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).
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Andrew J. Duncan, Ilya Kazachkov, Vladimir N. Remeslennikov, Automorphisms of partially commutative groups I: Linear subgroups. Groups Geom. Dyn. 4 (2010), no. 4, pp. 739–757DOI 10.4171/GGD/103