Automorphisms of Partially Commutative Groups I: Linear Subgroups

Duncan, Andrew J.; Kazachkov, Ilya V.; Remeslennikov, Vladimir N.

Automorphisms of Partially Commutative Groups I: Linear Subgroups

Authors

Duncan, Andrew J.

Kazachkov, Ilya V.

Remeslennikov, Vladimir N.

Description

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 $Γ$ 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 $Γ$. 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.
24 pages, 1 figure