# zbMATH — the first resource for mathematics

The stratum of random mapping classes. (English) Zbl 1409.37040
Summary: We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum. For such random walks, we show that mapping classes along almost every infinite sample path are eventually pseudo-Anosov, with invariant Teichmüller geodesics in the principal stratum. This provides an answer to a question of I. Kapovich and C. Pfaff [Int. J. Algebra Comput. 25, No. 5, 745–798 (2015; Zbl 1351.20025)].

##### MSC:
 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 37H10 Generation, random and stochastic difference and differential equations 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 05C81 Random walks on graphs 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
Full Text:
##### References:
 [1] Athreya, J., Bufetov, A., Eskin, A. and Mirzakhani, M.. Lattice point asymptotics and volume growth on Teichmüller space. Duke Math. J.161(6) (2012), 1055-1111. doi:10.1215/00127094-1548443 · Zbl 1246.37009 [2] Dowdall, S., Duchin, M. and Masur, H.. Statistical hyperbolicity in Teichmüller space. Geom. Funct. Anal.24(3) (2014), 748-795. doi:10.1007/s00039-014-0265-8 · Zbl 1302.30055 [3] Dahmani, F. and Horbez, C.. Spectral theorems for random walks on mapping class groups and $$\text{Out}(F_{N})$$. Preprint, 2015, arXiv:1506.06790 2015. · Zbl 1407.37046 [4] Delp, K., Hoffoss, D. and Manning, J.. Problems in groups, geometry and 3-manifolds. Preprint, 2015, arXiv:1512.04620. [5] Eskin, A. and Mirzakhani, M.. Counting closed geodesics in moduli space. J. Mod. Dyn.5(1) (2011), 71-105. doi:10.3934/jmd.2011.5.71 · Zbl 1219.37006 [6] Eskin, A., Mirzakhani, M. and Rafi, K.. Counting geodesics in a stratum. Invent. Math. (2012), to appear. · Zbl 1411.37033 [7] Gadre, V.. Harmonic measures for distributions with finite support on the mapping class group are singular. Duke Math. J.163(2) (2014), 309-368; doi:10.1215/00127094-2430368. doi:10.1215/00127094-2430368 · Zbl 1285.30025 [8] Gadre, V., Maher, J. and Tiozzo, G.. Word length statistics for Teichmüller geodesics and singularity of harmonic measure. Comment. Math. Helv. to appear. · Zbl 1362.32009 [9] Horbez, C.. Central limit theorems for mapping class groups and Out($$F_{N}$$). Preprint, 2015, arXiv:1506.07244 (2015). [10] Hubbard, J. and Masur, H.. Quadratic differentials and foliations. Acta Math.142(3-4) (1979), 221-274. doi:10.1007/BF02395062 · Zbl 0415.30038 [11] Kaimanovich, Vadim A. and Masur, H.. The Poisson boundary of the mapping class group. Invent. Math.125(2) (1996), 221-264. doi:10.1007/s002220050074 · Zbl 0864.57014 [12] Kapovich, I. and Pfaff, C.. A train track directed random walk on Out(F_{r}). Internat. J. Algebra Comput.25(5) (2015), 745-798. doi:10.1142/S0218196715500186 · Zbl 1351.20025 [13] Kerckhoff, S., Masur, H. and Smillie, J.. Ergodicity of billiard flows and quadratic differentials. Ann. of Math. (2)124(2) (1986), 293-311. doi:10.2307/1971280 · Zbl 0637.58010 [14] Klarreich, E.. The boundary at infinity of the curve complex and the relative Teichmüller space. Available at: https://pressfolios-production.s3.amazonaws.com/uploads/story/story_pdf/145710/1457101434403642.pdf. [15] Maher, J., Random walks on the mapping class group, Duke Math. J., 156, 3, 429-468, (2011) · Zbl 1213.37072 [16] Masur, H.. Uniquely ergodic quadratic differentials. Comment. Math. Helv.55(2) (1980), 255-266; doi:10.1007/BF02566685. doi:10.1007/BF02566685 · Zbl 0436.30034 [17] Masur, H., Interval exchange transformations and measured foliations, Ann. of Math. (2), 115, 1, 169-200, (1982) · Zbl 0497.28012 [18] Matelski, J. P., A compactness theorem for Fuchsian groups of the second kind, Duke Math. J., 43, 4, 829-840, (1976) · Zbl 0341.30020 [19] Masur, H. A. and Minsky, Y. N.. Geometry of the complex of curves. I. Hyperbolicity. Invent. Math.138(1) (1999), 103-149. doi:10.1007/s002220050343 · Zbl 0941.32012 [20] Maher, J. and Tiozzo, G.. Random walks on weakly hyperbolic groups. J. Reine Angew. Mathe. (2014), arXiv:1410.4173, to appear. · Zbl 1434.60015 [21] Rafi, K., Hyperbolicity in Teichmüller space, Geom. Topol., 18, 5, 3025-3053, (2014) · Zbl 1314.30082 [22] Rivin, I., Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms, Duke Math. J., 142, 2, 353-379, (2008) · Zbl 1207.20068 [23] Tiozzo, G., Sublinear deviation between geodesics and sample paths, Duke Math. J., 164, 3, 511-539, (2015) · Zbl 1314.30085 [24] Veech, W. A., Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115, 1, 201-242, (1982) · Zbl 0486.28014 [25] Wright, A., Translation surfaces and their orbit closures: an introduction for a broad audience, EMS Surv. Math. Sci., 2, 1, 63-108, (2015) · Zbl 1372.37090
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.