A surface-group action on real-hyperbolic space with no Cannon--Thurston map

Abstract

We present discrete, faithful and purely loxodromic representations of a closed surface group of genus 431, and of a free group, into the group of isometries of a finite-dimensional real-hyperbolic space for which no Cannon--Thurston map exists. The construction combines three ingredients: the small-cancellation strategy employed by Baker--Riley to construct a subgroup of an ambient hyperbolic group with no Cannon--Thurston map, the realization of these groups by octagon complexes, and a theorem of Desgroseilliers--Haglund that supplies a convex cocompact faithful representation of the ambient hyperbolic group on a real-hyperbolic space. These examples admit ray Cannon--Thurston maps, as defined by Halder, Mahan Mj and Sardar.