Quasi-Euclidean tilings over 2-dimensional Artin groups and their applications

Abstract

We describe the structure of quasiflats in two-dimensio\-nal Artin groups. We rely on the notion of metric systolicity developed in our previous work. Using this weak form of non-positive curvature and analyzing in details the combinatorics of tilings of the plane we describe precisely the building blocks for quasiflats in all two-dimensional Artin groups -- atomic sectors. This allows us to provide useful quasi-isometry invariants for such groups -- completions of atomic sectors, stable lines, and the intersection pattern of certain abelian subgroups. These are described combinatorially, in terms of the structure of the graph defining an Artin group. As an important tool, we introduce an analogue of the curve complex in the context of two-dimensional Artin groups -- the intersection graph. We show quasi-isometric invariance of the intersection graph under natural assumptions. As immediate consequences we present a number of results concerning quasi-isometric rigidity for the subclass of CLTTF Artin groups. We give a necessary and sufficient condition for such groups to be strongly rigid (self quasi-isometries are close to automorphisms), we describe quasi-isometry groups, we indicate when quasi-isometries imply isomorphisms for such groups. In particular, there exist many strongly rigid large-type Artin groups. In contrast, none of the right-angled Artin groups are strongly rigid by a previous work of Bestvina, Kleiner and Sageev.