×

On the rate of equidistribution of expanding horospheres in finite-volume quotients of \(\mathrm{SL}(2,\mathbb{C})\). (English) Zbl 1402.37043

Summary: 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 \(\Gamma\setminus G\). Our method of proof relies on the theory of unitary representations.

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37A17 Homogeneous flows
11F03 Modular and automorphic functions
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] J. Bernstein; A. Reznikov, Analytic continuation of representations and estimates of automorphic forms, Ann. of Math, 150, 329-352 (1999) · Zbl 0934.11023 · doi:10.2307/121105
[2] M. Burger, Horocycle flow on geometrically finite surfaces, Duke Mathematical Journal, 61, 779-803 (1990) · Zbl 0723.58041 · doi:10.1215/S0012-7094-90-06129-0
[3] J. Conway and N. Sloane, Sphere Packings, Lattices and Groups, Springer, 1988. · Zbl 0634.52002
[4] J. Elstrodt, F. Grunewald and J. Mennicke, Groups Acting on Hyperbolic Space, Springer Verlag, 1998. · Zbl 0888.11001
[5] L. Flaminio; G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119, 465-526 (2003) · Zbl 1044.37017 · doi:10.1215/S0012-7094-03-11932-8
[6] Automorphic Forms on Semisimple Lie Groups, Springer-Verlag (1968)
[7] 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, 839-853 (2000) · Zbl 1014.11038 · doi:10.4310/AJM.2000.v4.n4.a8
[8] R. Howe; C. Moore, Asymptotic properties of unitary representations, J. Funct. Anal., 32, 72-96 (1979) · Zbl 0404.22015 · doi:10.1016/0022-1236(79)90078-8
[9] 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. · Zbl 0843.22027
[10] A. Knapp, Lie Groups Beyond an Introduction, Second Edition, Birkhäuser, 2002. · Zbl 1075.22501
[11] A. Kontorovich; H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds, J. of AMS, 24, 603-648 (2011) · Zbl 1235.22015 · doi:10.1090/S0894-0347-2011-00691-7
[12] M. Lee; H. Oh, Effective circle count for Apollonian packings and closed horospheres, Geom. Funct. Anal., 23, 580-621 (2013) · Zbl 1276.52019 · doi:10.1007/s00039-013-0217-8
[13] M. Melián; D. Pestana, Geodesic excursions into cusps in finite-volume hyperbolic manifolds, Michigan Math. J., 40, 77-93 (1993) · Zbl 0793.53052 · doi:10.1307/mmj/1029004675
[14] P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and eisenstein series, Comm. Pure Appl. Math., 34, 719-739 (1981) · Zbl 0501.58027 · doi:10.1002/cpa.3160340602
[15] Y. Shalom, Rigidity, unitary representations of semisimple groups. and fundamental groups of manifolds with rank one transformation group, Ann. of Math., 152, 113-182 (2000) · Zbl 0970.22011 · doi:10.2307/2661380
[16] A. Strömbergsson, On the deviation of ergodic averages for horocycle flows, Journal of Modern Dynamics, 7, 291-328 (2013) · Zbl 1360.37088 · doi:10.3934/jmd.2013.7.291
[17] Strömbergsson A., On the uniform equidistribution of long closed horocycles, Duke Math.J., 123, 507-547 (2004) · Zbl 1060.37023 · doi:10.1215/S0012-7094-04-12334-6
[18] A. Södergren, On the uniform equidistribution of closed horospheres in hyperbolic manifolds, Proc. Lond. Math. Soc.(3), 105, 228-280 (2012) · Zbl 1257.37026 · doi:10.1112/plms/pdr052
[19] I. Vinogradov, Effective bisector estimate with application to apollonian circle packings, Int. Math. Res. Not. IMRN, 12, 3217-3262 (2014) · Zbl 1296.11088
[20] 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.
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.