Automorphisms of partially commutative groups I: Linear subgroups

Abstract

We construct and describe several arithmetic subgroups of the automorphism group of a partially commutative group. More precisely, given an arbitrary finite graph $\Gamma$ we construct arithmetic subgroups $\mathrm{St}(L_Y)$ and $\mathrm{St}(L^{\max })$, represented as subgroups of $\mathrm{GL}(n,Z)$, where $n$ is the number of vertices of the graph $\Gamma$. Here $L_Y$ and $L^{\max }$ are certain lattices of subsets of $X=V(\Gamma )$ and $\mathrm{St}(K)$ is the stabiliser of the subgroup generated by $K$. In addition we give a description of the decomposition of the group ${\mathrm{St}}^{\mathrm{conj}}(L_X)$, which stabilises $L_X$ up to conjugacy, as a semidirect product of the group of conjugating automorphisms and $\mathrm{St}(L_X)$.