Abstract
AbstractMahan 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
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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.
Funder
Zuckerman STEM Leadership Foundation
Israel Science Foundation
ETH Zurich Postdoctoral Fellowship
Deutsche Forschungsgemeinschaft
Azrieli Foundation
Publisher
Springer Science and Business Media LLC
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