Divergence, thick groups, and short conjugators

By Jason A. Behrstock, Cornelia Drutu. 2011

78 citations recorded by the source corpus.

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).

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/84eb99328267b44ea43a28a1f3caa1263f410392

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Citations in the corpus

Characterizations of Morse Quasi-Geodesics via Superlinear Divergence and Sublinear Contraction (2016, methodology): “[6] Jason A. Behrstock and Cornelia Druţu, Divergence, thick groups, and short conjugators, Illinois J. Math.”
Divergence in right-angled Coxeter groups (2012, methodology): “he induced map q: C !C with the inclusion C ,!C is actually an isometric embedding of C into C . Remark 4.8. Proposition 4.7 is a special case of the upper bound on divergence given by Theorem 4.9 of [3]. To see this, suppose is CFSand let Hbe the collection of special subgroups of W generated by th”