# American Institute of Mathematical Sciences

2021, 17: 337-352. doi: 10.3934/jmd.2021012

## Horospherically invariant measures and finitely generated Kleinian groups

 Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Jerusalem, 9190401, Israel

Received  September 09, 2020 Revised  February 28, 2021 Published  September 2021

Fund Project: This work was supported by ERC 2020 grant HomDyn (grant no. 833423)

Let $\Gamma < {\rm{PSL}}_2( \mathbb{C})$ be a Zariski dense finitely generated Kleinian group. We show all Radon measures on ${\rm{PSL}}_2( \mathbb{C}) / \Gamma$ which are ergodic and invariant under the action of the horospherical subgroup are either supported on a single closed horospherical orbit or quasi-invariant with respect to the geodesic frame flow and its centralizer. We do this by applying a result of Landesberg and Lindenstrauss [18] together with fundamental results in the theory of 3-manifolds, most notably the Tameness Theorem by Agol [2] and Calegari-Gabai [10].

Citation: Or Landesberg. Horospherically invariant measures and finitely generated Kleinian groups. Journal of Modern Dynamics, 2021, 17: 337-352. doi: 10.3934/jmd.2021012
##### References:
 [1] J. Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar [2] I. Agol, Tameness of hyperbolic 3-manifolds, preprint, arXiv: math/0405568. Google Scholar [3] J. W. Anderson, K. Falk and P. Tukia, Conformal measures associated to ends of hyperbolic $n$-manifolds, Q. J. Math., 58 (2007), 1-15.  doi: 10.1093/qmath/hal019.  Google Scholar [4] B. N. Apanasov, Cusp ends of hyperbolic manifolds, Ann. Global Anal. Geom., 3 (1985), no. 1, 1–11. doi: 10.1007/BF00054488.  Google Scholar [5] M. Babillot, On the classification of invariant measures for horosphere foliations on nilpotent covers of negatively curved manifolds, in Random Walks and Geometry, Walter de Gruyter, Berlin, 2004,319–335.  Google Scholar [6] M. Babillot and F. Ledrappier, Geodesic paths and horocycle flow on Abelian covers, in Lie Groups and Ergodic Theory (Mumbai, 1996), Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998, 1–32.  Google Scholar [7] A. Bellis, On the links between horocyclic and geodesic orbits on geometrically infinite surfaces, J. Éc. Polytech. Math., 5 (2018), 443–454. doi: 10.5802/jep.75.  Google Scholar [8] C. J. Bishop and P. W. Jones, The law of the iterated logarithm for Kleinian groups, in Lipa's Legacy (New York, 1995), Contemp. Math., 211, Amer. Math. Soc., Providence, RI, 1997, 17–50. doi: 10.1090/conm/211/02813.  Google Scholar [9] M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803.  doi: 10.1215/S0012-7094-90-06129-0.  Google Scholar [10] D. Calegari and D. Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Amer. Math. Soc., 19 (2006), 385-446.  doi: 10.1090/S0894-0347-05-00513-8.  Google Scholar [11] R. D. Canary, Ends of hyperbolic $3$-manifolds, J. Amer. Math. Soc., 6 (1993), 1-35.  doi: 10.2307/2152793.  Google Scholar [12] R. D. Canary, A covering theorem for hyperbolic $3$-manifolds and its applications, Topology, 35 (1996), 751-778.  doi: 10.1016/0040-9383(94)00055-7.  Google Scholar [13] S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., 47 (1978), 101-138.  doi: 10.1007/BF01578067.  Google Scholar [14] H. Furstenberg, The unique ergodicity of the horocycle flow, in Recent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Math., 318, Springer, Berlin, 1973, 95–115.  Google Scholar [15] G. Greschonig and K. Schmidt, Ergodic decomposition of quasi-invariant probability measures, Colloq. Math., 84/85 (2000), 495-514.  doi: 10.4064/cm-84/85-2-495-514.  Google Scholar [16] M. Hochman, A ratio ergodic theorem for multiparameter non-singular actions, J. Eur. Math. Soc. (JEMS), 12 (2010), 365-383.  doi: 10.4171/JEMS/201.  Google Scholar [17] M. Kapovich, Hyperbolic manifolds and discrete groups, in Modern Birkhäuser Classics, Birkhäuser Boston, Ltd., Boston, MA, 2009. doi: 10.1007/978-0-8176-4913-5.  Google Scholar [18] O. Landesberg and E. Lindenstrauss, On radon measures invariant under horospherical flows on geometrically infinite quotients, preprint, arXiv: 1910.08956. Google Scholar [19] C. Lecuire and M. Mj, Horospheres in degenerate 3-manifolds, Int. Math. Res. Not. IMRN, 2018 (2018), 816-861.  doi: 10.1093/imrn/rnw256.  Google Scholar [20] F. Ledrappier, Horospheres on abelian covers, Bol. Soc. Brasil. Mat. (N.S.), 29 (1998), 195-195.  doi: 10.1007/BF01245874.  Google Scholar [21] F. Ledrappier, Invariant measures for the stable foliation on negatively curved periodic manifolds, Ann. Inst. Fourier (Grenoble), 58 (2008), 85-105.  doi: 10.5802/aif.2345.  Google Scholar [22] F. Ledrappier and O. Sarig, Invariant measures for the horocycle flow on periodic hyperbolic surfaces, Israel J. Math., 160 (2007), 281-315.  doi: 10.1007/s11856-007-0064-0.  Google Scholar [23] M. Lee and H. Oh, Ergodic decompositions of geometric measures on Anosov homogeneous spaces, preprint, https://gauss.math.yale.edu/~ho2/doc/ED.pdf, 2020. Google Scholar [24] A. Marden, Hyperbolic Manifolds. An Introduction in 2 and 3 Dimensions, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316337776.  Google Scholar [25] S. Matsumoto, Horocycle flows without minimal sets, J. Math. Sci. Univ. Tokyo, 23 (2016), 661-673.   Google Scholar [26] K. Matsuzaki and M. Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar [27] D. McCullough, Compact submanifolds of $3$-manifolds with boundary, Quart. J. Math. Oxford Ser., 37 (1986), 299-307.  doi: 10.1093/qmath/37.3.299.  Google Scholar [28] Y. N. Minsky, End invariants and the classification of hyperbolic 3-manifolds, in Current Developments in Mathematics, 2002, Int. Press, Somerville, MA, 2003, 181–217.  Google Scholar [29] A. Mohammadi and H. Oh, Classification of joinings for Kleinian groups, Duke Math. J., 165 (2016), 2155-2223.  doi: 10.1215/00127094-3476807.  Google Scholar [30] H. Oh and W. Pan, Local mixing and invariant measures for horospherical subgroups on Abelian covers, International Mathematics Research Notices, 2019 (2019), 6036-6088.  doi: 10.1093/imrn/rnx292.  Google Scholar [31] J. G. Ratcliffe, Foundations of hyperbolic manifolds, second edition, Graduate Texts in Mathematics, 149, Springer, New York, 2006.  Google Scholar [32] M. Ratner, On Raghunathan's measure conjecture, Ann. of Math., 134 (1991), 545-607.  doi: 10.2307/2944357.  Google Scholar [33] T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), 95 (2003). doi: 10.24033/msmf.408.  Google Scholar [34] O. Sarig, Invariant Radon measures for horocycle flows on Abelian covers, Invent. Math., 157 (2004), 519-551.  doi: 10.1007/s00222-004-0357-4.  Google Scholar [35] O. Sarig, The horocyclic flow and the Laplacian on hyperbolic surfaces of infinite genus, Geom. Funct. Anal., 19 (2010), 1757-1812.  doi: 10.1007/s00039-010-0048-9.  Google Scholar [36] G. P. Scott, Compact submanifolds of $3$-manifolds, J. London Math. Soc., 7 (1973), 246-250.  doi: 10.1112/jlms/s2-7.2.246.  Google Scholar [37] H. Shimizu, On discontinuous groups operating on the product of the upper half planes, Ann. of Math. (2), 77 (1963), no. 1, 33–71. doi: 10.2307/1970201.  Google Scholar [38] A. N. Starkov, Parabolic fixed points of Kleinian groups and the horospherical foliation on hyperbolic manifolds, Internat. J. Math., 8 (1997), no. 2,289–299. doi: 10.1142/S0129167X97000135.  Google Scholar [39] D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math., 153 (1984), 259-277.  doi: 10.1007/BF02392379.  Google Scholar [40] D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom., 25 (1987), 327-351.   Google Scholar [41] W. P. Thurston, The Geometry and Topology of Three-Manifolds, Princeton University, Princeton, NJ, 1979. Google Scholar [42] W. P. Thurston, Three-Dimensional Geometry and Topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997.  Google Scholar [43] W. A. Veech, Unique ergodicity of horospherical flows, Amer. J. Math., 99 (1977), 827-859.  doi: 10.2307/2373868.  Google Scholar [44] D. Winter, Mixing of frame flow for rank one locally symmetric spaces and measure classification, Israel J. Math., 210 (2015), 467-507.  doi: 10.1007/s11856-015-1258-5.  Google Scholar

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##### References:
 [1] J. Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar [2] I. Agol, Tameness of hyperbolic 3-manifolds, preprint, arXiv: math/0405568. Google Scholar [3] J. W. Anderson, K. Falk and P. Tukia, Conformal measures associated to ends of hyperbolic $n$-manifolds, Q. J. Math., 58 (2007), 1-15.  doi: 10.1093/qmath/hal019.  Google Scholar [4] B. N. Apanasov, Cusp ends of hyperbolic manifolds, Ann. Global Anal. Geom., 3 (1985), no. 1, 1–11. doi: 10.1007/BF00054488.  Google Scholar [5] M. Babillot, On the classification of invariant measures for horosphere foliations on nilpotent covers of negatively curved manifolds, in Random Walks and Geometry, Walter de Gruyter, Berlin, 2004,319–335.  Google Scholar [6] M. Babillot and F. Ledrappier, Geodesic paths and horocycle flow on Abelian covers, in Lie Groups and Ergodic Theory (Mumbai, 1996), Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998, 1–32.  Google Scholar [7] A. Bellis, On the links between horocyclic and geodesic orbits on geometrically infinite surfaces, J. Éc. Polytech. Math., 5 (2018), 443–454. doi: 10.5802/jep.75.  Google Scholar [8] C. J. Bishop and P. W. Jones, The law of the iterated logarithm for Kleinian groups, in Lipa's Legacy (New York, 1995), Contemp. Math., 211, Amer. Math. Soc., Providence, RI, 1997, 17–50. doi: 10.1090/conm/211/02813.  Google Scholar [9] M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803.  doi: 10.1215/S0012-7094-90-06129-0.  Google Scholar [10] D. Calegari and D. Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Amer. Math. Soc., 19 (2006), 385-446.  doi: 10.1090/S0894-0347-05-00513-8.  Google Scholar [11] R. D. Canary, Ends of hyperbolic $3$-manifolds, J. Amer. Math. Soc., 6 (1993), 1-35.  doi: 10.2307/2152793.  Google Scholar [12] R. D. Canary, A covering theorem for hyperbolic $3$-manifolds and its applications, Topology, 35 (1996), 751-778.  doi: 10.1016/0040-9383(94)00055-7.  Google Scholar [13] S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., 47 (1978), 101-138.  doi: 10.1007/BF01578067.  Google Scholar [14] H. Furstenberg, The unique ergodicity of the horocycle flow, in Recent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Math., 318, Springer, Berlin, 1973, 95–115.  Google Scholar [15] G. Greschonig and K. Schmidt, Ergodic decomposition of quasi-invariant probability measures, Colloq. Math., 84/85 (2000), 495-514.  doi: 10.4064/cm-84/85-2-495-514.  Google Scholar [16] M. Hochman, A ratio ergodic theorem for multiparameter non-singular actions, J. Eur. Math. Soc. (JEMS), 12 (2010), 365-383.  doi: 10.4171/JEMS/201.  Google Scholar [17] M. Kapovich, Hyperbolic manifolds and discrete groups, in Modern Birkhäuser Classics, Birkhäuser Boston, Ltd., Boston, MA, 2009. doi: 10.1007/978-0-8176-4913-5.  Google Scholar [18] O. Landesberg and E. Lindenstrauss, On radon measures invariant under horospherical flows on geometrically infinite quotients, preprint, arXiv: 1910.08956. Google Scholar [19] C. Lecuire and M. Mj, Horospheres in degenerate 3-manifolds, Int. Math. Res. Not. IMRN, 2018 (2018), 816-861.  doi: 10.1093/imrn/rnw256.  Google Scholar [20] F. Ledrappier, Horospheres on abelian covers, Bol. Soc. Brasil. Mat. (N.S.), 29 (1998), 195-195.  doi: 10.1007/BF01245874.  Google Scholar [21] F. Ledrappier, Invariant measures for the stable foliation on negatively curved periodic manifolds, Ann. Inst. Fourier (Grenoble), 58 (2008), 85-105.  doi: 10.5802/aif.2345.  Google Scholar [22] F. Ledrappier and O. Sarig, Invariant measures for the horocycle flow on periodic hyperbolic surfaces, Israel J. Math., 160 (2007), 281-315.  doi: 10.1007/s11856-007-0064-0.  Google Scholar [23] M. Lee and H. Oh, Ergodic decompositions of geometric measures on Anosov homogeneous spaces, preprint, https://gauss.math.yale.edu/~ho2/doc/ED.pdf, 2020. Google Scholar [24] A. Marden, Hyperbolic Manifolds. An Introduction in 2 and 3 Dimensions, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316337776.  Google Scholar [25] S. Matsumoto, Horocycle flows without minimal sets, J. Math. Sci. Univ. Tokyo, 23 (2016), 661-673.   Google Scholar [26] K. Matsuzaki and M. Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar [27] D. McCullough, Compact submanifolds of $3$-manifolds with boundary, Quart. J. Math. Oxford Ser., 37 (1986), 299-307.  doi: 10.1093/qmath/37.3.299.  Google Scholar [28] Y. N. Minsky, End invariants and the classification of hyperbolic 3-manifolds, in Current Developments in Mathematics, 2002, Int. Press, Somerville, MA, 2003, 181–217.  Google Scholar [29] A. Mohammadi and H. Oh, Classification of joinings for Kleinian groups, Duke Math. J., 165 (2016), 2155-2223.  doi: 10.1215/00127094-3476807.  Google Scholar [30] H. Oh and W. Pan, Local mixing and invariant measures for horospherical subgroups on Abelian covers, International Mathematics Research Notices, 2019 (2019), 6036-6088.  doi: 10.1093/imrn/rnx292.  Google Scholar [31] J. G. Ratcliffe, Foundations of hyperbolic manifolds, second edition, Graduate Texts in Mathematics, 149, Springer, New York, 2006.  Google Scholar [32] M. Ratner, On Raghunathan's measure conjecture, Ann. of Math., 134 (1991), 545-607.  doi: 10.2307/2944357.  Google Scholar [33] T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), 95 (2003). doi: 10.24033/msmf.408.  Google Scholar [34] O. Sarig, Invariant Radon measures for horocycle flows on Abelian covers, Invent. Math., 157 (2004), 519-551.  doi: 10.1007/s00222-004-0357-4.  Google Scholar [35] O. Sarig, The horocyclic flow and the Laplacian on hyperbolic surfaces of infinite genus, Geom. Funct. Anal., 19 (2010), 1757-1812.  doi: 10.1007/s00039-010-0048-9.  Google Scholar [36] G. P. Scott, Compact submanifolds of $3$-manifolds, J. London Math. Soc., 7 (1973), 246-250.  doi: 10.1112/jlms/s2-7.2.246.  Google Scholar [37] H. Shimizu, On discontinuous groups operating on the product of the upper half planes, Ann. of Math. (2), 77 (1963), no. 1, 33–71. doi: 10.2307/1970201.  Google Scholar [38] A. N. Starkov, Parabolic fixed points of Kleinian groups and the horospherical foliation on hyperbolic manifolds, Internat. J. Math., 8 (1997), no. 2,289–299. doi: 10.1142/S0129167X97000135.  Google Scholar [39] D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math., 153 (1984), 259-277.  doi: 10.1007/BF02392379.  Google Scholar [40] D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom., 25 (1987), 327-351.   Google Scholar [41] W. P. Thurston, The Geometry and Topology of Three-Manifolds, Princeton University, Princeton, NJ, 1979. Google Scholar [42] W. P. Thurston, Three-Dimensional Geometry and Topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997.  Google Scholar [43] W. A. Veech, Unique ergodicity of horospherical flows, Amer. J. Math., 99 (1977), 827-859.  doi: 10.2307/2373868.  Google Scholar [44] D. Winter, Mixing of frame flow for rank one locally symmetric spaces and measure classification, Israel J. Math., 210 (2015), 467-507.  doi: 10.1007/s11856-015-1258-5.  Google Scholar
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