Abdul Zalloum (Queen's University)

Title: Globally stable cylinders and fine structures for hyperbolic groups.

Abstract: A hyperbolic group G is said to have globally stable cylinders if there exist integers K,m and a collection of equivariant (1,K)-quasi-geodesics connecting every pair of points in G such that the following holds. For any x,y,z in G; the (aforementioned) quasi-geodesics connecting them form a tripod, up to removing at most m balls of radius K centered along such quasi-geodesics. In short, the property asks for the existence of a bicombing on G satisfying some very strong properties making the group look as close to a tree as possible. In 1995, Rips and Sela asked if torsion-free hyperbolic groups have this property, and in 2022 Sageev and Lazarovich established it for cubulated hyperbolic groups (hyperbolic groups with a geometric action on a CAT(0) cube complex). I will discuss recent work with Petyt and Spriano showing that residually finite hyperbolic groups admit globally stable cylinders.