On subgroups of right angled Artin groups with few generators

TL;DR

This paper constructs specific subgroups of right angled Artin groups with prescribed cohomological dimensions, demonstrating limitations of universality and embedding properties, and explores finitely presented subgroups of direct products of limit groups.

Contribution

It introduces new constructions of subgroups with controlled cohomological dimensions and analyzes their embedding properties within right angled Artin groups and limit groups.

Findings

Constructed 3-generated groups with arbitrary cohomological dimension as subgroups of free groups.

Showed that certain special HNN-extensions force containment of quotient groups in right angled Artin groups.

Proved bounds on embeddings of finitely presented subgroups into direct powers of free groups.

Abstract

For each natural number $d$ we construct a $3$-generated group $H_d$, which is a subdirect product of free groups, such that the cohomological dimension of $H_d$ is $d$. Given a group $F$ and a normal subgroup $N \lhd F$ we prove that any right angled Artin group containing the special HNN-extension of $F$ with respect to $N$ must also contain $F/N$. We apply this to construct, for every $d \in \mathbb{N}$, a $4$-generated group $G_d$, embeddable into a right angled Artin group, such that the cohomological dimension of $G_d$ is $2$ but the cohomological dimension of any right angled Artin group, containing $G_d$, is at least $d$. These examples are used to show the non-existence of certain "universal" right angled Artin groups. We also investigate finitely presented subgroups of direct products of limit groups. In particular we show that for every $n\in \mathbb{N}$ there exists $\delta(n) \in \mathbb{N}$ such that any $n$-generated finitely presented subgroup of a direct product of finitely many free groups embeds into the $\delta(n)$-th direct power of the free group of rank $2$. As another corollary we derive that any $n$-generated finitely presented residually free group embeds into the direct product of at most $\delta(n)$ limit groups.