Statistical limit laws for hyperbolic groups

Cantrell, Stephen

doi:10.1090/tran/8266

Statistical limit laws for hyperbolic groups
HTML articles powered by AMS MathViewer

by Stephen Cantrell PDF

Trans. Amer. Math. Soc. 374 (2021), 2687-2732 Request permission

Abstract:

Using techniques from ergodic theory and symbolic dynamics, we derive statistical limit laws for real valued functions on hyperbolic groups. In particular, our results apply to convex cocompact group actions on $\text {CAT}(-1)$ spaces, and provide a precise statistical comparison between word length and displacement. After generalising our methods to the multidimensional setting, we prove that the abelianisation map satisfies a non-degenerate multidimensional central limit theorem. We also obtain local limit theorems for group homomorphisms and for the displacement function associated to certain actions.

References

Jean Barge and Étienne Ghys, Surfaces et cohomologie bornée, Invent. Math. 92 (1988), no. 3, 509–526 (French). MR 939473, DOI 10.1007/BF01393745
Rajendra Bhatia and K. R. Parthasarathy, Lectures on functional analysis. Part I, ISI Lecture Notes, vol. 3, Macmillan Co. of India, Ltd., New Delhi, 1978. Perturbation by bounded operators. MR 506158
Robert Brooks, Some remarks on bounded cohomology, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 53–63. MR 624804
Danny Calegari, The ergodic theory of hyperbolic groups, Geometry and topology down under, Contemp. Math., vol. 597, Amer. Math. Soc., Providence, RI, 2013, pp. 15–52. MR 3186668, DOI 10.1090/conm/597/11762
Danny Calegari and Koji Fujiwara, Combable functions, quasimorphisms, and the central limit theorem, Ergodic Theory Dynam. Systems 30 (2010), no. 5, 1343–1369. MR 2718897, DOI 10.1017/S0143385709000662
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
Zaqueu Coelho and William Parry, Central limit asymptotics for shifts of finite type, Israel J. Math. 69 (1990), no. 2, 235–249. MR 1045376, DOI 10.1007/BF02937307
Michel Coornaert, Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993), no. 2, 241–270 (French, with French summary). MR 1214072
Rémi Coulon, Françoise Dal’Bo, and Andrea Sambusetti, Growth gap in hyperbolic groups and amenability, Geom. Funct. Anal. 28 (2018), no. 5, 1260–1320. MR 3856793, DOI 10.1007/s00039-018-0459-6
Françoise Dal’bo, Remarques sur le spectre des longueurs d’une surface et comptages, Bol. Soc. Brasil. Mat. (N.S.) 30 (1999), no. 2, 199–221 (French, with English and French summaries). MR 1703039, DOI 10.1007/BF01235869
David B. A. Epstein and Koji Fujiwara, The second bounded cohomology of word-hyperbolic groups, Topology 36 (1997), no. 6, 1275–1289. MR 1452851, DOI 10.1016/S0040-9383(96)00046-8
William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
Ilya Gekhtman, Samuel J. Taylor, and Giulio Tiozzo, Counting loxodromics for hyperbolic actions, J. Topol. 11 (2018), no. 2, 379–419. MR 3789828, DOI 10.1112/topo.12053
Ilya Gekhtman, Samuel J. Taylor, and Giulio Tiozzo, A central limit theorem for random closed geodesics: proof of the Chas-Li-Maskit conjecture, Adv. Math. 358 (2019), 106852, 18. MR 4020454, DOI 10.1016/j.aim.2019.106852
Ilya Gekhtman, Samuel J. Taylor, and Giulio Tiozzo, Counting problems in graph products and relatively hyperbolic groups, Israel J. Math. 237 (2020), no. 1, 311–371. MR 4111874, DOI 10.1007/s11856-020-2008-x
É. Ghys and P. de la Harpe, Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics 83, 1990.
Sébastien Gouëzel, Frédéric Mathéus, and François Maucourant, Entropy and drift in word hyperbolic groups, Invent. Math. 211 (2018), no. 3, 1201–1255. MR 3763407, DOI 10.1007/s00222-018-0788-y
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}
Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
M. Horsham, Central limit theorems for quasimorphisms of surface groups, PhD thesis, Manchester, 2008.
Matthew Horsham and Richard Sharp, Lengths, quasi-morphisms and statistics for free groups, Spectral analysis in geometry and number theory, Contemp. Math., vol. 484, Amer. Math. Soc., Providence, RI, 2009, pp. 219–237. MR 1500150, DOI 10.1090/conm/484/09477
Ilya Kapovich and Tatiana Nagnibeda, The Patterson-Sullivan embedding and minimal volume entropy for outer space, Geom. Funct. Anal. 17 (2007), no. 4, 1201–1236. MR 2373015, DOI 10.1007/s00039-007-0621-z
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1980.
Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625, DOI 10.1090/chel/340
George Kenison and Richard Sharp, Statistics in conjugacy classes in free groups, Geom. Dedicata 198 (2019), 57–70. MR 3933450, DOI 10.1007/s10711-018-0329-2
Steven P. Lalley, Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math. 163 (1989), no. 1-2, 1–55. MR 1007619, DOI 10.1007/BF02392732
William Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc. 112 (1964), 55–66. MR 161372, DOI 10.1090/S0002-9947-1964-0161372-1
William Parry and Mark Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990), 268 (English, with French summary). MR 1085356
Mark Pollicott and Richard Sharp, Comparison theorems and orbit counting in hyperbolic geometry, Trans. Amer. Math. Soc. 350 (1998), no. 2, 473–499. MR 1376553, DOI 10.1090/S0002-9947-98-01756-5
Mark Pollicott and Richard Sharp, Poincaré series and comparison theorems for variable negative curvature, Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2, vol. 202, Amer. Math. Soc., Providence, RI, 2001, pp. 229–240. MR 1819191, DOI 10.1090/trans2/202/16
Mark Pollicott and Richard Sharp, Statistics of matrix products in hyperbolic geometry, Dynamical numbers—interplay between dynamical systems and number theory, Contemp. Math., vol. 532, Amer. Math. Soc., Providence, RI, 2010, pp. 213–230. MR 2762142, DOI 10.1090/conm/532/10492
Igor Rivin, Growth in free groups (and other stories)—twelve years later, Illinois J. Math. 54 (2010), no. 1, 327–370. MR 2776999
J. Rousseau-Egele, Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux, Ann. Probab. 11 (1983), no. 3, 772–788 (French, with English summary). MR 704569
Caroline Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature, Ergodic Theory Dynam. Systems 6 (1986), no. 4, 601–625. MR 873435, DOI 10.1017/S0143385700003722
Richard Sharp, Local limit theorems for free groups, Math. Ann. 321 (2001), no. 4, 889–904. MR 1872533, DOI 10.1007/s002080100258
Richard Sharp, Relative growth series in some hyperbolic groups, Math. Ann. 312 (1998), no. 1, 125–132. MR 1645953, DOI 10.1007/s002080050214
Matthew Sunderland, Linear progress with exponential decay in weakly hyperbolic groups, Groups Geom. Dyn. 14 (2020), no. 2, 539–566. MR 4118628, DOI 10.4171/GGD/554