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Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of $ \operatorname{SL}_2(\mathbb{Z})$


Authors: Hee Oh and Dale Winter
Journal: J. Amer. Math. Soc. 29 (2016), 1069-1115
MSC (2010): Primary 37D35, 22E40, 37A25, 37D40, 11F72; Secondary 37F30, 11N45
DOI: https://doi.org/10.1090/jams/849
Published electronically: November 2, 2015
MathSciNet review: 3522610
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Abstract: Let $ \Gamma <\operatorname {SL}_2(\mathbb{Z})$ be a non-elementary finitely generated subgroup and let $ \Gamma (q)$ be its congruence subgroup of level $ q$ for each $ q\in \mathbb{N}$. We obtain an asymptotic formula for the matrix coefficients of $ L^2(\Gamma (q) \backslash \operatorname {SL}_2(\mathbb{R}))$ with a uniform exponential error term for all square free $ q$ with no small prime divisors. As an application we establish a uniform resonance free half plane for the resolvent of the Laplacian on $ \Gamma (q)\backslash \mathbb{H}^2$ over $ q$ as above. Our approach is to extend Dolgopyat's dynamical proof of exponential mixing of the geodesic flow uniformly over congruence covers, by establishing uniform spectral bounds for congruence transfer operators associated to the geodesic flow. One of the key ingredients is the expander theory due to Bourgain-Gamburd-Sarnak.


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  • [1] Artur Avila, Sébastien Gouëzel, and Jean-Christophe Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci. 104 (2006), 143-211. MR 2264836 (2007j:37049), https://doi.org/10.1007/s10240-006-0001-5
  • [2] Martine Babillot, On the mixing property for hyperbolic systems, Israel J. Math. 129 (2002), 61-76. MR 1910932 (2003g:37008), https://doi.org/10.1007/BF02773153
  • [3] Rufus Bowen and David Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181-202. MR 0380889 (52 #1786)
  • [4] David Borthwick, Spectral theory of infinite-area hyperbolic surfaces, Progress in Mathematics, vol. 256, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2344504 (2008h:58056)
  • [5] Rufus Bowen, Markov partitions for Axiom $ {\rm A}$ diffeomorphisms, Amer. J. Math. 92 (1970), 725-747. MR 0277003 (43 #2740)
  • [6] B. H. Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. J. 77 (1995), no. 1, 229-274. MR 1317633 (96b:53056), https://doi.org/10.1215/S0012-7094-95-07709-6
  • [7] Jean Bourgain and Alex Gamburd, Uniform expansion bounds for Cayley graphs of $ {\rm SL}_2(\mathbb{F}_p)$, Ann. of Math. (2) 167 (2008), no. 2, 625-642. MR 2415383 (2010b:20070), https://doi.org/10.4007/annals.2008.167.625
  • [8] Jean Bourgain, Alex Gamburd, and Peter Sarnak, Generalization of Selberg's $ \frac {3}{16}$ theorem and affine sieve, Acta Math. 207 (2011), no. 2, 255-290. MR 2892611, https://doi.org/10.1007/s11511-012-0070-x
  • [9] Jean Bourgain, Alex Gamburd, and Peter Sarnak, Affine linear sieve, expanders, and sum-product, Invent. Math. 179 (2010), no. 3, 559-644. MR 2587341 (2011d:11018), https://doi.org/10.1007/s00222-009-0225-3
  • [10] Jean Bourgain and Alex Kontorovich, On Zaremba's conjecture, Ann. of Math. (2) 180 (2014), no. 1, 137-196. MR 3194813, https://doi.org/10.4007/annals.2014.180.1.3
  • [11] Jean Bourgain, Alex Kontorovich, and Michael Magee, Thermodynamic expansion to arbitrary moduli. Preprint.
  • [12] Jean Bourgain, Alex Kontorovich, and Peter Sarnak, Sector estimates for hyperbolic isometries, Geom. Funct. Anal. 20 (2010), no. 5, 1175-1200. MR 2746950, https://doi.org/10.1007/s00039-010-0092-5
  • [13] Jack Button, All Fuchsian Schottky groups are classical Schottky groups, The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, 1998, pp. 117-125 (electronic). MR 1668339 (2000e:20078), https://doi.org/10.2140/gtm.1998.1.117
  • [14] Jean Bourgain and Péter P. Varjú, Expansion in $ SL_d({\bf Z}/q{\bf Z}),\,q$ arbitrary, Invent. Math. 188 (2012), no. 1, 151-173. MR 2897695, https://doi.org/10.1007/s00222-011-0345-4
  • [15] N. Chernov, Invariant measures for hyperbolic dynamical systems, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 321-407. MR 1928521 (2003g:37047), https://doi.org/10.1016/S1874-575X(02)80006-6
  • [16] Yves Colin De Verdiere, Théorie spectrale des surfaces de Riemann d'aire infinite, Astérisque 132 (1985), 259-275.
  • [17] Dmitry Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 357-390. MR 1626749 (99g:58073), https://doi.org/10.2307/121012
  • [18] Boris Hasselblatt and Anatole Katok (eds.), Handbook of Dynamical Systems, 1A, North-Holland, Amsterdam, 2002.
  • [19] A. Salehi Golsefidy and Péter P. Varjú, Expansion in perfect groups, Geom. Funct. Anal. 22 (2012), no. 6, 1832-1891. MR 3000503, https://doi.org/10.1007/s00039-012-0190-7
  • [20] Colin Guillarmou and Frédéric Naud, Wave decay on convex co-compact hyperbolic manifolds, Comm. Math. Phys. 287 (2009), no. 2, 489-511. MR 2481747 (2009m:58060), https://doi.org/10.1007/s00220-008-0706-z
  • [21] Laurent Guillopé, Kevin K. Lin, and Maciej Zworski, The Selberg zeta function for convex co-compact Schottky groups, Comm. Math. Phys. 245 (2004), no. 1, 149-176. MR 2036371 (2005f:11193), https://doi.org/10.1007/s00220-003-1007-1
  • [22] Min Lee and Hee Oh, Effective circle count for Apollonian packings and closed horospheres, Geom. Funct. Anal. 23 (2013), no. 2, 580-621. MR 3053757, https://doi.org/10.1007/s00039-013-0217-8
  • [23] Peter D. Lax and Ralph S. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Funct. Anal. 46 (1982), no. 3, 280-350. MR 661875 (83j:10057), https://doi.org/10.1016/0022-1236(82)90050-7
  • [24] Michael Magee, Hee Oh, and Dale Winter, Expanding maps and continued fractions, available at arXiv:1412.4284.
  • [25] Rafe R. Mazzeo and Richard B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), no. 2, 260-310. MR 916753 (89c:58133), https://doi.org/10.1016/0022-1236(87)90097-8
  • [26] Gregory Margulis, Amir Mohammadi, and Hee Oh, Closed geodesics and holonomies for Kleinian manifolds, Geom. Funct. Anal. 24 (2014), no. 5, 1608-1636. MR 3261636, https://doi.org/10.1007/s00039-014-0299-y
  • [27] Amir Mohammadi and Hee Oh, Matrix coefficients, counting and primes for orbits of geometrically finite groups, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 4, 837-897. MR 3336838, https://doi.org/10.4171/JEMS/520
  • [28] Frédéric Naud, Expanding maps on Cantor sets and analytic continuation of zeta functions, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), no. 1, 116-153 (English, with English and French summaries). MR 2136484 (2006e:37033), https://doi.org/10.1016/j.ansens.2004.11.002
  • [29] Hee Oh and Nimish A. Shah, Equidistribution and counting for orbits of geometrically finite hyperbolic groups, J. Amer. Math. Soc. 26 (2013), no. 2, 511-562. MR 3011420, https://doi.org/10.1090/S0894-0347-2012-00749-8
  • [30] 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 (92f:58141)
  • [31] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), no. 3-4, 241-273. MR 0450547 (56 #8841)
  • [32] S. J. Patterson, On a lattice-point problem in hyperbolic space and related questions in spectral theory, Ark. Mat. 26 (1988), no. 1, 167-172. MR 948288 (89g:11093), https://doi.org/10.1007/BF02386116
  • [33] Mark Pollicott, On the rate of mixing of Axiom A flows, Invent. Math. 81 (1985), no. 3, 413-426. MR 807065 (87i:58148), https://doi.org/10.1007/BF01388579
  • [34] Vesselin Pektov and Luchezar Stoyanov, Spectral estimates for Ruelle transfer operators with two parameters and applications, available at arXiv:1409.0721.
  • [35] Thomas Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.) 95 (2003), 96p.
  • [36] M. Ratner, Markov partitions for Anosov flows on $ n$-dimensional manifolds, Israel J. Math. 15 (1973), 92-114. MR 0339282 (49 #4042)
  • [37] Daniel J. Rudolph, Ergodic behaviour of Sullivan's geometric measure on a geometrically finite hyperbolic manifold, Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 491-512 (1983). MR 721736 (85i:58101), https://doi.org/10.1017/S0143385700001735
  • [38] Barbara Schapira, Equidistribution of the horocycles of a geometrically finite surface, Int. Math. Res. Not. IMRN 40 (2005), 2447-2471. MR 2180113 (2006i:37073), https://doi.org/10.1155/IMRN.2005.2447
  • [39] Luchezar Stoyanov, Spectra of Ruelle transfer operators for axiom A flows, Nonlinearity 24 (2011), no. 4, 1089-1120. MR 2776112 (2012f:37060), https://doi.org/10.1088/0951-7715/24/4/005
  • [40] Luchezar Stoyanov, On the Ruelle-Perron-Frobenius theorem, Asymptot. Anal. 43 (2005), no. 1-2, 131-150. MR 2148129 (2007d:37023)
  • [41] Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171-202. MR 556586 (81b:58031)
  • [42] Dennis Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984), no. 3-4, 259-277. MR 766265 (86c:58093), https://doi.org/10.1007/BF02392379
  • [43] Ilya Vinogradov, Effective bisector estimate with application to Apollonian circle packings, Int. Math. Res. Not. IMRN 12 (2014), 3217-3262. MR 3217660
  • [44] David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, NJ, 1941. MR 0005923 (3,232d)

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Additional Information

Hee Oh
Affiliation: Mathematics Department, Yale University, New Haven, Connecticut 06511 and Korea Institute for Advanced Study, Seoul, Korea
Email: hee.oh@yale.edu

Dale Winter
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02906
Email: dale_winter@brown.edu

DOI: https://doi.org/10.1090/jams/849
Received by editor(s): April 24, 2015
Received by editor(s) in revised form: August 26, 2015
Published electronically: November 2, 2015
Additional Notes: The first author was supported in part by NSF Grant 1361673.
Dedicated: Dedicated to Peter Sarnak on the occasion of his sixty-first birthday.
Article copyright: © Copyright 2015 American Mathematical Society

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