Divergence of Geodesics in Teichmüller Space and the Mapping Class Group

By Moon Duchin, Kasra Rafi. 2006

51 citations recorded by the source corpus.

We show that both Teichmüller space (with the Teichmüller metric) and the mapping class group (with a word metric) have geodesic divergence that is intermediate between the linear rate of flat spaces and the exponential rate of hyperbolic spaces. For every two geodesic rays in Teichmüller space, we find that their divergence is at most quadratic. Furthermore, this estimate is shown to be sharp via examples of pairs of rays with exactly quadratic divergence. The same statements are true for geodesic rays in the mapping class group. We explicitly describe efficient paths “near infinity” in both spaces.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bc830d3090f2c3833740c5e23797d626da466464

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Divergence and quasimorphisms of right-angled Artin groups (2010, result): “p have super-linear divergence. This also implies that these directions are quasi-geodesically stable, or equivalently, Morse geodesics. [Beh]. (Alternate proofs have since been obtained by [DMS] and [DR]). More recently, Dru¸tu–Mozes–Sapir showed in [DMS] that a group has superlinear divergence if ”
Infinitely presented graphical small cancellation groups are acylindrically hyperbolic (2014, background): “It was first studied in [24] and [23], and in recent years, for example, in [8, 9, 10, 19, 21, 37, 45].”
Divergence in right-angled Coxeter groups (2012, background): “More recently, Duchin–Rafi [13] established that the divergence of Teichmüller space and the mapping class group is quadratic (for mapping class groups this was also obtained by Behrstock in [5]).”
Growth tight actions (2014, background): “All of these spaces are quasi-isometric, and Duchin and Rafi [28] show that pseudo-Anosov elements are contracting for the action of a mapping class group on any one of its Cayley graphs, but we do not know whether one of these actions admits a strongly contacting element.”
Convex cocompactness and stability in mapping class groups (2014, methodology): “The proof of the contracting property uses the following theorem, which can be derived from work of Behrstock [1], though the version we use appears in [7].”