Finiteness of Cannon--Thurston fibers

Indranil Bhattacharyya; Rakesh Halder; Nir Lazarovich; Mahan Mj

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arXiv · 2603.22428

Finiteness of Cannon--Thurston fibers

Indranil Bhattacharyya·Rakesh Halder·Nir Lazarovich·Mahan Mj

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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.

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Indranil Bhattacharyya, Rakesh Halder, Nir Lazarovich, Mahan Mj. 2026-03-23. Finiteness of Cannon--Thurston fibers. https://arxiv.org/abs/2603.22428

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