Automorphisms of partially commutative groups. I: Linear subgroups. (Q531910)

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