Pullbacks of metric bundles and Cannon–Thurston maps

Krishna, Swathi; Sardar, Pranab

doi:10.2140/agt.2025.25.2667

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Pullbacks of metric bundles and Cannon–Thurston maps

Swathi Krishna and Pranab Sardar

Algebraic & Geometric Topology 25 (2025) 2667–2755

DOI: 10.2140/agt.2025.25.2667

Abstract

Metric (graph) bundles were defined by Mj and Sardar (Geom. Funct. Anal. 22 (2012) 1636–1707). In this paper, we introduce the notion of morphisms and pullbacks of metric (graph) bundles. Given a metric (graph) bundle $X$ over $B$ where $X$ and all the fibers are uniformly (Gromov) hyperbolic and nonelementary, and a Lipschitz quasiisometric embedding $i : A \to B$ , we show that the pullback $i^{*} X$ is hyperbolic and the map $i^{*} : i^{*} X \to X$ admits a continuous boundary extension, ie the Cannon–Thurston (CT) map $\partial i^{*} : \partial (i^{*} X) \to \partial X$ . As an application of our theorem, we show that given a short exact sequence of nonelementary hyperbolic groups $1 \to N \to G \to_{}^{π} Q \to 1$ and a finitely generated quasiisometrically embedded subgroup $Q_{1} < Q$ , $G_{1} : = π^{- 1} (Q_{1})$ is hyperbolic and the inclusion $G_{1} \to G$ admits the CT map $\partial G_{1} \to \partial G$ . We then derive several interesting properties of the CT map.

Keywords

pullback, metric bundles, Cannon–Thurston map

Mathematical Subject Classification

Primary: 20F65

References

Publication

Received: 3 June 2021

Revised: 4 June 2024

Accepted: 23 July 2024

Published: 17 September 2025

Authors

Swathi Krishna
Department of Mathematics Indian Institute of Science Education and Research Mohali Mohali India

Pranab Sardar
Department of Mathematics Indian Institute of Science Education and Research Mohali Mohali India

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Volume 25, issue 5 (2025)