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**Convexity of strata in diagonal pants graphs of surfaces.**
*(English)*
Zbl 1288.57015

The pants graph \({\mathcal P}(S)\) of a connected orientable surface \(S\) with empty boundary and negative Euler characteristic is the graph whose vertices correspond to homotopy classes of pants decompositions of the surface \(S\), and the edges are defined by an elementary move. By a result of J. F. Brock [J. Am. Math. Soc. 16, No. 3, 495–535 (2003; Zbl 1059.30036)] the pants graph \({\mathcal P}(S)\) is quasi-isometric to the Teichmüller space \({\mathcal T}(S)\) of \(S\) equipped with the Weil-Petersson metric, so it serves as a combinatorial model for Teichmüller space.

Teichmüller space is not complete; its completion (by surfaces with nodes) admits a natural stratification whose strata correspond to multicurves on the surface and are known to be convex subsets of the completion. The pants graph admits an analogous stratification corresponding to multicurves (considering pants containing a given multicurve). Certain families of strata in \({\mathcal P}(S)\) are known to be totally geodesic, and conjecturally this is the case for all strata.

In the present paper, the diagonal pants graph \({\mathcal DP}(S)\) and the cubical pants graph \({\mathcal CP}(S)\) are studied which are obtained by adding an edge of length 1 resp. \(\sqrt {k}\) between any two pants decompositions that differ by \(k\) disjoint elementary moves; both are quasi-isometric to \({\mathcal P}(S)\) and hence to \({\mathcal T}(S)\). The main results of the paper state then that, if \(S\) is a sphere with punctures, the stratum defined by any multicurve on \(S\) is convex in \({\mathcal DP}(S)\) (but in general not totally geodesic), and that for any multicurve of deficiency 1 (one curve less than a pants decomposition) on a general surface \(S\) the associated stratum is convex in \({\mathcal DP}(S)\) and totally geodesic in \({\mathcal CP}(S)\). “As a consequence, we exhibit convex flat subgraphs of every possible rank inside the diagonal pants graph.”

Teichmüller space is not complete; its completion (by surfaces with nodes) admits a natural stratification whose strata correspond to multicurves on the surface and are known to be convex subsets of the completion. The pants graph admits an analogous stratification corresponding to multicurves (considering pants containing a given multicurve). Certain families of strata in \({\mathcal P}(S)\) are known to be totally geodesic, and conjecturally this is the case for all strata.

In the present paper, the diagonal pants graph \({\mathcal DP}(S)\) and the cubical pants graph \({\mathcal CP}(S)\) are studied which are obtained by adding an edge of length 1 resp. \(\sqrt {k}\) between any two pants decompositions that differ by \(k\) disjoint elementary moves; both are quasi-isometric to \({\mathcal P}(S)\) and hence to \({\mathcal T}(S)\). The main results of the paper state then that, if \(S\) is a sphere with punctures, the stratum defined by any multicurve on \(S\) is convex in \({\mathcal DP}(S)\) (but in general not totally geodesic), and that for any multicurve of deficiency 1 (one curve less than a pants decomposition) on a general surface \(S\) the associated stratum is convex in \({\mathcal DP}(S)\) and totally geodesic in \({\mathcal CP}(S)\). “As a consequence, we exhibit convex flat subgraphs of every possible rank inside the diagonal pants graph.”

Reviewer: Bruno Zimmermann (Trieste)

### MSC:

57M50 | General geometric structures on low-dimensional manifolds |

30F60 | Teichmüller theory for Riemann surfaces |