Automorphisms of Partially Commutative Groups I: Linear Subgroups

TL;DR

This paper constructs and describes specific arithmetic subgroups within the automorphism group of a partially commutative group, focusing on their structure and decomposition related to the underlying graph.

Contribution

It introduces a method to construct arithmetic subgroups of automorphisms for any finite graph and describes their decomposition as a semidirect product.

Findings

Construction of arithmetic subgroups as subgroups of GL(n,Z)

Description of automorphism group decomposition

Relation to previous theoretical results

Abstract

The goal of this paper is to construct and describe certain arithmetic subgroups of the automorphism group of a partially commutative group. More precisely, given an arbitrary finite graph $\Gamma$ we construct an arithmetic subgroup $St(L(G))$, represented as a subgroup of $GL(n,Z)$, where $n$ is the number of vertices of the graph $\Gamma$. In the last section of the paper we give a description of the decomposition of the group of automorphisms $St^{conj}(L(G))$ as a semidirect product of the group of conjugating automorphisms $Conj(G)$ and $St(L(G))$. This result is closely related to Theorem 1.4 of the paper arXiv:0710.2573v1.