Cannon–Thurston maps for {{,textrm{CAT},}}(0) groups with isolated flats

Abstract

Mahan Mitra (Mj) proved Cannon–Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group (Mitra in Topology, 37(3):527–538, 1998). We prove that Cannon–Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic {{,textrm{CAT},}}(0) groups with isolated flats with respect to the visual boundaries. We also show Cannon–Thurston maps do not exist for infinite infinite-index normal {{,textrm{CAT},}}(0) subgroups with isolated flats in non-hyperbolic {{,textrm{CAT},}}(0) groups with isolated flats. We obtain a structure theorem for the normal subgroups in these settings and show that outer automorphism groups of hyperbolic groups have no purely atoroidal mathbb {Z}^2 subgroups.