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Refined horoball counting and conformal measure for Kleinian group actions

Jonathan M. Fraser, Liam Stuart

Abstract

Parabolic fixed points form a countable dense subset of the limit set of a non-elementary geometrically finite Kleinian group with at least one parabolic element. Given such a group, one may associate a standard set of pairwise disjoint horoballs, each tangent to the boundary at a parabolic fixed point. The diameter of such a horoball can be thought of as the `inverse cost' of approximating an arbitrary point in the limit set by the associated parabolic point. A result of Stratmann and Velani allows one to count horoballs of a given size and, roughly speaking, for small $r>0$ there are $r^{-δ}$ many horoballs of size approximately $r$, where $δ$ is the Poincaré exponent of the group. We investigate localisations of this result, where we seek to count horoballs of size approximately $r$ inside a given ball $B(z,R)$. Roughly speaking, if $r \lesssim R^2$, then we obtain an analogue of the Stratmann-Velani result (normalised by the Patterson-Sullivan measure of $B(z,R)$). However, for larger values of $r$, the count depends in a subtle way on $z$. Our counting results have several applications, especially to the geometry of conformal measures supported on the limit set. For example, we compute or estimate several `fractal dimensions' of certain $s$-conformal measures for $s>δ$ and use this to examine continuity properties of $s$-conformal measures at $s=δ$.

Refined horoball counting and conformal measure for Kleinian group actions

Abstract

Parabolic fixed points form a countable dense subset of the limit set of a non-elementary geometrically finite Kleinian group with at least one parabolic element. Given such a group, one may associate a standard set of pairwise disjoint horoballs, each tangent to the boundary at a parabolic fixed point. The diameter of such a horoball can be thought of as the `inverse cost' of approximating an arbitrary point in the limit set by the associated parabolic point. A result of Stratmann and Velani allows one to count horoballs of a given size and, roughly speaking, for small there are many horoballs of size approximately , where is the Poincaré exponent of the group. We investigate localisations of this result, where we seek to count horoballs of size approximately inside a given ball . Roughly speaking, if , then we obtain an analogue of the Stratmann-Velani result (normalised by the Patterson-Sullivan measure of ). However, for larger values of , the count depends in a subtle way on . Our counting results have several applications, especially to the geometry of conformal measures supported on the limit set. For example, we compute or estimate several `fractal dimensions' of certain -conformal measures for and use this to examine continuity properties of -conformal measures at .
Paper Structure (17 sections, 15 theorems, 100 equations, 4 figures)

This paper contains 17 sections, 15 theorems, 100 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Kleinian groups, parabolic points and horoballs
  3. Notation
  4. Counting horoballs
  5. The Patterson-Sullivan measure $\mu_\delta$ and the global measure formula
  6. The $s$-conformal measures $\mu_s$
  7. Results and applications
  8. Main results: refined horoball counting
  9. Application: Diophantine approximation and counting rationals
  10. Application: global measure formulae for conformal measures
  11. Application: dimension theory of conformal measures
  12. Remaining proofs
  13. Proof of Theorem \ref{['localvelani']}
  14. Proof of Theorem \ref{['theoretical']}
  15. Proof of Theorem \ref{['intermediate']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

There exists $\tau \in(0,1)$ such that, for all sufficiently large $k \in \mathbb{N}$,

Figures (4)

  • Figure 1: An illustration showing how $\iota^{-1}(V_\lambda) \cap B(z,\tau^{k/2})$ is squeezed between two spheres tangent at $p_0$ (in this picture we have $\mathbf{k}(p_0)=1$).
  • Figure 2: An illustration in the case where $d=1$ showing how the horoballs in the set $B(x,R/3)$ are moved into the set $B(z,R)$ by using a parabolic map $f$ fixing 0. The dashed arc represents the circle that the horoball $H_p$ is dragged along by repeated applications of $f$.
  • Figure 3: A picture of the associated right-angled triangle.
  • Figure 4: Estimating $\rho(p_n, -\log R_n)$ from above by the hyperbolic distance between $(p_n)_{-\log R_n}$ and the 'tip' of $H_{p_n}$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Corollary 2.7
  • Theorem 2.8
  • proof
  • ...and 9 more