Divergence, thick groups, and short conjugators

Abstract

The notion of thickness, introduced in (Math. Ann. 344 (2009) 543–595), is one of the first tools developed to study the quasi-isometric behavior of weakly relatively hyperbolic groups. In this paper, we further this exploration through a relationship between thickness and the divergence of geodesics. We construct examples, for every positive integer $n$, of $\operatorname{CAT}(0)$ groups which are thick of order $n$ and with polynomial divergence of order $n+1$. With respect to thickness, these examples show the non-triviality at each level of the thickness hierarchy defined in (Math. Ann. 344 (2009) 543–595). With respect to divergence, our examples provide an answer to questions of Gromov (In Geometric Group Theory (1993) 1–295 Cambridge Univ. Press) and Gersten (Geom. Funct. Anal. 4 (1994) 633–647; Geom. Funct. Anal. 4 (1994) 37–51). The divergence questions were independently answered by Macura in ($\operatorname{CAT}(0)$ spaces with polynomial divergence of geodesics (2011) Preprint). We also provide tools for obtaining both lower and upper bounds on the divergence of geodesics and spaces, and we prove an effective quadratic lower bound for Morse quasi-geodesics in $\operatorname{CAT}(0)$ spaces, generalizing results of Kapovich–Leeb and Bestvina–Fujiwara (Geom. Funct. Anal. 8 (1998) 841–852; Geom. Funct. Anal. 19 (2009) 11–40). In the final section, we obtain linear and quadratic bounds on the length of the shortest conjugators for various families of groups. For general $3$-manifold groups, sharp estimates are provided. We also consider mapping class groups, where we provide a new streamlined proof of the length of shortest conjugators which contains the corresponding results of Masur–Minsky in the pseudo-Anosov case (Geom. Funct. Anal. 10 (2000) 902–974) and Tao in the reducible case (Geom. Funct. Anal. 23 (2013) 415–466).