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Dynamics, Geometry and Groups Seminar

Wednesday, May 10th, 2023

Time: 3:30 p.m.  Place: Jeffery Hall, Room 422

Speaker: Rémi Coulon (Université de Rennes)

Title: Growth problems in negatively curved groups

Abstract: Given a group G acting by isometries on a metric space X, its exponential growth rate provides a way to measure the "size" of its orbits. More precisely it quantifies te asymptotic behavior of the number of orbit points in a ball. When G is the fundamental group of a compact hyperbolic manifold M acting on the universal cover X of M, then this rate has numerous interpretations both of geometric and dynamical nature. In particular it is the entropy of the geodesic flow on the unit tangent bundle of M. In this configuration the study of the geodesic flow on M (and its covers) provides useful informations on the growth rates of G and its subgroups.

In this talk we will present a work in progress whose aim is to exploit similar techniques, while strongly relaxing the hypothesis on the curvature of X. For our purpose, it suffices to assume that the group G contains a "contracting element', which can be thought of as a "hyperbolic direction". Besides fundamental groups of hyperbolic manifolds, this general framework encompasses (relatively) hyperbolic groups, modular groups of surfaces, many right angled Artin groups, etc. As an application we will provide an amenability criterion for the quotients of G.