New Garside structures and applications to Artin groups

Garside groups are combinatorial generalizations of braid groups which enjoy many nice algebraic, geometric, and algorithmic properties. In this article, we propose a method for turning the direct product of a group G by Z into a Garside group, under simple assumptions on G. This method gives many new examples of Garside groups, including groups satisfying a certain small cancellation condition (including surface groups) and groups with a systolic presentation. Our method also works for an infinite class of Artin groups, leading to many new group-theoretic, geometric, and topological consequences for them. In particular, we prove new cases of the K(π,1)-conjecture for some hyperbolic type Artin groups.