The automorphism group of a graph product with no SIL

TL;DR

This paper investigates the automorphism groups of graph products of cyclic groups, revealing structural properties and geometric implications, especially when the defining graph lacks SILs and the groups are finite.

Contribution

It demonstrates that automorphisms generated by partial conjugations form a graph product of cyclic groups without SILs, and shows the automorphism group is virtually CAT(0) for finite cyclic groups.

Findings

Automorphism group generated by partial conjugations is a graph product of cyclic groups.

Automorphism group is virtually CAT(0) when cyclic groups are finite.

Automorphism group acts on a right-angled building.

Abstract

We study the automorphisms of graph products of cyclic groups, a class of groups that includes all right-angled Coxeter and right-angled Artin groups. We show that the group of automorphism generated by partial conjugations is itself a graph product of cyclic groups providing its defining graph does not contain any separating intersection of links (SIL). In the case that all the cyclic groups are finite, this implies that the automorphism group is virtually CAT(0); it has a finite index subgroup which acts geometrically on a right-angled building.