Uniform exponential growth in groups with nonpositive curvature

Thomas Ng (Temple)

Abstract: A finitely generated group is said to have exponential growth if the number of elements that can be spelled with words of bounded length grow exponentially fast. Such a group has uniform exponential growth if a single exponential function bounds the growth for every generating set. Uniform exponential growth is closely related to entropy, characterizing generic elements, and subgroup structure. In 1981, Gromov asked whether exponential growth implies uniform exponential growth for finitely generated groups. While this was shown to be false in general, it has been verified for many natural classes of groups such as hyperbolic groups, linear groups, and the mapping class group of surfaces. In this talk I will explain the role of nonpositive curvature in building uniformly short words that generate free subgroups in several different contexts including relative, hierarchical, and acylidrical hyperbolicity. This is based on joint work with Carolyn Abbott and Davide Spriano.