Markov chains on hyperbolic-like groups and quasi-isometries

We propose the study of Markov chains on groups as a “quasi-isometry invariant” theory that encompasses random walks. In particular, we focus on certain classes of groups acting on hyperbolic spaces including (non-elementary) hyperbolic and relatively hyperbolic groups, acylindrically hyperbolic 3-manifold groups, as well as fundamental groups of certain graphs of groups with edge groups of subexponential growth. For those, we prove a linear progress result and various applications, and these lead to a Central Limit Theorem for random walks on groups quasi-isometric to the ones we consider.

[1]  Combination of convergence groups , 2002, math/0203258.

[2]  D. Vere-Jones.  Markov Chains , 1972, Nature.

[3]  P. Couturier.  Japan , 1988, The Lancet.

[4]  G. Tiozzo,et al.  Random walks on weakly hyperbolic groups , 2014, Journal für die reine und angewandte Mathematik (Crelles Journal).

[5]  Denis V. Osin.  Elementary Subgroups of Relatively Hyperbolic Groups and Bounded Generation , 2006, Int. J. Algebra Comput..

[6]  Takuya Kon-no,et al.  Transactions of the American Mathematical Society , 1996 .

[7]  8. , 2020, When Bad Things Happen to Rich People.

[8]  D. Osin,et al.  Correction to: Acylindrical hyperbolicity of groups acting on trees , 2018, Mathematische Annalen.

[9]  M. Sapir,et al.  Divergence in lattices in semisimple Lie groups and graphs of groups , 2008, 0801.4141.

[10]  D. V. Osin.  Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems , 2004 .

[11]  Kevin Whyte,et al.  Amenability, bilipschitz equivalence, and the von Neumann conjecture , 1999 .

[12]  A. Sisto,et al.  Random subgroups of acylindrically hyperbolic groups and hyperbolic embeddings , 2017, 1701.00253.

[13]  P. Mathieu,et al.  Large deviations for random walks on hyperbolic spaces , 2020 .

[14]  Jian Ding,et al.  Sensitivity of mixing times , 2013 .

[15]  Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth , 1991 .

[16]  Geometriae Dedicata,et al.  Geometriae Dedicata , 2003 .

[17]  Samuel J. Taylor,et al.  Counting loxodromics for hyperbolic actions , 2016, 1605.02103.

[18]  C. Pittet,et al.  On the stability of the behavior of random walks on groups , 2000 .

[19]  STAT , 2019, Springer Reference Medizin.

[20]  Combable functions, quasimorphisms, and the central limit theorem , 2008, Ergodic Theory and Dynamical Systems.

[21]  B. M. Fulk.  MATH , 1992 .

[22]  D. Osin.  Acylindrically hyperbolic groups , 2013, 1304.1246.

[23]  K. Fujiwara,et al.  Constructing group actions on quasi-trees and applications to mapping class groups , 2010, 1006.1939.

[24]  Mahlon M. Day.  Convolutions, means, and spectra , 1964 .

[25]  Ilya Kapovich,et al.  On hyperbolicity of free splitting and free factor complexes , 2012, 1206.3626.

[26]  P. Zalesskii,et al.  Profinite properties of graph manifolds , 2008, 0807.3727.

[27]  G. Christopher Hruska,et al.  Relative hyperbolicity and relative quasiconvexity for countable groups , 2008, 0801.4596.

[28]  Samuel J. Taylor,et al.  Largest projections for random walks and shortest curves in random mapping tori , 2016, Mathematical Research Letters.

[29]  Bruce Kleiner,et al.  Notes on Perelman's papers , 2006 .

[30]  G. Perelman.  Ricci flow with surgery on three-manifolds , 2003, math/0303109.

[31]  M. Yamasaki,et al.  Parabolic index and rough isometries , 1993 .

[32]  Y. Peres,et al.  On sensitivity of mixing times and cutoff , 2016, 1610.04357.

[33]  Jason A. Behrstock.  Asymptotic Geometry of the Mapping Class Group and Teichmuller Space , 2005, math/0502367.

[34]  A. Sisto.  Contracting elements and random walks , 2011, Journal für die reine und angewandte Mathematik (Crelles Journal).

[35]  Jack W Drone.  Means. , 2013, Journal.

[36]  P. Scott,et al.  The geometries of 3-manifolds , 1983 .

[37]  Carolyn R. Abbott,et al.  Random walks and quasi‐convexity in acylindrically hyperbolic groups , 2019, Journal of Topology.

[38]  A. Sisto.  Tracking rates of random walks , 2013, 1305.5472.

[39]  D. Osin,et al.  Acylindrical hyperbolicity of groups acting on trees , 2013, Mathematische Annalen.

[40]  Matt Sunderland.  Linear progress with exponential decay in weakly hyperbolic groups , 2017, 1710.05107.

[41]  A. Minasyan.  SOME PROPERTIES OF SUBSETS OF HYPERBOLIC GROUPS , 2004, math/0404473.

[42]  M. Kapovich,et al.  3-manifold Groups and Nonpositive Curvature , 1998 .

[43]  Jason A. Behrstock,et al.  Divergence, thick groups, and short conjugators , 2011, 1110.5005.

[44]  Mathematical Research Letters , 2022 .

[45]  S. Gersten.  Divergence in 3-manifold groups , 1994 .

[46]  Harry Kesten,et al.  Symmetric random walks on groups , 1959 .

[47]  M. Kanai.  Rough isometries and the parabolicity of riemannian manifolds , 1986 .

[48]  S. Schleimer,et al.  Quotients of the curve complex , 2020, Groups, Geometry, and Dynamics.

[49]  Laurent Saloff-Coste,et al.  Variétés riemanniennes isométriques à l'infini , 1995 .

[50]  P. Mathieu,et al.  Deviation inequalities for random walks , 2014 .

[51]  K. Fujiwara,et al.  Acylindrical actions on projection complexes , 2017, L’Enseignement Mathématique.

[52]  Huai-Dong Cao,et al.  A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow , 2006 .

[53]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[54]  Divergence of Geodesics in Teichmüller Space and the Mapping Class Group , 2006, math/0611359.

[55]  Matthew I. Roberts,et al.  Mixing Time Bounds via Bottleneck Sequences , 2016, 1610.07874.

[56]  G. Perelman.  The entropy formula for the Ricci flow and its geometric applications , 2002, math/0211159.

[57]  J. Morgan,et al.  Ricci Flow and the Poincare Conjecture , 2006, math/0607607.

[58]  R. Tennant.  Algebra , 1941, Nature.

[59]  D. Osin,et al.  Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces , 2011, 1111.7048.

[60]  Susan Hermiller,et al.  Algorithms and Geometry for Graph Products of Groups , 1995 .

[61]  藤原 耕二.  Bounded cohomology of subgroups of mapping class groups (双曲空間及び離散群の研究 研究集会報告集) , 2001 .

[62]  M. Bridson,et al.  Metric Spaces of Non-Positive Curvature , 1999 .

[63]  S'ebastien Gouezel.  Exponential bounds for random walks on hyperbolic spaces without moment conditions , 2021 .