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

TL;DR

This paper investigates the existence of Cannon--Thurston maps in CAT(0) groups with isolated flats, demonstrating their non-existence in certain non-hyperbolic cases and analyzing subgroup structures and automorphism groups.

Contribution

It extends the understanding of Cannon--Thurston maps by proving their non-existence in specific non-hyperbolic CAT(0) groups with isolated flats and provides a structure theorem for their normal subgroups.

Findings

Cannon--Thurston maps do not exist for certain non-hyperbolic CAT(0) groups with isolated flats.

Normal subgroup structures are characterized in these settings.

Outer automorphism groups lack purely atoroidal Z^2 subgroups.

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.