Automorphisms of Partially Commutative Groups II: Combinatorial Subgroups

TL;DR

This paper explores the structure of automorphism groups of partially commutative groups by defining standard subgroups and providing decomposition theorems based on the properties of the commutation graph.

Contribution

It introduces new decompositions of Aut(G) using standard subgroups and characterizes graphs where Aut(G) splits into a product of subgroups.

Findings

Decomposition of Aut(G) based on connected components of the commutation graph

Semi-direct decomposition when the graph has no dominated vertices

Characterization of graphs with Aut(G) as a direct product

Abstract

We define several "standard" subgroups of the automorphism group Aut(G) of a partially commutative (right-angled Artin) group and use these standard subgroups to describe decompositions of Aut(G). If C is the commutation graph of G, we show how Aut(G) decomposes in terms of the connected components of C: obtaining a particularly clear decomposition theorem in the special case where C has no isolated vertices. If C has no vertices of a type we call dominated then we give a semi-direct decompostion of Aut(G) into a subgroup of locally conjugating automorphisms by the subgroup stabilising a certain lattice of "admissible subsets" of the vertices of C. We then characterise those graphs for which Aut(G) is a product (not necessarily semi-direct) of two such subgroups.