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

Abstract

Mahan Mitra (Mj) proved Cannon--Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group. We prove that Cannon--Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic 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 CAT(0) subgroups with isolated flats in non-hyperbolic 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.

arXiv commentv2: 20 pages, 1 figure. The main theorems now cover normal subgroups that have torsion

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• JSJ decompositions of groups
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