Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T06:31:46.939Z Has data issue: false hasContentIssue false
  • English
  • Français

Directions in orbits of geometrically finite hyperbolic subgroups

Part of: Ergodic theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  23 November 2020

CHRISTOPHER LUTSKO
CHRISTOPHER LUTSKO*
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, U.S.A. e-mail: chris.lutsko@rutgers.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove a theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case with two recent infinite-volume exceptions by Zhang for Apollonian circle packings and certain Schottky groups. Our results hold for general Zariski dense, non-elementary, geometrically finite subgroups in any dimension. Unlike in the lattice case orbits of geometrically finite subgroups do not necessarily equidistribute on the whole boundary of hyperbolic space. But rather they may equidistribute on a fractal subset. Understanding the behavior of these orbits near the boundary is central to Patterson–Sullivan theory and much further work. Our theorem characterises the higher order spatial statistics and thus addresses a very natural question. As a motivating example our work applies to sphere packings (in any dimension) which are invariant under the action of such discrete subgroups. At the end of the paper we show how this statistical characterization can be used to prove convergence of moments and to write down the limiting formula for the two-point correlation function and nearest neighbor distribution. Moreover we establish a formula for the 2 dimensional limiting gap distribution (and cumulative gap distribution) which also applies in the lattice case.

MSC classification

Primary: 11F06: Structure of modular groups and generalizations; arithmetic groups
Secondary: 37A17: Homogeneous flows
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 2 , September 2021 , pp. 277 - 316
Copyright
© Cambridge Philosophical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

A subgroup is geometrically finite if the unit neighborhood of its convex core has finite Riemannian volume. Discrete groups whose fundamental domain is a finite-sided polygon are geometrically finite.

References

REFERENCES

Boca, F., Popa, A. and Zaharescu, A.. Pair correlation of hyperbolic lattice angles. Int. J. Number Theory, 10(8), (2014), 1955–1989.CrossRefGoogle Scholar
Borthwick, D.. Spectral theory of infinite-area hyperbolic surfaces. Prog. Math. (Birkhäuser Boston, 2007).Google Scholar
Bourgain, J., Gamburd, A. and Sarnak, P.. Generalisation of Selberg’s 3/16 theorem and affine sieve. Acta Mathematica, 207(2), (2011), 255–290.CrossRefGoogle Scholar
Bourgain, J., Kontorovich, A. and Sarnak, P.. Sector estimates for hyperbolic isometries. Geometric and Functional Analysis, 20(5), (2010), 1175–1200.CrossRefGoogle Scholar
El-Baz, D., Marklof, J. and Vinogradov, I.. The distribution of directions in an affine lattice: Two-point correlations and mixed moments. Internat. Math. Res. Not., 2015(5), 1371–1400.Google Scholar
Good, A.. Local analysis of Selberg’s trace formula . Lecture Notes in Math. (Springer-Verlag, 1983).Google Scholar
Kelmer, D. and Kontorovich, A.. On the pair correlation density for hyperbolic angles. Duke Math. J. 164(3), (2015), 473–509.CrossRefGoogle Scholar
Kontorovich, A.. The local-global principle for integral soddy sphere packings. arXiv:1208.5441, (2017).Google Scholar
Margulis, G.. On some aspects of the theory of Anosov systems . Springer Monographs in Mathematics. (Springer-Verlag, Berlin, 2004). With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska.Google Scholar
J. Marklof. Distribution modulo one and Ratner’s theorem. In Equidistribution in Number Theory, an Introduction, pages 217–244, (Dordrecht, 2007. Springer, Netherlands).Google Scholar
Marklof, J. and Strömbergsson, A.. The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math., 172, (2010), 1949–2033.CrossRefGoogle Scholar
Marklof, J. and Strömbergsson, A.. Free path lengths in quasicrystals. Communications in Mathematical Physics, 330(2), (2014), 723–755.CrossRefGoogle Scholar
Marklof, J. and Vinogradov, I.. Directions in hyperbolic lattices. J. Reine Angew. Math., 740, (2018), 161–186.CrossRefGoogle Scholar
Mohammadi, A.. Ergodic properties of the Burger-Roblin measure. In Thin groups and superstrong approximation, vol 61 (MSRI Publications, 2013).Google Scholar
Mohammadi, A. and Oh, H.. Matrix coefficients, counting and primes for orbits of geometrically finite groups. Journal of the EMS, 17, (2015), 837–897.Google Scholar
Oh, H.. Harmonic analysis, ergodic theory and counting for thin groups. In Breuillard, E. and Oh, H., editors, Thin groups and superstrong approximation. (Cambridge Univ. Press., 2014).Google Scholar
Oh, H. and Shah, N.. The asymptotic distribution of circles in the orbits of kleinian groups. Inventiones mathematicae, 187(1), (2012), 1–35.CrossRefGoogle Scholar
Oh, H. and Shah, N.. Equidistribution and counting for orbits of geometrically finite hyperbolic groups. J. Amer. Math. Soc., 26(2), 2013, 511–562.CrossRefGoogle Scholar
Patterson, S.. The limit set of a Fuchsian group. Acta Math., 136, (1976), 241–273.CrossRefGoogle Scholar
Risager, M. and Södergren, A.. Angles in hyperbolic lattices: The pair correlation density. Trans. Amer. Math. Soc., 369(4), (2017), 2807–2841.CrossRefGoogle Scholar
Rudnick, Z. and Zhang, X.. Gap distributions in circle packings. Münster J. Mathematics, 10, (2017), 131–170.Google Scholar
Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publi. Math. Inst. Houkes Étude Sc, 50, (1979), 171–202.CrossRefGoogle Scholar
Sullivan, D.. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math., 153, (1984), 259–277.CrossRefGoogle Scholar
Waterman, P.. Möbius tranformations in several dimensions. Adv. in Math., 101, (1993), 87–113.CrossRefGoogle Scholar
Winter, D.. Mixing of frame flow for rank one locally symmetric spaces and measure classification. Israel J. Math., 210(1), (2015), 467–507.CrossRefGoogle Scholar
Zhang, X.. The gap distribution of directions in some Schottky groups. J. Modern Dynamics, 11, (2017), 477.CrossRefGoogle Scholar
Zhang, X.. Statistical regularity of Apollonian gaskets. arXiv:1709.00676, (2018).Google Scholar