Automorphisms of graph products of groups from a geometric perspective

TL;DR

This paper characterizes automorphisms of graph products of groups using geometric methods, computes their automorphism groups in specific cases, and explores their properties like acylindrical hyperbolicity.

Contribution

It provides a complete geometric description of automorphisms preserving conjugacy classes and computes automorphism groups for certain graph products, revealing their geometric and algebraic properties.

Findings

Automorphisms preserving conjugacy classes decompose into elementary automorphisms.

Automorphism groups of certain graph products are acylindrically hyperbolic.

These groups do not satisfy Kazhdan's property (T).

Abstract

This article studies automorphism groups of graph products of arbitrary groups. We completely characterise automorphisms that preserve the set of conjugacy classes of vertex groups as those automorphisms that can be decomposed as a product of certain elementary automorphisms (inner automorphisms, partial conjugations, automorphisms associated to symmetries of the underlying graph). This allows us to completely compute the automorphism group of certain graph products, for instance in the case where the underlying graph is finite, connected, leafless and of girth at least $5$. If in addition the underlying graph does not contain separating stars, we can understand the geometry of the automorphism groups of such graph products of groups further: we show that such automorphism groups do not satisfy Kazhdan's property (T) and are acylindrically hyperbolic. Applications to automorphism groups of graph products of finite groups are also included. The approach in this article is geometric and relies on the action of graph products of groups on certain complexes with a particularly rich combinatorial geometry. The first such complex is a particular Cayley graph of the graph product that has a quasi-median geometry, a combinatorial geometry reminiscent of (but more general than) CAT(0) cube complexes. The second (strongly related) complex used is the Davis complex of the graph product, a CAT(0) cube complex that also has a structure of right-angled building.