting of inner automorphisms. The outer automorphism group of $A_\Gamma$ is defined as the quotient $\textrm{Out}(A_\Gamma )={\textrm{Aut}}(A_\Gamma )/\textrm{Inn}(A_\Gamma )$ . If $S\subseteq V(\Gamma )$ , then the subgroup $H$ generated by $S$ is called a special subgroup of $A_\Gamma$ . It can be proven that, in fact, $H$ is the rightangled Artin group $A_S$ associated with the full subgraph induced by $S$ in $\Gamma$ .
If $\Delta$ is a full subgraph of $\Gamma$ , we denote by $A_{\Delta }$ the special subgroup generated by the vertices contained in $\Delta$ . An outer automorphism $F$ of $A_\Gamma$ preserves $A_\Delta$ if there exists a representative $f\in F$ that restricts to an automorphism of $A_\Delta$ . An outer automorphism $F$ acts trivially on $A_{\Delta }$ if there exists representative $f\in F$ that acts as the identity on $A_\Delta$ .
Definition 4.9. Let $\mathcal G$ , $\mathcal H$ be two collections of special subgroups of $A_\Gamma$ . The relative outer automorphism group $\textrm{Out}(A_{\Gamma };\;{\mathcal G},{\mathcal H}^t)$ consists of automorphisms that preserve each $A_\Delta \in{\mathcal G}$ and act trivially on each $A_\Delta \in{\mathcal H}.$
Proposition 4.10. Let $A_{\Gamma }=A_{\Delta _1}*A_{\Delta _1}*\cdots *A_{\Delta _k}*F_n$ be a free factor decomposition of a rightangled Artin group with $k\geq 1$ . Then $\textrm{gd}_{{\mathcal F}_k}(\textrm{Out}(A_{\Gamma };\; \{A_{\Delta _i}\}^{t}))\geq \textrm{vcd}(\textrm{Out}(A_{\Gamma };\; \{A_{\Delta _i}\}^{t}))+k$ for all $0\leq k\lt \textrm{vcd}(\textrm{Out}(A_{\Gamma };\; \{A_{\Delta _i}\}^{t})).$
Proof. By [Reference Day and Wade9, Theorem A] $\textrm{Out}(A_{\Gamma };\; \{A_{\Delta _i}\}^{t})$ has a free abelian subgroup of $\textrm{rank}$ equal to $\textrm{vcd}(\textrm{Out}(A_{\Gamma };\; \{A_{\Delta _i}\}^{t}))$ . The inequality follows from Corollary 1.2.
Let $F_n$ be the free group in $n$ generators. The group $F_n$ can be seen as the RAAG associated with the graph that has $n$ vertices and no edges. In [Reference Culler and Vogtmann6] was proved that $\textrm{vcd}(\textrm{Out}(F_n))=2n3$ for $n\geq 2$ and that $\textrm{Out}(F_n)$ has a subgroup ismorphic to $\mathbb{Z}^{\textrm{vcd}(\textrm{Out}(F_n))}$ . From Corollary 1.2, we get the following
Proposition 4.11. Let $n\geq 2$ . Let $F_n$ be the free group in $n$ generators. Then $\textrm{gd}_{{\mathcal F}_k}(\textrm{Out}(F_n))\geq 2n+k3$ for all $0\leq k\lt 2n3$ .
Let $A_d$ be the rightangled Artin group given by a string of $d$ diamonds. In [Reference Degrijse and Petrosyan10, Proposition 6.5] was proved that $\textrm{vcd}(Out(A_d))=4d1$ and $Out(A_d)$ has a subgroup isomorphic to $\mathbb{Z}^{\textrm{vcd}(\textrm{Out}(A_d))}$ , from Corollary 1.2 we have
Proposition 4.12. Let $A_d$ be the rightangled Artin group given by a string of $d$ diamonds. Then $\textrm{gd}_{{\mathcal F}_k}(\textrm{Out}(A_d))\geq 4d +k1$ for all $0\leq k\lt 4d1$ .
4.3. The ${\mathcal F}_k$ geometric dimension for graphs of groups of finitely generated virtually abelian groups
The objective of this section is to explicitly calculate the ${\mathcal F}_n$ geometric dimension of the fundamental group of a graph of groups whose vertex groups are finitely generated virtually abelian groups, and whose edge groups are finite groups.
Bass–Serre theory
We recall some basic notions about Bass–Serre theory, for further details see [Reference Saldaña24]. A graph of groups $\mathbf{Y}$ consists of a graph $Y$ , a group $Y_v$ for each $v\in V(Y)$ , and a group $Y_e$ for each $e=\{v,w\}\in E(Y)$ , together with monomorphisms $\varphi \colon Y_e \to Y_{i}$ $i=v,w$ .
Given a graph of groups $\mathbf{Y}$ , one of the classic theorems of Bass–Serre theory provides the existence of a group $G=\pi _1(\mathbf{Y})$ , called the fundamental group of the graph of groups $\mathbf{Y}$ and the tree $T$ (a graph with no cycles), called the Bass–Serre tree of $\mathbf{Y}$ , such that $G$ acts on $T$ without inversions, and the induced graph of groups is isomorphic to $\mathbf{Y}$ . The identification $G=\pi _1(\mathbf{Y})$ is called a splitting of $G$ .
Definition 4.13. Let $Y$ be a graph of groups with fundamental group $G$ . The splitting $G=\pi _1(Y)$ is acylindrical if there is an integer $k$ such that, for every path $\gamma$ of length $k$ in the Bass–Serre tree $T$ of $Y$ , the stabilizer of $\gamma$ is finite.
Recall a geodesic line of a simplicial tree $T$ , is a simplicial embedding of $\mathbb{R}$ in $T$ , where $\mathbb{R}$ has as vertex set $\mathbb{Z}$ and an edge joining any two consecutive integers.
Theorem 4.14 ([Reference Lück16], Theorem 6.3). Let $Y$ be a graph of groups with finitely generated fundamental group $G$ and Bass–Serre tree $T$ . Consider the collection $\mathcal{A}$ of all the geodesics of $T$ that admit a cocompact action of an infinite virtually cyclic subgroup of $G$ . Then the space $\widetilde{T}$ given by the following homotopy $G$ pushout
is a model $\widetilde{T}$ for $E_{\textrm{Iso}_{G}(\widetilde{T})}G$ where $\textrm{Iso}_{G}(\widetilde{T})$ is the family generated by the isotropy groups of $\widetilde{T}$ , i.e., by coningoff on $T$ the geodesics in $\mathcal{A}$ we obtain a model for $E_{\textrm{Iso}_{G}(\widetilde{T})}G$ . Moreover, if the splitting $G=\pi _1(Y)$ is acylindrical, then the family $\textrm{Iso}_{G}(\widetilde{T})$ contains the family ${\mathcal F}_n$ of $G$ for all $n\ge 0$ .
The following theorem is mild generalization of [Reference Lück16, Proposition 7.4]. We include a proof for the sake of completeness.
Theorem 4.15. Let $Y$ be a graph of groups with finitely generated fundamental group $G$ and Bass–Serre tree $T$ . Suppose that the splitting of $G$ is acylindrical. Then for all $k\geq 1$ we have
and
Proof. For each $s\in V(Y)\cup E(Y)$ we have that $G_s$ is a subgroup of $G$ , then the first inequality follows. Now we prove the second inequality. The splitting of $G$ is acylindrical, then we can use Theorem 4.14 to obtain a 2dimensional space $\widetilde{T}$ that is obtained from $T$ coningoff some geodesics of $T$ , see Figure 1, the space $\widetilde{T}$ is a model for $E_{\textrm{Iso}_{G}(\widetilde{T})}G$ and ${\mathcal F}_k\subseteq \textrm{Iso}_{G}(\widetilde{T})$ . By Proposition 2.12, we have
Let $\sigma$ be a cell of $\widetilde{T}$ , we compute $\textrm{gd}_{{\mathcal F}_k\cap G_\sigma }(G_{\sigma })+\dim (\sigma )$ .

• If $\sigma$ is 0cell, we have two cases $\sigma \in T$ or $\sigma \in \widetilde{T}T$ , in the first case we have $G_\sigma =G_v$ for some $v\in V(Y)$ , in the other case we have $G_\sigma$ is virtually cyclic, then $\textrm{gd}_{{\mathcal F}_k\cap G_\sigma }(G_{\sigma })+\dim (\sigma )= \textrm{gd}_{{\mathcal F}_k\cap G_v}(G_v)$ or $0$ .

• If $\sigma$ is 1cell, we have two cases $\sigma \in T$ or $\sigma$ has a vertex in $\widetilde{T}T$ , in the first case we have $G_\sigma =G_e$ for some $e\in E(Y)$ , in the other case we have $G_\sigma$ is virtually cyclic, then $\textrm{gd}_{{\mathcal F}_k\cap G_\sigma }(G_{\sigma })+\dim (\sigma )= \textrm{gd}_{{\mathcal F}_k\cap G_e}(G_e)+1$ or $1$ .

• If $\sigma$ is 2cell, then $\sigma$ has a vertex in $\widetilde{T}T$ , then $G_\sigma$ is virtually cyclic, it follows that $\textrm{gd}_{{\mathcal F}_k\cap G_\sigma }(G_{\sigma })+\dim (\sigma )= 2$ .
The inequality follows.
Proposition 4.16. Let $Y$ be a finite graph of groups such that for each $v\in V(Y)$ the group $G_v$ is infinite finitely generated virtually abelian, with $\textrm{rank}(G_e)\lt \textrm{rank}(G_v)$ . Suppose that the splitting of $G=\pi _1(Y)$ is acylindrical. Let $m=\max \{\textrm{rank}(G_v) v\in V(Y) \}$ . Then for $1\leq k\lt m$ we have $\textrm{gd}_{{\mathcal F}_k}(G)=m+k.$
Proof. First, we prove that $\textrm{gd}_{{\mathcal F}_k}(G)\geq m+k$ . The splitting of $G$ is acylindrical, then by Theorem 4.15 we have
Also by Theorem 4.15, we have
Corollary 4.17. Let $Y$ be a finite graph of groups such that for each $v\in V(Y)$ the group $G_v$ is infinite finitely generated virtually abelian, and for each $e\in E(Y)$ the group $G_e$ is a finite group.
Let $m=\max \{rank(G_v) v\in V(Y) \}$ . Then for $1\leq k\lt m$ we have $\textrm{gd}_{{\mathcal F}_k}(G)=m+k.$
Acknowledgements
I was supported by a doctoral scholarship of the Mexican Council of Humanities, Science and Technology (CONAHCyT). I would like to thank Luis Jorge Sánchez Saldaña for several useful discussions during the preparation of this article. I also thank Rita Jiménez Rolland for comments on a draft of the present article. I am grateful for the financial support of DGAPAUNAM grant PAPIIT IA106923 and CONACyT grant CF 2019217392. I thank the anonymous referee for corrections and comments that improved the exposition.