Large-scale rank and rigidity of the Weil–Petersson metric

Abstract

Please refer to published version for the most recent bibliographic citation information. If a published version is known of, the repository item page linked to above, will contain details on accessing it.

The (non)-relative hyperbolicity of the separating curve graph

“…For a connected, compact, orientable surface S " S g,b with genus g and b boundary components, the separating curve graph, SeppSq, is the metric graph whose vertices are all isotopy classes of separating curves, with edges of length 1 corresponding to disjointness. The separating curve graph arises naturally in the study of the Johnson kernel of the mapping class group [BM04,Kid13], the coarse geometry of the Weil-Petersson metric on Teichmüller space [BM08,Bow15,Sul15], and the algebraic topology of the moduli space of a surface [Loo13].…”

Notes on coarse median spaces

“…While the delails are (significantly) different, related arguments can be made to work for quasi-isometries of T and W, giving the rigidity results for these spaces [Bo5,Bo7]. We remark that the rigidity of T is independently proven in [EMR2] using quite different arguments of coarse differentiation.…”

“…So are limit groups, as defined by Sela. (5) Mapping class groups, Teichmüller space in either the Teichmüller metric or the Weil-Petersson metric [Bo6,Bo5,Bo7], and the separating curve graphs, [Vo]. (See Section 5).…”

“…It is a relatively simple matter to account for the low complexity cases (ξ ≤ 1), and so this give a compete answer for M, T and C. However, [Bo7] leaves unresolved about a dozen cases for W.…”

“…Theorem 5.6 (Bowditch). [Bo6,Bo5,Bo7]. M and T are coarse median of rank ξ, W is coarse median of rank ξ 0 and C is coarse median of rank 1.…”