Automorphisms of graph products of groups from a geometric perspective

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.

arXiv comment36 pages. The article subsumes and vastly generalises our preprint arXiv:1803.07536. To appear in Proc. Lond. Math. Soc

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Cited by in corpus (9)

• Cross ratios and cubulations of hyperbolic groups
• Quasi-isometrically rigid subgroups in right-angled Coxeter groups
• Coarse cubical rigidity
• Outer automorphism groups of graph products: subgroups and quotients
• A note on morphisms to wreath products
• Rigidity for von Neumann algebras of graph product groups. I. Structure of automorphisms
• A note on separability in outer automorphism groups
• The quasi-isometry invariance of the Coset Intersection Complex
• Flat braid groups, right-angled Artin groups, and commensurability