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Singularity of Cannon-Thurston maps

Vaibhav Gadre, Joseph Maher, Catherine Pfaff, Caglar Uyanik

TL;DR

This work studies Cannon–Thurston maps for fibered hyperbolic 3–manifolds by comparing pushforwards of natural boundary measures on the fiber with natural boundary measures on the 3–manifold. The central achievement is establishing the singularity of these pushforward measures: surface measures on $S^1_ fty$ pushed forward via the Cannon–Thurston map become singular with respect to Lebesgue and hitting measures on $S^2_ fty$ associated to geometric/random walks on $pi_1(M)$. The authors develop a robust CT-metric framework, introducing height and radius functions and uniform quasi-geodesics to analyze geodesic behavior under different boundary measures. By proving that typical geodesics sampled from surface measures spend a definite positive proportion near the base fiber while those from 3–manifold measures do not, they derive mutual singularity and provide effective bounds for the restricted Lebesgue/hitting cases. The results shed light on the statistical geometry of Cannon–Thurston maps, offering explicit geometric criteria and a toolkit (ladders, extended laminations, bottlenecks) for broader Cannon–Thurston-type phenomena in hyperbolic groups and $3$-manifolds.

Abstract

In a closed fibered hyperbolic 3-manifold M, the inclusion of a fiber S, with S and M lifted to the universal covers gives an exponentially distorted embedding of the hyperbolic plane into hyperbolic 3-space. Nevertheless, Cannon and Thurston showed that there is a map from the circle at infinity of the hyperbolic plane to the 2-sphere at infinity of hyperbolic 3-space. The Cannon-Thurston map is surjective, finite-to-one, and gives a space-filling curve. Here we prove that many natural measures on the circle when pushed forward by the Cannon-Thurston map become singular with respect to many natural measures on the 2-sphere. The circle measures we consider are the Lebesgue measure and stationary measures that arise from fully supported random walks on the surface group. Whereas the measures on the sphere we consider are the Lebesgue measure and stationary measures that arise from geometric random walks on the 3-manifold group. The singularity of measures is ultimately derived from the following geometric result. We prove that a hyperbolic geodesic sampled with respect to a pushforward measure asymptotically spends a definite proportion of its time close to a fiber. On the other hand, we show that a hyperbolic geodesic sampled with respect to a natural measure on the sphere spends an asymptotically negligible proportion of its time close to a fiber. For a more restricted class of measures, namely the Lebesgue measure and stationary measures from geometric random walks on the surface group, we also prove an effective result for the proportion of time spent close to a fiber. To this end, we give precise descriptions of quasi-geodesics in the Cannon-Thurston metric, which may be of independent interest.

Singularity of Cannon-Thurston maps

TL;DR

This work studies Cannon–Thurston maps for fibered hyperbolic 3–manifolds by comparing pushforwards of natural boundary measures on the fiber with natural boundary measures on the 3–manifold. The central achievement is establishing the singularity of these pushforward measures: surface measures on pushed forward via the Cannon–Thurston map become singular with respect to Lebesgue and hitting measures on associated to geometric/random walks on . The authors develop a robust CT-metric framework, introducing height and radius functions and uniform quasi-geodesics to analyze geodesic behavior under different boundary measures. By proving that typical geodesics sampled from surface measures spend a definite positive proportion near the base fiber while those from 3–manifold measures do not, they derive mutual singularity and provide effective bounds for the restricted Lebesgue/hitting cases. The results shed light on the statistical geometry of Cannon–Thurston maps, offering explicit geometric criteria and a toolkit (ladders, extended laminations, bottlenecks) for broader Cannon–Thurston-type phenomena in hyperbolic groups and -manifolds.

Abstract

In a closed fibered hyperbolic 3-manifold M, the inclusion of a fiber S, with S and M lifted to the universal covers gives an exponentially distorted embedding of the hyperbolic plane into hyperbolic 3-space. Nevertheless, Cannon and Thurston showed that there is a map from the circle at infinity of the hyperbolic plane to the 2-sphere at infinity of hyperbolic 3-space. The Cannon-Thurston map is surjective, finite-to-one, and gives a space-filling curve. Here we prove that many natural measures on the circle when pushed forward by the Cannon-Thurston map become singular with respect to many natural measures on the 2-sphere. The circle measures we consider are the Lebesgue measure and stationary measures that arise from fully supported random walks on the surface group. Whereas the measures on the sphere we consider are the Lebesgue measure and stationary measures that arise from geometric random walks on the 3-manifold group. The singularity of measures is ultimately derived from the following geometric result. We prove that a hyperbolic geodesic sampled with respect to a pushforward measure asymptotically spends a definite proportion of its time close to a fiber. On the other hand, we show that a hyperbolic geodesic sampled with respect to a natural measure on the sphere spends an asymptotically negligible proportion of its time close to a fiber. For a more restricted class of measures, namely the Lebesgue measure and stationary measures from geometric random walks on the surface group, we also prove an effective result for the proportion of time spent close to a fiber. To this end, we give precise descriptions of quasi-geodesics in the Cannon-Thurston metric, which may be of independent interest.

Paper Structure

This paper contains 54 sections, 127 theorems, 219 equations, 27 figures.

Table of Contents

  1. Introduction
  2. Statistics for typical geodesics
  3. Remarks on Related Work
  4. Acknowledgments
  5. Singularity of measures
  6. Background
  7. Measured laminations
  8. Flow sets and ladders
  9. The Cannon-Thurston metric
  10. Separation for ladders
  11. Quasigeodesics
  12. Nearest point projections and fellow traveling
  13. Lebesgue measures and the geodesic flow
  14. Random walks and hitting measures
  15. The pseudo-metric on the surface and the flat metric
  16. ...and 39 more sections

Key Result

Theorem 4

Suppose that $M$ is a closed hyperbolic $3$-manifold that fibers over the circle. Then pushforwards to $S^2_\infty$ under the Cannon-Thurston map of any full surface measure from def:surface general (using any hyperbolic structure $S_h$ on the fiber $S$) is mutually singular with respect to any $3$-

Figures (27)

  • Figure 1: Rescaling arising from the vertical flow in the Cannon-Thurston metric.
  • Figure 2: Ladders over leaves are quasi-isometric to $\mathbb{H}^2$.
  • Figure 3: Notation for the complements of the leaves $\ell_i$ and ladders $F(\ell_i)$.
  • Figure 4: Extended leaves in a complementary region with four sides.
  • Figure 5: Two ideal polygons intersecting in a non-rectangular polygon.
  • ...and 22 more figures

Theorems & Definitions (249)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 4
  • Corollary 5
  • proof
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Definition 9
  • ...and 239 more