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Finiteness of Cannon--Thurston fibers

Indranil Bhattacharyya, Rakesh Halder, Nir Lazarovich, Mahan Mj

Abstract

Let $Y\to X$ be a proper map between proper hyperbolic metric spaces. A Cannon--Thurston map is a continuous extension $\partial Y \to \partial X$. We prove that in most known settings in which a Cannon--Thurston map exists it is uniformly finite-to-one. This answers a question due to Swarup from Bestvina's problem list and generalizes previous results of Cannon--Thurston, Kapovich--Lustig, Dowdall--Kapovich--Taylor and Ghosh.

Finiteness of Cannon--Thurston fibers

Abstract

Let be a proper map between proper hyperbolic metric spaces. A Cannon--Thurston map is a continuous extension . We prove that in most known settings in which a Cannon--Thurston map exists it is uniformly finite-to-one. This answers a question due to Swarup from Bestvina's problem list and generalizes previous results of Cannon--Thurston, Kapovich--Lustig, Dowdall--Kapovich--Taylor and Ghosh.
Paper Structure (14 sections, 29 theorems, 13 equations)

This paper contains 14 sections, 29 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic metric spaces and barycenter maps
  4. Cannon--Thurston maps
  5. Trees of spaces
  6. Trees of hyperbolic metric spaces
  7. Boundary flows
  8. Finite fibers of Cannon--Thurston maps for trees of spaces
  9. Metric graph bundles
  10. Preliminaries on Metric graph bundles
  11. Cannon--Thurston fibers For metric graph bundles
  12. Applications and generalizations
  13. New proofs of old theorems
  14. Subtrees of spaces

Key Result

Theorem A

Let $X,Y$ be bounded valence graphs, satisfying one of the following: Then the Cannon--Thurston map $\partial Y\to \partial X$ of the natural inclusion $Y\to X$ is uniformly finite-to-one.

Theorems & Definitions (57)

  • Theorem A
  • Lemma 2.1
  • Definition 2.2: Barycenter map
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • Definition 3.1
  • Theorem 3.2
  • ...and 47 more