Cubical-like geometry of quasi-median graphs and applications to geometric group theory

Abstract

The class of quasi-median graphs is a generalisation of median graphs, or equivalently of CAT(0) cube complexes. The purpose of this thesis is to introduce these graphs in geometric group theory. In the first part of our work, we extend the definition of hyperplanes from CAT(0) cube complexes, and we show that the geometry of a quasi-median graph essentially reduces to the combinatorics of its hyperplanes. In the second part, we exploit the specific structure of the hyperplanes to state combination results. The main idea is that if a group acts in a suitable way on a quasi-median graph so that clique-stabilisers satisfy some non-positively curved property $\mathcal{P}$, then the whole group must satisfy $\mathcal{P}$ as well. The properties we are interested in are mainly (relative) hyperbolicity, (equivariant) $\ell^p$-compressions, CAT(0)-ness and cubicality. In the third part, we apply our general criteria to several classes of groups, including graph products, Guba and Sapir's diagram products, some wreath products, and some graphs of groups. Graph products are our most natural examples, where the link between the group and its quasi-median graph is particularly strong and explicit; in particular, we are able to determine precisely when a graph product is relatively hyperbolic.