AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Hyperbolic groupoids and duality
About this Title
Volodymyr V. Nekrashevych, Department of Mathematics, Texas A & M University, College Station, Texas 77843-3368
Publication: Memoirs of the American Mathematical Society
Publication Year:
2015; Volume 237, Number 1122
ISBNs: 978-1-4704-1544-0 (print); 978-1-4704-2511-1 (online)
DOI: https://doi.org/10.1090/memo/1122
Published electronically: February 16, 2015
Keywords: Hyperbolic groupoids,
Smale spaces,
Smale quasi-flows,
Gromov hyperbolic graphs
MSC: Primary 37D20, 20L05; Secondary 20F67
Table of Contents
Chapters
- Introduction
- 1. Technical preliminaries
- 2. Preliminaries on groupoids and pseudogroups
- 3. Hyperbolic groupoids
- 4. Smale quasi-flows and duality
- 5. Examples of hyperbolic groupoids and their duals
Abstract
We introduce a notion of hyperbolic groupoids, generalizing the notion of a Gromov hyperbolic group. Examples of hyperbolic groupoids include actions of Gromov hyperbolic groups on their boundaries, pseudogroups generated by expanding self-coverings, natural pseudogroups acting on leaves of stable (or unstable) foliation of an Anosov diffeomorphism, etc.
We describe a duality theory for hyperbolic groupoids. We show that for every hyperbolic groupoid $\mathfrak {G}$ there is a naturally defined dual groupoid $\mathfrak {G}^\top$ acting on the Gromov boundary of a Cayley graph of $\mathfrak {G}$. The groupoid $\mathfrak {G}^\top$ is also hyperbolic and such that $(\mathfrak {G}^\top )^\top$ is equivalent to $\mathfrak {G}$.
Several classes of examples of hyperbolic groupoids and their applications are discussed.
- Laurent Bartholdi, Rostislav Grigorchuk, and Volodymyr Nekrashevych, From fractal groups to fractal sets, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, pp. 25–118. MR 2091700
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486
- Laurent Bartholdi and Volodymyr Nekrashevych, Thurston equivalence of topological polynomials, Acta Math. 197 (2006), no. 1, 1–51. MR 2285317, DOI 10.1007/s11511-006-0007-3
- Rufus Bowen, Markov partitions for Axiom $\textrm {A}$ diffeomorphisms, Amer. J. Math. 92 (1970), 725–747. MR 277003, DOI 10.2307/2373370
- Michael Brin and Garrett Stuck, Introduction to dynamical systems, Cambridge University Press, Cambridge, 2002. MR 1963683
- James W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), no. 2, 123–148. MR 758901, DOI 10.1007/BF00146825
- Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
- Michel Coornaert and Athanase Papadopoulos, Symbolic dynamics and hyperbolic groups, Lecture Notes in Mathematics, vol. 1539, Springer-Verlag, Berlin, 1993. MR 1222644
- Daniele D’Angeli, Alfredo Donno, Michel Matter, and Tatiana Nagnibeda, Schreier graphs of the Basilica group, J. Mod. Dyn. 4 (2010), no. 1, 167–205. MR 2643891, DOI 10.3934/jmd.2010.4.167
- Heath Emerson, Noncommutative Poincaré duality for boundary actions of hyperbolic groups, J. Reine Angew. Math. 564 (2003), 1–33. MR 2021032, DOI 10.1515/crll.2003.090
- David Fried, Métriques naturelles sur les espaces de Smale, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 1, 77–79 (French, with English summary). MR 719952
- Martin Gardner, Mathematical games, Scientific American (1977), 110–121.
- Götz Gelbrich, Fractal Penrose tiles. II. Tiles with fractal boundary as duals of Penrose triangles, Aequationes Math. 54 (1997), no. 1-2, 108–116. MR 1466298, DOI 10.1007/BF02755450
- É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648
- É. Ghys, A. Haefliger, and A. Verjovsky (eds.), Group theory from a geometrical viewpoint, World Scientific Publishing Co., Inc., River Edge, NJ, 1991. MR 1170362
- M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
- André Haefliger, Some remarks on foliations with minimal leaves, J. Differential Geometry 15 (1980), no. 2, 269–284 (1981). MR 614370
- André Haefliger, Foliations and compactly generated pseudogroups, Foliations: geometry and dynamics (Warsaw, 2000) World Sci. Publ., River Edge, NJ, 2002, pp. 275–295. MR 1882774, DOI 10.1142/9789812778246_{0}013
- Peter Haïssinsky and Kevin M. Pilgrim, Coarse expanding conformal dynamics, Astérisque 325 (2009), viii+139 pp. (2010) (English, with English and French summaries). MR 2662902
- Vadim A. Kaimanovich, Random walks on Sierpiński graphs: hyperbolicity and stochastic homogenization, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, pp. 145–183. MR 2091703
- John L. Kelley, General topology, Springer-Verlag, New York-Berlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.]; Graduate Texts in Mathematics, No. 27. MR 0370454
- Jerome Kaminker and Ian Putnam, $K$-theoretic duality of shifts of finite type, Comm. Math. Phys. 187 (1997), no. 3, 509–522. MR 1468312, DOI 10.1007/s002200050147
- Jerome Kaminker, Ian F. Putnam, and Michael F. Whittaker, K-theoretic duality for hyperbolic dynamical systems, (preprint \verb!arXiv:1009.4999v1!), 2010.
- Harold Marston Morse, A One-to-One Representation of Geodesics on a Surface of Negative Curvature, Amer. J. Math. 43 (1921), no. 1, 33–51. MR 1506428, DOI 10.2307/2370306
- Paul S. Muhly, Jean N. Renault, and Dana P. Williams, Equivalence and isomorphism for groupoid $C^\ast$-algebras, J. Operator Theory 17 (1987), no. 1, 3–22. MR 873460
- Volodymyr Nekrashevych, Hyperbolic spaces from self-similar group actions, Algebra Discrete Math. 1 (2003), 77–86. MR 2051640
- Volodymyr Nekrashevych, Self-similar groups, Mathematical Surveys and Monographs, vol. 117, American Mathematical Society, Providence, RI, 2005. MR 2162164
- Volodymyr Nekrashevych, Self-similar inverse semigroups and Smale spaces, Internat. J. Algebra Comput. 16 (2006), no. 5, 849–874. MR 2274718, DOI 10.1142/S0218196706003153
- Volodymyr Nekrashevych, Combinatorial models of expanding dynamical systems, Ergodic Theory Dynam. Systems 34 (2014), no. 3, 938–985. MR 3199801, DOI 10.1017/etds.2012.163
- Volodymyr Nekrashevych, Symbolic dynamics and self-similar groups, Holomorphic dynamics and renormalization, Fields Inst. Commun., vol. 53, Amer. Math. Soc., Providence, RI, 2008, pp. 25–73. MR 2477417
- Volodymyr Nekrashevych, $C^*$-algebras and self-similar groups, J. Reine Angew. Math. 630 (2009), 59–123. MR 2526786, DOI 10.1515/CRELLE.2009.035
- Volodymyr Nekrashevych, Hyperbolic groupoids: metric and measure, Groups Geom. Dyn. 8 (2014), no. 3, 883–932. MR 3267528, DOI 10.4171/GGD/252
- Volodymyr Nekrashevych, The Julia set of a post-critically finite endomorphism of $\Bbb P\Bbb C^2$, J. Mod. Dyn. 6 (2012), no. 3, 327–375. MR 2988812, DOI 10.3934/jmd.2012.6.327
- R. Penrose, Pentaplexity: a class of nonperiodic tilings of the plane, Geometrical combinatorics (Milton Keynes, 1984) Res. Notes in Math., vol. 114, Pitman, Boston, MA, 1984, pp. 55–65. MR 777156
- Kevin M. Pilgrim, Julia sets as Gromov boundaries following V. Nekrashevych, Topology Proc. 29 (2005), no. 1, 293–316. Spring Topology and Dynamical Systems Conference. MR 2182937
- Ian F. Putnam, $C^*$-algebras from Smale spaces, Canad. J. Math. 48 (1996), no. 1, 175–195. MR 1382481, DOI 10.4153/CJM-1996-008-2
- Ian F. Putnam, Smale spaces and $c^*$-algebras, Lecture Notes, Univ. of Victoria, 2006.
- David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655
- David Ruelle, Noncommutative algebras for hyperbolic diffeomorphisms, Invent. Math. 93 (1988), no. 1, 1–13. MR 943921, DOI 10.1007/BF01393685
- William P. Thurston, Groups, tilings and finite state automata, (AMS Colloqium Lecture Notes), 1989.
- R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 169–203. MR 348794