Relatively hyperbolic metric bundles and Cannon-Thurston map

Given a metric (graph) bundle \(X\) over \(B\) where all the fibres are strongly relatively hyperbolic and nonelementary we show that, under certain conditions, \(X\) is strongly hyperbolic relative to a collection of maximal cone-subbundles of horosphere-like spaces. Further, given a coarsely Lipsc...

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Bibliographic Details
Published in:arXiv.org 2022-04
Main Author: Swathi Krishna
Format: Article
Language:English
Online Access:Get full text
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Summary:Given a metric (graph) bundle \(X\) over \(B\) where all the fibres are strongly relatively hyperbolic and nonelementary we show that, under certain conditions, \(X\) is strongly hyperbolic relative to a collection of maximal cone-subbundles of horosphere-like spaces. Further, given a coarsely Lipschitz qi embedding \(i: A\to B\), we show that the pullback \(Y\) is strongly relatively hyperbolic and the map \(Y\to X\) admits a Cannon-Thurston (CT) map. As an application, we prove a group-theoretic analogue of this result for a relatively hyperbolic extension of groups.
ISSN:2331-8422