Finiteness of Cannon--Thurston fibers

TL;DR

This paper proves that in most known cases where a Cannon--Thurston map exists between hyperbolic spaces, it is uniformly finite-to-one, answering a longstanding question and extending previous results.

Contribution

It establishes the finiteness property of Cannon--Thurston fibers in a broad class of hyperbolic space maps, generalizing prior work.

Findings

Cannon--Thurston maps are uniformly finite-to-one in most known settings.

Answers Swarup's question on the finiteness of fibers.

Extends previous results by Cannon--Thurston, Kapovich--Lustig, Dowdall--Kapovich--Taylor, and Ghosh.

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.