Fractal models for normal subgroups of Schottky groups
HTML articles powered by AMS MathViewer
- by Johannes Jaerisch PDF
- Trans. Amer. Math. Soc. 366 (2014), 5453-5485 Request permission
Abstract:
For a normal subgroup $N$ of the free group $\mathbb {F}_{d}$ with at least two generators, we introduce the radial limit set $\Lambda _{r}(N,\Phi )$ of $N$ with respect to a graph directed Markov system $\Phi$ associated to $\mathbb {F}_{d}$. These sets are shown to provide fractal models of radial limit sets of normal subgroups of Kleinian groups of Schottky type. Our main result states that if $\Phi$ is symmetric and linear, then we have that $\dim _{H}(\Lambda _{r}(N,\Phi ))=\dim _{H}(\Lambda _{r}(\mathbb {F}_d,\Phi ))$ if and only if the quotient group $\mathbb {F}_{d}/N$ is amenable, where $\dim _{H}$ denotes the Hausdorff dimension. This extends a result of Brooks for normal subgroups of Kleinian groups to a large class of fractal sets. Moreover, we show that if $\mathbb {F}_{d}/N$ is non-amenable, then $\dim _{H}(\Lambda _{r}(N,\Phi ))>\dim _{H}(\Lambda _{r}(\mathbb {F}_d,\Phi ))/2$, which extends results by Falk and Stratmann and by Roblin.References
- Jon Aaronson and Manfred Denker, On exact group extensions, Sankhyā Ser. A 62 (2000), no. 3, 339–349. Ergodic theory and harmonic analysis (Mumbai, 1999). MR 1803461
- Jon Aaronson and Manfred Denker, Group extensions of Gibbs-Markov maps, Probab. Theory Related Fields 123 (2002), no. 1, 28–40. MR 1906436, DOI 10.1007/s004400100173
- Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original. MR 1393195
- George M. Bergman, An invitation to general algebra and universal constructions, Henry Helson, Berkeley, CA, 1998. MR 1650275
- Christopher J. Bishop and Peter W. Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997), no. 1, 1–39. MR 1484767, DOI 10.1007/BF02392718
- Alan F. Beardon and Bernard Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974), 1–12. MR 333164, DOI 10.1007/BF02392106
- Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989, DOI 10.1007/BFb0081279
- Robert Brooks, The bottom of the spectrum of a Riemannian covering, J. Reine Angew. Math. 357 (1985), 101–114. MR 783536, DOI 10.1515/crll.1985.357.101
- Petra Bonfert-Taylor, Katsuhiko Matsuzaki, and Edward C. Taylor, Large and small covers of a hyperbolic manifold, J. Geom. Anal. 22 (2012), no. 2, 455–470. MR 2891734, DOI 10.1007/s12220-010-9204-6
- Joel M. Cohen, Cogrowth and amenability of discrete groups, J. Funct. Anal. 48 (1982), no. 3, 301–309. MR 678175, DOI 10.1016/0022-1236(82)90090-8
- M. M. Day, Means on semigroups and groups, Bull. Amer. Math. Soc. 55 (1949), 1054–1055.
- Mahlon Marsh Day, Convolutions, means, and spectra, Illinois J. Math. 8 (1964), 100–111. MR 159230
- J. Dodziuk and W. S. Kendall, Combinatorial Laplacians and isoperimetric inequality, From local times to global geometry, control and physics (Coventry, 1984/85) Pitman Res. Notes Math. Ser., vol. 150, Longman Sci. Tech., Harlow, 1986, pp. 68–74. MR 894523
- Jozef Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284 (1984), no. 2, 787–794. MR 743744, DOI 10.1090/S0002-9947-1984-0743744-X
- Erling Følner, On groups with full Banach mean value, Math. Scand. 3 (1955), 243–254. MR 79220, DOI 10.7146/math.scand.a-10442
- Kurt Falk and Bernd O. Stratmann, Remarks on Hausdorff dimensions for transient limit sets of Kleinian groups, Tohoku Math. J. (2) 56 (2004), no. 4, 571–582. MR 2097162
- Peter Gerl, Random walks on graphs with a strong isoperimetric property, J. Theoret. Probab. 1 (1988), no. 2, 171–187. MR 938257, DOI 10.1007/BF01046933
- Y. Guivarc’h, Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire, Conference on Random Walks (Kleebach, 1979) Astérisque, vol. 74, Soc. Math. France, Paris, 1980, pp. 47–98, 3 (French, with English summary). MR 588157
- Johannes Jaerisch, Thermodynamic formalism for group-extended Markov systems with applications to Fuchsian groups, Doctoral dissertation at the University Bremen (2011).
- Johannes Jaerisch, A lower bound for the exponent of convergence of normal subgroups of Kleinian groups, J. Geom. Anal., online first (2013).
- Johannes Jaerisch, Group-extended Markov systems, amenability, and the Perron-Frobenius operator, Proc. Amer. Math. Soc. (2014) to appear.
- Johannes Jaerisch, Recurrence and pressure for group extensions, Ergodic Theory and Dynamical Systems (2014) to appear.
- Johannes Jaerisch, Conformal fractals for normal subgroups of free groups, Conform. Geom. Dyn. 18 (2014), 31–55. MR 3175016, DOI 10.1090/S1088-4173-2014-00263-X
- Johannes Jaerisch, Marc Kesseböhmer, and Sanaz Lamei, Induced topological pressure for countable state Markov shifts, Stoch. Dyn. 14 (2014), no. 2, 1350016, 31. MR 3190211, DOI 10.1142/S0219493713500160
- Vadim A. Kaimanovich, Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators, Potential Anal. 1 (1992), no. 1, 61–82. MR 1245225, DOI 10.1007/BF00249786
- Harry Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156. MR 112053, DOI 10.7146/math.scand.a-10568
- Harry Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336–354. MR 109367, DOI 10.1090/S0002-9947-1959-0109367-6
- B. O. Koopman, Hamiltonian systems and transformation in Hilbert space, Proceedings of the National Academy of Sciences of the United States of America, vol. 17, 1931.
- Andrzej Lasota and Michael C. Mackey, Chaos, fractals, and noise, 2nd ed., Applied Mathematical Sciences, vol. 97, Springer-Verlag, New York, 1994. Stochastic aspects of dynamics. MR 1244104, DOI 10.1007/978-1-4612-4286-4
- Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
- Bojan Mohar, Isoperimetric inequalities, growth, and the spectrum of graphs, Linear Algebra Appl. 103 (1988), 119–131. MR 943998, DOI 10.1016/0024-3795(88)90224-8
- Katsuhiko Matsuzaki and Masahiko Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. Oxford Science Publications. MR 1638795
- R. Daniel Mauldin and Mariusz Urbański, Graph directed Markov systems, Cambridge Tracts in Mathematics, vol. 148, Cambridge University Press, Cambridge, 2003. Geometry and dynamics of limit sets. MR 2003772, DOI 10.1017/CBO9780511543050
- J. v. Neumann, Zur allgemeinen Theorie des Masses, Fund. Math. 13 (1929), 73–116.
- Peter J. Nicholls, The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, vol. 143, Cambridge University Press, Cambridge, 1989. MR 1041575, DOI 10.1017/CBO9780511600678
- Ronald Ortner and Wolfgang Woess, Non-backtracking random walks and cogrowth of graphs, Canad. J. Math. 59 (2007), no. 4, 828–844. MR 2338235, DOI 10.4153/CJM-2007-035-1
- Georg Pólya, Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz, Math. Ann. 84 (1921), no. 1-2, 149–160 (German). MR 1512028, DOI 10.1007/BF01458701
- John G. Ratcliffe, Foundations of hyperbolic manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 149, Springer, New York, 2006. MR 2249478
- Thomas Roblin, Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative, Israel J. Math. 147 (2005), 333–357 (French, with French summary). MR 2166367, DOI 10.1007/BF02785371
- Mario Roy and Mariusz Urbański, Real analyticity of Hausdorff dimension for higher dimensional hyperbolic graph directed Markov systems, Math. Z. 260 (2008), no. 1, 153–175. MR 2413348, DOI 10.1007/s00209-007-0267-4
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
- David Ruelle, Statistical mechanics: Rigorous results, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0289084
- E. Seneta, Non-negative matrices and Markov chains, Springer Series in Statistics, Springer, New York, 2006. Revised reprint of the second (1981) edition [Springer-Verlag, New York; MR0719544]. MR 2209438
- Richard Sharp, Critical exponents for groups of isometries, Geom. Dedicata 125 (2007), 63–74. MR 2322540, DOI 10.1007/s10711-007-9137-9
- Manuel Stadlbauer, An extension of Kesten’s criterion for amenability to topological Markov chains, Adv. Math. 235 (2013), 450–468. MR 3010065, DOI 10.1016/j.aim.2012.12.004
- Bernd O. Stratmann, The exponent of convergence of Kleinian groups; on a theorem of Bishop and Jones, Fractal geometry and stochastics III, Progr. Probab., vol. 57, Birkhäuser, Basel, 2004, pp. 93–107. MR 2087134
- Bernd O. Stratmann, Fractal geometry on hyperbolic manifolds, Non-Euclidean geometries, Math. Appl. (N. Y.), vol. 581, Springer, New York, 2006, pp. 227–247. MR 2191250, DOI 10.1007/0-387-29555-0_{1}2
- D. Vere-Jones, Geometric ergodicity in denumerable Markov chains, Quart. J. Math. Oxford Ser. (2) 13 (1962), 7–28. MR 141160, DOI 10.1093/qmath/13.1.7
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
- Wolfgang Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, vol. 138, Cambridge University Press, Cambridge, 2000. MR 1743100, DOI 10.1017/CBO9780511470967
Additional Information
- Johannes Jaerisch
- Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043 Japan
- Email: jaerisch@cr.math.sci.osaka-u.ac.jp
- Received by editor(s): December 15, 2012
- Published electronically: May 20, 2014
- Additional Notes: The author was supported by the research fellowship JA 2145/1-1 of the German Research Foundation (DFG)
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 5453-5485
- MSC (2010): Primary 37C45, 30F40; Secondary 37C85, 43A07
- DOI: https://doi.org/10.1090/S0002-9947-2014-06095-9
- MathSciNet review: 3240930