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Mahan Mitra (Mj) proved Cannon--Thurston maps exist for normal hyperbolicsubgroups of a hyperbolic group. We prove that Cannon--Thurston maps do notexist for infinite normal hyperbolic subgroups of non-hyperbolic CAT(0) groupswith isolated flats with respect to the visual boundaries. We also showCannon--Thurston maps do not exist for infinite infinite-index normal CAT(0)subgroups with isolated flats in non-hyperbolic CAT(0) groups with isolatedflats. We obtain a structure theorem for the normal subgroups in these settingsand show that outer automorphism groups of hyperbolic groups have no purelyatoroidal $\mathbb{Z}^2$ subgroups.

Cannon--Thurston maps for CAT(0) groups with isolated flats