×

Classification of joinings for Kleinian groups. (English) Zbl 1362.37009

In the 1980s, Ratner classified the ergodic measures for the action of unipotent subgroups on the quotient \(\Gamma\backslash G\) of a connected Lie group \(G\) by a lattice \(\Gamma\). This impressive work generalizes Ratner’s results to the case where \(\Gamma\) is no longer a lattice. More precisely, it focuses on \(G=\mathrm{PSL}_2(\mathbb R)\), \(\mathrm{PSL}_2(\mathbb C)\), takes \(\Gamma\) to be a geometrically finite Zariski dense subgroup of \(G\), and classifies (locally finite) ergodic measures for the action of a horospherical subgroup on \(\Gamma\backslash G\).

MSC:

37A17 Homogeneous flows
11N45 Asymptotic results on counting functions for algebraic and topological structures
57M60 Group actions on manifolds and cell complexes in low dimensions
20F67 Hyperbolic groups and nonpositively curved groups
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
22E40 Discrete subgroups of Lie groups
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] J. Aaronson, An Introduction to Infinite Ergodic Theory , Math. Survey Monogr. 50 , Amer. Math. Soc., Providence, 1997. · Zbl 0882.28013
[2] M. Babillot, “On the classification of invariant measures for horosphere foliations on nilpotent covers of negatively curved manifolds” in Random Walks and Geometry , de Gruyter, Berlin, 2004, 319-335. · Zbl 1069.37022
[3] B. H. Bowditch, Geometrical finiteness for hyperbolic groups , J. Funct. Anal. 113 (1993), 245-317. · Zbl 0789.57007
[4] M. Burger, Horocycle flow on geometrically finite surfaces , Duke Math. J. 61 (1990), 779-803. · Zbl 0723.58041
[5] S. G. Dani and G. A. Margulis, “Limit distributions of orbits of unipotent flows and values of quadratic forms” in I. M. Gel’fand Seminar , Adv. Soviet Math. 16 , Amer. Math. Soc., Providence, 1993, 91-137. · Zbl 0814.22003
[6] S. G. Dani and J. Smillie, Uniform distribution of horocycle orbits for Fuchsian groups , Duke Math. J. 51 (1984), 185-194. · Zbl 0547.20042
[7] M. Einsiedler and E. Lindenstrauss, Joinings of higher-rank diagonalizable actions on locally homogeneous spaces , Duke Math. J. 138 (2007), 203-232. · Zbl 1118.37008
[8] L. Flaminio and R. Spatzier, Geometrically finite groups, Patterson-Sullivan measures and Ratner’s theorem , Invent. Math. 99 (1990), 601-626. · Zbl 0667.57014
[9] A. Furman, Measurable rigidity of actions on infinite measure homogeneous spaces, II , J. Amer. Math. Soc. 21 (2008), 479-512. · Zbl 1210.37002
[10] H. Furtstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation , Math. Systems Theory 1 (1967), 1-49. · Zbl 0146.28502
[11] M. Hochman, A ratio ergodic theorem for multiparameter non-singular actions , J. Eur. Math. Soc. (JEMS) 12 (2010), 365-383. · Zbl 1190.22007
[12] M. Hochman, On the ratio ergodic theorem for group actions , J. Lond. Math. Soc. (2) 88 (2013), 465-482. · Zbl 1310.37005
[13] R. A. Johnson, Atomic and nonatomic measures , Proc. Amer. Math. Soc. 25 (1970), 650-655. · Zbl 0201.06201
[14] D. Kleinbock, E. Lindenstrauss, and B. Weiss, On fractal measures and Diophantine approximation , Selecta Math. (N.S.) 10 (2004), 479-523. · Zbl 1130.11039
[15] D. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds , Ann. of Math. (2) 148 (1998), 339-360. · Zbl 0922.11061
[16] A. Kontorovich and H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds , J. Amer. Math. Soc. 24 (2011), 603-648. · Zbl 1235.22015
[17] U. Krengel, Ergodic Theorems , De Gruyter Stud. Math. 6 , de Gruyter, New York, 1985.
[18] F. Ledrappier, Invariant measures for the stable foliation on negatively curved periodic manifolds , Ann. Inst. Fourier (Grenoble) 58 (2008), 85-105. · Zbl 1149.37022
[19] F. Ledrappier and O. Sarig, Invariant measures for the horocycle flow on periodic hyperbolic surfaces , Israel J. Math. 160 (2007), 281-315. · Zbl 1137.37016
[20] G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces , Invent. Math. 116 (1994), 347-392. · Zbl 0816.22004
[21] F. Maucourant and B. Schapira, Distribution of orbits in \(\mathbb{R}^{2}\) of a finitely generated subgroup of \(\mathrm{SL}(2,\mathbb{R})\) , Amer. J. Math. 136 (2014), 1497-1542. · Zbl 1348.37052
[22] A. Mohammadi and H. Oh, Ergodicity of unipotent flows and Kleinian groups , J. Amer. Math. Soc. 28 (2015), 531-577. · Zbl 1318.37001
[23] A. Mohammadi and H. Oh, Matrix coefficients, counting and primes for orbits of geometrically finite groups , J. Eur. Math. Soc. (JEMS) 17 (2015), 837-897. · Zbl 1391.11137
[24] H. Oh and N. A. Shah, The asymptotic distribution of circles in the orbits of Kleinian groups , Invent. Math. 187 (2012), 1-35. · Zbl 1235.52033
[25] H. Oh and N. A. Shah, Equidistribution and counting for orbits of geometrically finite hyperbolic groups , J. Amer. Math. Soc. 26 (2013), 511-562. · Zbl 1334.22011
[26] S. J. Patterson, The limit set of a Fuchsian group , Acta Math. 136 (1976), 241-273. · Zbl 0336.30005
[27] M. S. Raghunathan, Discrete Subgroups of Lie Groups , Ergeb. Math. Grenzgeb. (3) 68 , New York, Springer, 1972. · Zbl 0254.22005
[28] M. Ratner, Factors of horocycle flows , Ergodic Theory Dynam. Systems 2 (1982), 465-489. · Zbl 0536.58029
[29] M. Ratner, Rigidity of horocycle flows , Ann. of Math. (2) 115 (1982), 597-614. · Zbl 0506.58030
[30] M. Ratner, Horocycle flows, joinings and rigidity of products , Ann. of Math. (2) 118 (1983), 277-313. · Zbl 0556.28020
[31] M. Ratner, On Raghunathan’s measure conjecture , Ann. of Math. (2) 134 (1991), 545-607. · Zbl 0763.28012
[32] T. Roblin, Ergodicité et équidistribution en courbure négative , Mém. Soc. Math. Fr. (N.S.) 95 , Soc. Math. France, Paris, 2003.
[33] V. A. Rohlin, On the fundamental ideas of measure theory , Mat. Sbornik 67 (1949), 107-150.
[34] D. J. Rudolph, Ergodic behavior of Sullivan’s geometric measure on a geometrically finite hyperbolic manifold , Ergodic Theory Dynam. Systems 2 (1982), 491-512. · Zbl 0525.58025
[35] O. Sarig, Invariant Radon measures for the horocycle flow on abelian covers , Invent. Math. 157 (2004), 519-551. · Zbl 1052.37004
[36] O. Sarig, “Unique ergodicity for infinite measures” in Proceedings of the International Congress of Mathematicians, III , Hindustan Book Agency, New Delhi, 2010, 1777-1803. · Zbl 1246.37018
[37] B. Schapira, Lemme de l’ombre et non divergence des horosphères d’une variété géométriquement finie , Ann. Inst. Fourier (Grenoble) 54 (2004), 939-987. · Zbl 1063.37029
[38] B. Schapira, Equidistribution of the horocycles of a geometrically finite surface , Int. Math. Res. Not. IMRN 2005 , no. 40, 2447-2471. · Zbl 1094.37015
[39] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions , Publ. Math. Inst. Hautes Études Sci. 50 (1979), 171-202. · Zbl 0439.30034
[40] D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups , Acta Math. 153 (1984), 259-277. · Zbl 0566.58022
[41] D. Winter, Mixing of frame flow for rank one locally symmetric manifold and measure classification , Israel J. Math. 210 (2015), 467-507. · Zbl 1352.37011
[42] D. Witte, Measurable quotients of unipotent translations on homogeneous spaces , Trans. Amer. Math. Soc. 345 , no. 2 (1994), 577-594. · Zbl 0831.28010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.