Automorphisms of graph products of groups from a geometric perspective

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.Comment: 36 pages. The article subsumes and vastly generalises our preprint arXiv:1803.07536. To appear in Proc. Lond. Math. So

Coarse-median preserving automorphisms

This paper has three main goals. First, we study fixed subgroups of automorphisms of right-angled Artin and Coxeter groups. If $\varphi$ is an untwisted automorphism of a RAAG, or an arbitrary automorphism of a RACG, we prove that ${\rm Fix}~\varphi$ is finitely generated and undistorted. Up to replacing $\varphi$ with a power, we show that ${\rm Fix}~\varphi$ is quasi-convex with respect to the standard word metric. This implies that ${\rm Fix}~\varphi$ is separable and a special group in the sense of Haglund-Wise. By contrast, there exist "twisted" automorphisms of RAAGs for which ${\rm Fix}~\varphi$ is undistorted but not of type $F$ (hence not special), of type $F$ but distorted, or even infinitely generated. Secondly, we introduce the notion of "coarse-median preserving" automorphism of a coarse median group, which plays a key role in the above results. We show that automorphisms of RAAGs are coarse-median preserving if and only if they are untwisted. On the other hand, all automorphisms of Gromov-hyperbolic groups and right-angled Coxeter groups are coarse-median preserving. These facts also yield new or more elementary proofs of Nielsen realisation for RAAGs and RACGs. Finally, we show that, for every special group $G$ (in the sense of Haglund-Wise), every infinite-order, coarse-median preserving outer automorphism of $G$ can be realised as a homothety of a finite-rank median space $X$ equipped with a "moderate" isometric $G$-action. This generalises the classical result, due to Paulin, that every infinite-order outer automorphism of a hyperbolic group $H$ projectively stabilises a small $H$-tree.Comment: 70 pages, 5 figures; v3: added application to Nielsen realisation (Corollaries F and G) and reference