On Geometry of Coned-Off Spaces and Cannon-Thurston Maps

Abstract

A typical question addressed in this paper is the following. Suppose $Z\subset Y\subset X$ are hyperbolic spaces where $Z$ is quasiconvex in both $Y$ and $X$. Let $\HAT{Y}$ and $\HAT{X}$ denote the spaces obtained from $Y$ and $X$ respectively by coning off $Z$ as defined by Farb. {\em If the inclusion of the coned-off spaces $\HAT{Y}\map \HAT{X}$ admits the Cannon-Thurston (CT) map then does the inclusion $Y\map X$ also admit the Cannon-Thurston map?} The main result of this paper answers this question affirmatively provided $\HAT{Y}\map \HAT{X}$ satisfies Mitra's criterion for the existence of CT maps, although the answer in general is negative. The main application of our theorem is in the context of acylindrical complexes of hyperbolic groups. A. Martin proved a combination theorem for developable, acylindrical complexes of hyperbolic groups. Suppose $(\mathcal G, \YY)$ is an acylindrical complex of hyperbolic groups with universal cover $B$ which satisfy the hypotheses of Martin's theorem. Suppose $\YY_1\subset \YY$ is a connected subcomplex such that the subcomplex of groups $(\mathcal G, \YY_1)$ also satisfies the hypotheses of Martin's theorem, it has universal cover $B_1$ and the natural homomorphism $\pi_1(\mathcal G, \YY_1)\map \pi_1(\mathcal G, \YY)$ is injective. It follows from the main theorem of this paper that the inclusion $\pi_1(\mathcal G, \YY_1)\map \pi_1(\mathcal G, \YY)$ admits the CT map if the inclusion $B_1\rightarrow B$ satisfies Mitra's criterion. Also $\pi_1(\mathcal G, \YY_1)$ is quasiconvex in $\pi_1(\mathcal G, \YY)$ if in addition $B_1$ is qi embedded in $B$.