Cannon–thurston Maps Do Not Always exist

Abstract

We construct a hyperbolic group with a hyperbolic subgroup for which inclusion does not induce a continuous map of the boundaries.

Cannon–thurston Maps

“…A simple and basic criterion for the existence of Cannon-Thurston maps was established in [Mit98a,Mit98b]: In the generality above Question 2.7 turns out to have a negative answer. An explicit counterexample to Question 2.7 was recently found by Baker and Riley [BR13] in the context of small cancellation theory. The counterexample uses Lemma 2.8 to rule out the existence of Cannon-Thurston maps.…”

“…Can the small cancellation group of Baker-Riley in [BR13] act geometrically on a rank one symmetric space thus giving a negative answer to Question 6.13 with surface group replaced by free group? Work of Wise [Wis12,Wis04] guarantees linearity of such small cancellation groups.…”

“…However no systematic theory exists. In the light of the counterexample in [BR13], the general answer to Question 2.7 is negative. Are there necessary and/or sufficient conditions beyond Lemma 2.8 to guarantee existence of Cannon-Thurston maps?…”

Cannon–Thurston maps for hyperbolic free group extensions

“…is a short exact sequence of three infinite word-hyperbolic groups, then the Cannon-Thurston map ∂ι : ∂H → ∂G exists and is surjective. Only recently did the work of Baker and Riley [BR1] produce the first example of a word-hyperbolic subgroup H of a word-hyperbolic group G for which the inclusion H ≤ G does not extend to a Cannon-Thurston map. Analogs and generalizations of the Cannon-Thurston map have been studied in many other contexts, see for example [Kla,McM,Miy,LLR,LMS,Ger,Bow1,Bow2,MP,Mj2,Mj1,JKLO].…”

On Cannon–Thurston maps for relatively hyperbolic groups

“…Also it follows from [7, Theorem 0.1 (2)] that G 0 is hyperbolic relative to H 0 [ ¹K G 0 j K D g 0 L 0 g 0 1 for some g 0 2 G 0 º: Since L 0 is hyperbolic, we have condition (i) by [16,Theorem 2.40]. …”