Divergence, thick groups, and short conjugators

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).