Pullbacks of metric (graph) bundles and Cannon-Thurston maps

Abstract: Metric (graph) bundles were first defined by Mahan Mj and Pranab Sardar. Given a metric (graph) bundle $X$ over a hyperbolic space $B$ and a qi embedding $i: A\to B$, when $X$ and all the fibers are uniformly (Gromov) hyperbolic and nonelementary, the pullback $Y$ of $X$ exists and is hyperbolic. Moreover, a Cannon-Thurston map exists for the pullback map $i^*:Y \to X$. In this talk, we discuss an application of this theorem which shows that given a short exact sequence of nonelementary hyperbolic groups $1\to N\to G\stackrel{\pi}{\to} Q\to 1$ and a finitely generated qi embedded subgroup $Q_1 < Q$, $G_1:= \pi^{-1}(Q_1)$ is hyperbolic and the inclusion $G_1 \hookrightarrow G$ admits a Cannon Thurston map $\partial G_1\to \partial G$. This is part of joint work with Pranab Sardar.

group theory

Audience: researchers in the topic

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