Groups, Geometry, and Dynamics

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Volume 4, Issue 4, 2010, pp. 739–757
DOI: 10.4171/GGD/103

Published online: 2010-10-15

Automorphisms of partially commutative groups I: Linear subgroups

Andrew J. Duncan[1], Ilya V. Kazachkov[2] and Vladimir N. Remeslennikov[3]

(1) University of Newcastle, Great Britain
(2) McGill University, Montreal, Canada
(3) Omsk State University, Russian Federation

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(LY) and St(Lmax), represented as subgroups of GL(n,ℤ), where n is the number of vertices of the graph Γ. Here LY and Lmax 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 Stconj(LX), which stabilises LX up to conjugacy, as a semidirect product of the group of conjugating automorphisms and St(LX).

Keywords: Partially commutative groups, right-angled Artin groups, automorphism groups, arithmetic groups

Duncan Andrew, Kazachkov Ilya, Remeslennikov Vladimir: Automorphisms of partially commutative groups I: Linear subgroups. Groups Geom. Dyn. 4 (2010), 739-757. doi: 10.4171/GGD/103