Divergence, thick groups, and short conjugators

TL;DR

This paper investigates the relationship between divergence and thickness in groups, introduces new examples of CAT(0) groups with specific properties, and provides bounds on shortest conjugator lengths in various groups.

Contribution

It constructs novel CAT(0) groups with polynomial divergence and thickness, and offers new bounds on conjugator lengths in groups like 3-manifold and mapping class groups.

Findings

Examples of CAT(0) groups with polynomial divergence of order n+1 and thickness of order n.

Resolution of divergence questions posed by Gromov and Gersten.

Linear and quadratic bounds on shortest conjugator lengths in certain groups.

Abstract

In this paper we explore relationships between divergence and thick groups, and with the same techniques we estimate lengths of shortest conjugators. We produce examples, for every positive integer n, of CAT(0) groups which are thick of order n and with polynomial divergence of order n+1, both these phenomena are new. With respect to thickness, these examples show the non-triviality at each level of the thickness hierarchy defined by Behrstock-Drutu-Mosher. With respect to divergence our examples resolve questions of Gromov and Gersten (the divergence questions were also recently and independently answered by Macura. We also provide general tools for obtaining both lower and upper bounds on the divergence of geodesics and spaces, and we give the definitive lower bound for Morse geodesics in the CAT(0) spaces, generalizing earlier results of Kapovich-Leeb and Bestvina-Fujiwara. In the final section, we turn to the question of bounding the length of the shortest conjugators in several interesting classes of groups. We obtain linear and quadratic bounds on such lengths for classes of groups including 3-manifold groups and mapping class groups (the latter gives new proofs of corresponding results of Masur-Minsky in the pseudo-Anosov case and Tao in the reducible case).