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The stratum of random mapping classes

Published online by Cambridge University Press:  02 May 2017

VAIBHAV GADRE  and
JOSEPH MAHER
VAIBHAV GADRE
Affiliation:
School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, UK email [email protected]
JOSEPH MAHER
Affiliation:
Department of Mathematics, College of Staten Island, CUNY, 2800 Victory Boulevard, Staten Island, NY 10314, USA email [email protected] Department of Mathematics, 4307 Graduate Center, CUNY, 365 5th Avenue, New York, NY 10016, USA
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Abstract

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 Kapovich and Pfaff [Internat. J. Algebra Comput.25, 2015 (5) 745–798].

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 7 , October 2018 , pp. 2666 - 2682
Copyright
© Cambridge University Press, 2017 

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References

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.Google Scholar
Dowdall, S., Duchin, M. and Masur, H.. Statistical hyperbolicity in Teichmüller space. Geom. Funct. Anal. 24(3) (2014), 748–795.Google Scholar
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.Google Scholar
Delp, K., Hoffoss, D. and Manning, J.. Problems in groups, geometry and 3-manifolds. Preprint, 2015, arXiv:1512.04620.Google Scholar
Eskin, A. and Mirzakhani, M.. Counting closed geodesics in moduli space. J. Mod. Dyn. 5(1) (2011), 71–105.Google Scholar
Eskin, A., Mirzakhani, M. and Rafi, K.. Counting geodesics in a stratum. Invent. Math. (2012), to appear.Google Scholar
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.Google Scholar
Gadre, V., Maher, J. and Tiozzo, G.. Word length statistics for Teichmüller geodesics and singularity of harmonic measure. Comment. Math. Helv. to appear.Google Scholar
Horbez, C.. Central limit theorems for mapping class groups and Out( $F_{N}$ ). Preprint, 2015, arXiv:1506.07244 (2015).Google Scholar
Hubbard, J. and Masur, H.. Quadratic differentials and foliations. Acta Math. 142(3–4) (1979), 221–274.Google Scholar
Kaimanovich, Vadim A. and Masur, H.. The Poisson boundary of the mapping class group. Invent. Math. 125(2) (1996), 221–264.Google Scholar
Kapovich, I. and Pfaff, C.. A train track directed random walk on Out(F r ). Internat. J. Algebra Comput. 25(5) (2015), 745–798.Google Scholar
Kerckhoff, S., Masur, H. and Smillie, J.. Ergodicity of billiard flows and quadratic differentials. Ann. of Math. (2) 124(2) (1986), 293–311.Google Scholar
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.Google Scholar
Maher, J.. Random walks on the mapping class group. Duke Math. J. 156(3) (2011), 429–468.Google Scholar
Masur, H.. Uniquely ergodic quadratic differentials. Comment. Math. Helv. 55(2) (1980), 255–266; doi:10.1007/BF02566685.Google Scholar
Masur, H.. Interval exchange transformations and measured foliations. Ann. of Math. (2) 115(1) (1982), 169–200.Google Scholar
Matelski, J. P.. A compactness theorem for Fuchsian groups of the second kind. Duke Math. J. 43(4) (1976), 829–840.Google Scholar
Masur, H. A. and Minsky, Y. N.. Geometry of the complex of curves. I. Hyperbolicity. Invent. Math. 138(1) (1999), 103–149.Google Scholar
Maher, J. and Tiozzo, G.. Random walks on weakly hyperbolic groups. J. Reine Angew. Mathe. (2014), arXiv:1410.4173, to appear.Google Scholar
Rafi, K.. Hyperbolicity in Teichmüller space. Geom. Topol. 18(5) (2014), 3025–3053.Google Scholar
Rivin, I.. Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms. Duke Math. J. 142(2) (2008), 353–379.Google Scholar
Tiozzo, G.. Sublinear deviation between geodesics and sample paths. Duke Math. J. 164(3) (2015), 511–539.Google Scholar
Veech, W. A.. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115(1) (1982), 201–242.Google Scholar
Wright, A.. Translation surfaces and their orbit closures: an introduction for a broad audience. EMS Surv. Math. Sci. 2(1) (2015), 63–108.Google Scholar