# Automorphisms of partially commutative groups I: Linear subgroups

### Andrew J. Duncan

University of Newcastle, Great Britain### Ilya Kazachkov

McGill University, Montreal, Canada### Vladimir N. Remeslennikov

Omsk State University, Russian Federation

## Abstract

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

## Cite this article

Andrew J. Duncan, Ilya Kazachkov, Vladimir N. Remeslennikov, Automorphisms of partially commutative groups I: Linear subgroups. Groups Geom. Dyn. 4 (2010), no. 4, pp. 739–757

DOI 10.4171/GGD/103