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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    $${{\,\mathrm{CAT}\,}}(0)$$  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    $${{\,\mathrm{CAT}\,}}(0)$$  CAT ( 0 ) subgroups with isolated flats in non-hyperbolic    $${{\,\mathrm{CAT}\,}}(0)$$  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    $$\mathbb {Z}^2$$  Z 2 subgroups.

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