A generalization of the distortion function and the asymptotic geometry of subgroups

Abstract

This thesis introduces and studies a natural generalization of the distortion function that applies to not necessarily finitely generated subgroups of finitely generated groups. We begin by computing this function in several natural cases, and provides an example of a group with an uncountable collection of incom- parably distorted subgroups.We then show that when we restrict this function to the case of finitely generated subgroups H of finitely generated groups G, the generalized distortion function characterizes when a natural subspace of the asymptotic cone of G corresponding to H is connected. We denote this subspace by Coneω G(H) and show that the ordinary distortion function is not sufficient to detect this subspace’s connectedness. We then study the convexity properties of Coneω G(H). We show that a subgroup H of a finitely generated group G is strongly quasi-convex if and only if Coneω G(H) satisfies a natural convexity property in Coneω (G). G acts on Coneω (G) in a natural way. We show that the stabilizer of Coneω G(G) is the same as the commensurator of H in G whenever H is strongly quasi-convex in G. We conclude by providing several applications of this result to groups with Morse elements.