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.

[2]

I. Agol, Tameness of hyperbolic 3-manifolds, preprint, arXiv: math/0405568.

[3]

J. W. AndersonK. 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.

[4]

B. N. Apanasov, Cusp ends of hyperbolic manifolds, Ann. Global Anal. Geom., 3 (1985), no. 1, 1–11. doi: 10.1007/BF00054488.

[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.

[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.

[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.

[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.

[9]

M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803.  doi: 10.1215/S0012-7094-90-06129-0.

[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.

[11]

R. D. Canary, Ends of hyperbolic $3$-manifolds, J. Amer. Math. Soc., 6 (1993), 1-35.  doi: 10.2307/2152793.

[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.

[13]

S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., 47 (1978), 101-138.  doi: 10.1007/BF01578067.

[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.

[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.

[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.

[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.

[18]

O. Landesberg and E. Lindenstrauss, On radon measures invariant under horospherical flows on geometrically infinite quotients, preprint, arXiv: 1910.08956.

[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.

[20]

F. Ledrappier, Horospheres on abelian covers, Bol. Soc. Brasil. Mat. (N.S.), 29 (1998), 195-195.  doi: 10.1007/BF01245874.

[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.

[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.

[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.

[24]

A. Marden, Hyperbolic Manifolds. An Introduction in 2 and 3 Dimensions, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316337776.

[25]

S. Matsumoto, Horocycle flows without minimal sets, J. Math. Sci. Univ. Tokyo, 23 (2016), 661-673. 

[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.

[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.

[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.

[29]

A. Mohammadi and H. Oh, Classification of joinings for Kleinian groups, Duke Math. J., 165 (2016), 2155-2223.  doi: 10.1215/00127094-3476807.

[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.

[31]

J. G. Ratcliffe, Foundations of hyperbolic manifolds, second edition, Graduate Texts in Mathematics, 149, Springer, New York, 2006.

[32]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math., 134 (1991), 545-607.  doi: 10.2307/2944357.

[33]

T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), 95 (2003). doi: 10.24033/msmf.408.

[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.

[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.

[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.

[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.

[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.

[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.

[40]

D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom., 25 (1987), 327-351. 

[41]

W. P. Thurston, The Geometry and Topology of Three-Manifolds, Princeton University, Princeton, NJ, 1979.

[42]

W. P. Thurston, Three-Dimensional Geometry and Topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997.

[43]

W. A. Veech, Unique ergodicity of horospherical flows, Amer. J. Math., 99 (1977), 827-859.  doi: 10.2307/2373868.

[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.

show all references

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.

[2]

I. Agol, Tameness of hyperbolic 3-manifolds, preprint, arXiv: math/0405568.

[3]

J. W. AndersonK. 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.

[4]

B. N. Apanasov, Cusp ends of hyperbolic manifolds, Ann. Global Anal. Geom., 3 (1985), no. 1, 1–11. doi: 10.1007/BF00054488.

[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.

[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.

[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.

[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.

[9]

M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803.  doi: 10.1215/S0012-7094-90-06129-0.

[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.

[11]

R. D. Canary, Ends of hyperbolic $3$-manifolds, J. Amer. Math. Soc., 6 (1993), 1-35.  doi: 10.2307/2152793.

[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.

[13]

S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., 47 (1978), 101-138.  doi: 10.1007/BF01578067.

[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.

[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.

[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.

[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.

[18]

O. Landesberg and E. Lindenstrauss, On radon measures invariant under horospherical flows on geometrically infinite quotients, preprint, arXiv: 1910.08956.

[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.

[20]

F. Ledrappier, Horospheres on abelian covers, Bol. Soc. Brasil. Mat. (N.S.), 29 (1998), 195-195.  doi: 10.1007/BF01245874.

[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.

[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.

[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.

[24]

A. Marden, Hyperbolic Manifolds. An Introduction in 2 and 3 Dimensions, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316337776.

[25]

S. Matsumoto, Horocycle flows without minimal sets, J. Math. Sci. Univ. Tokyo, 23 (2016), 661-673. 

[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.

[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.

[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.

[29]

A. Mohammadi and H. Oh, Classification of joinings for Kleinian groups, Duke Math. J., 165 (2016), 2155-2223.  doi: 10.1215/00127094-3476807.

[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.

[31]

J. G. Ratcliffe, Foundations of hyperbolic manifolds, second edition, Graduate Texts in Mathematics, 149, Springer, New York, 2006.

[32]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math., 134 (1991), 545-607.  doi: 10.2307/2944357.

[33]

T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), 95 (2003). doi: 10.24033/msmf.408.

[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.

[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.

[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.

[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.

[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.

[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.

[40]

D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom., 25 (1987), 327-351. 

[41]

W. P. Thurston, The Geometry and Topology of Three-Manifolds, Princeton University, Princeton, NJ, 1979.

[42]

W. P. Thurston, Three-Dimensional Geometry and Topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997.

[43]

W. A. Veech, Unique ergodicity of horospherical flows, Amer. J. Math., 99 (1977), 827-859.  doi: 10.2307/2373868.

[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.

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