Helly groups

Helly graphs are graphs in which every family of pairwise intersecting balls has a non-empty intersection. This is a classical and widely studied class of graphs. In this article we focus on groups acting geometrically on Helly graphs -- Helly groups. We provide numerous examples of such groups: all (Gromov) hyperbolic, CAT(0) cubical, finitely presented graphical C(4)$-$T(4) small cancellation groups, and type-preserving uniform lattices in Euclidean buildings of type $C_n$ are Helly; free products of Helly groups with amalgamation over finite subgroups, graph products of Helly groups, some diagram products of Helly groups, some right-angled graphs of Helly groups, and quotients of Helly groups by finite normal subgroups are Helly. We show many properties of Helly groups: biautomaticity, existence of finite dimensional models for classifying spaces for proper actions, contractibility of asymptotic cones, existence of EZ-boundaries, satisfiability of the Farrell-Jones conjecture and of the coarse Baum-Connes conjecture. This leads to new results for some classical families of groups (e.g. for FC-type Artin groups) and to a unified approach to results obtained earlier

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