Algebraic and measurable embeddings between square-free right-angled Artin groups

Abstract

Let $\Gamma ,\Delta$ be finite simplicial graphs, and assume that $\Gamma$ has no induced squares. We prove that the right-angled Artin group $A(\Delta )$ measurably embeds into $A(\Gamma )$ if and only if $A(\Delta )$ embeds as a subgroup in a graph product of free abelian groups over $\Gamma$. This in turn has a graph-theoretical characterisation which can be checked algorithmically. If additionally $A(\Gamma )$ and $A(\Delta )$ have cohomological dimension equal to two and $A(\Gamma )$ is not isomorphic to $\mathbb{Z}\times F_n$, we get that $A(\Delta )$ measurably embeds into $A(\Gamma )$ if and only if it embeds as a subgroup in $A(\Gamma )$. Our proof relies on the following algebraic statement. Let $K_1,\dots ,K_n$ be a family of pairwise disjoint cliques in the square-free graph $\Gamma$ (or more generally in its extension graph ${\Gamma }^{\mathrm{ext}}$), and let $g_1,\dots ,g_n$ be elements of $A(\Gamma )$ with respective parabolic supports $A(K_1),\dots ,A(K_n)$. Then the subgroup of $A(\Gamma )$ generated by $g_1,\dots ,g_n$ is a right-angled Artin group, where the only relations impose that $g_i$ and $g_j$ commute if $K_i\cup K_j$ is contained in a clique of $\Gamma$ (or of ${\Gamma }^{\mathrm{ext}}$).

arXiv comment38 pages, 1 figure