August 2017, 11: 155-188. doi: 10.3934/jmd.2017008

On the rate of equidistribution of expanding horospheres in finite-volume quotients of SL(2, ${\mathbb{C}}$)

Department of Mathematics, Box 480, Uppsala University, SE-75106 Uppsala, Sweden

Received  December 22, 2015 Revised  December 02, 2016 Published  February 2017

Let $\Gamma$ be a lattice in $G=\mathrm{SL}(2, \mathbb{C})$. We give an effective equidistribution result with precise error terms for expanding translates of pieces of horospherical orbits in $\Gamma\backslash G$. Our method of proof relies on the theory of unitary representations.

Citation: Samuel C. Edwards. On the rate of equidistribution of expanding horospheres in finite-volume quotients of SL(2, ${\mathbb{C}}$). Journal of Modern Dynamics, 2017, 11: 155-188. doi: 10.3934/jmd.2017008
References:
[1]

J. Bernstein and A. Reznikov, Analytic continuation of representations and estimates of automorphic forms, Ann. of Math, 150 (1999), 329-352. doi: 10.2307/121105.

[2]

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

[3]

J. Conway and N. Sloane, Sphere Packings, Lattices and Groups, Springer, 1988. doi: 10.1007/978-1-4757-2016-7.

[4]

J. Elstrodt, F. Grunewald and J. Mennicke, Groups Acting on Hyperbolic Space, Springer Verlag, 1998. doi: 10.1007/978-3-662-03626-6.

[5]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.

[6] G. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995.
[7]

Harish-Chandra, Automorphic Forms on Semisimple Lie Groups, Springer-Verlag, (1968).

[8]

D. Hejhal, On the uniform equidistribution of long closed horocycles. in Loo-Keng Hua: a great mathematician of the twentieth century, Asian J. Math., 4 (2000), 839-853. doi: 10.4310/AJM.2000.v4.n4.a8.

[9]

R. Howe and C. Moore, Asymptotic properties of unitary representations, J. Funct. Anal., 32 (1979), 72-96. doi: 10.1016/0022-1236(79)90078-8.

[10]

D. Kleinbock and G. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Sinai's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, Amer.Math. Soc., Providence, RI, 171 (1996), 141–172. doi: 10.1090/trans2/171/11.

[11] A. Knapp, Representation Theory of Semisimple Lie Groups, Princeton University Press, 1986. doi: 10.1515/9781400883974.
[12]

A. Knapp, Lie Groups Beyond an Introduction, Second Edition, Birkhäuser, 2002.

[13]

A. Kontorovich and H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds, J. of AMS, 24 (2011), 603-648. doi: 10.1090/S0894-0347-2011-00691-7.

[14]

M. Lee and H. Oh, Effective circle count for Apollonian packings and closed horospheres, Geom. Funct. Anal., 23 (2013), 580-621. doi: 10.1007/s00039-013-0217-8.

[15]

M. Melián and D. Pestana, Geodesic excursions into cusps in finite-volume hyperbolic manifolds, Michigan Math. J., 40 (1993), 77-93. doi: 10.1307/mmj/1029004675.

[16]

P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and eisenstein series, Comm. Pure Appl. Math., 34 (1981), 719-739. doi: 10.1002/cpa.3160340602.

[17]

Y. Shalom, Rigidity, unitary representations of semisimple groups. and fundamental groups of manifolds with rank one transformation group, Ann. of Math., 152 (2000), 113-182. doi: 10.2307/2661380.

[18]

A. Strömbergsson, On the deviation of ergodic averages for horocycle flows, Journal of Modern Dynamics, 7 (2013), 291-328. doi: 10.3934/jmd.2013.7.291.

[19]

Strömbergsson A., On the uniform equidistribution of long closed horocycles, Duke Math.J., 123 (2004), 507-547. doi: 10.1215/S0012-7094-04-12334-6.

[20]

A. Södergren, On the uniform equidistribution of closed horospheres in hyperbolic manifolds, Proc. Lond. Math. Soc.(3), 105 (2012), 228-280. doi: 10.1112/plms/pdr052.

[21]

I. Vinogradov, Effective bisector estimate with application to apollonian circle packings, Int. Math. Res. Not. IMRN, 12 (2014), 3217-3262.

[22] N. Wallach, Real Reductive Groups Ⅱ, Academic Press, 1992.
[23]

D. Zagier, Eisenstein series and the Riemann zeta function, in Automorphic Forms, Repre sentation Theory and Arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., 10, Tata Inst. Fundamental Res., Bombay, 1981,275–301.

show all references

References:
[1]

J. Bernstein and A. Reznikov, Analytic continuation of representations and estimates of automorphic forms, Ann. of Math, 150 (1999), 329-352. doi: 10.2307/121105.

[2]

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

[3]

J. Conway and N. Sloane, Sphere Packings, Lattices and Groups, Springer, 1988. doi: 10.1007/978-1-4757-2016-7.

[4]

J. Elstrodt, F. Grunewald and J. Mennicke, Groups Acting on Hyperbolic Space, Springer Verlag, 1998. doi: 10.1007/978-3-662-03626-6.

[5]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.

[6] G. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995.
[7]

Harish-Chandra, Automorphic Forms on Semisimple Lie Groups, Springer-Verlag, (1968).

[8]

D. Hejhal, On the uniform equidistribution of long closed horocycles. in Loo-Keng Hua: a great mathematician of the twentieth century, Asian J. Math., 4 (2000), 839-853. doi: 10.4310/AJM.2000.v4.n4.a8.

[9]

R. Howe and C. Moore, Asymptotic properties of unitary representations, J. Funct. Anal., 32 (1979), 72-96. doi: 10.1016/0022-1236(79)90078-8.

[10]

D. Kleinbock and G. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Sinai's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, Amer.Math. Soc., Providence, RI, 171 (1996), 141–172. doi: 10.1090/trans2/171/11.

[11] A. Knapp, Representation Theory of Semisimple Lie Groups, Princeton University Press, 1986. doi: 10.1515/9781400883974.
[12]

A. Knapp, Lie Groups Beyond an Introduction, Second Edition, Birkhäuser, 2002.

[13]

A. Kontorovich and H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds, J. of AMS, 24 (2011), 603-648. doi: 10.1090/S0894-0347-2011-00691-7.

[14]

M. Lee and H. Oh, Effective circle count for Apollonian packings and closed horospheres, Geom. Funct. Anal., 23 (2013), 580-621. doi: 10.1007/s00039-013-0217-8.

[15]

M. Melián and D. Pestana, Geodesic excursions into cusps in finite-volume hyperbolic manifolds, Michigan Math. J., 40 (1993), 77-93. doi: 10.1307/mmj/1029004675.

[16]

P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and eisenstein series, Comm. Pure Appl. Math., 34 (1981), 719-739. doi: 10.1002/cpa.3160340602.

[17]

Y. Shalom, Rigidity, unitary representations of semisimple groups. and fundamental groups of manifolds with rank one transformation group, Ann. of Math., 152 (2000), 113-182. doi: 10.2307/2661380.

[18]

A. Strömbergsson, On the deviation of ergodic averages for horocycle flows, Journal of Modern Dynamics, 7 (2013), 291-328. doi: 10.3934/jmd.2013.7.291.

[19]

Strömbergsson A., On the uniform equidistribution of long closed horocycles, Duke Math.J., 123 (2004), 507-547. doi: 10.1215/S0012-7094-04-12334-6.

[20]

A. Södergren, On the uniform equidistribution of closed horospheres in hyperbolic manifolds, Proc. Lond. Math. Soc.(3), 105 (2012), 228-280. doi: 10.1112/plms/pdr052.

[21]

I. Vinogradov, Effective bisector estimate with application to apollonian circle packings, Int. Math. Res. Not. IMRN, 12 (2014), 3217-3262.

[22] N. Wallach, Real Reductive Groups Ⅱ, Academic Press, 1992.
[23]

D. Zagier, Eisenstein series and the Riemann zeta function, in Automorphic Forms, Repre sentation Theory and Arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., 10, Tata Inst. Fundamental Res., Bombay, 1981,275–301.

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