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

2018/10/31 by Benjamin Beeker, Beeker, Benjamin, Matthew Cordes +7 · 1 citation
Mathematics · #20E07 #20F65 #20F67 #57M07 #57M27 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1810.13285

openalex publication_date 2018/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ℤ² subgroups.

Citations

• Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle

Cited by

• Entropy for Generalized Parabolic Dynamics

Related

• Non-existence of Cannon-Thurston maps for hierarchically hyperbolic groups
• Cannon-Thurston maps do not always exist
• Cannon-Thurston Maps
• Cannon-Thurston maps, subgroup distortion, and hyperbolic hydra
• On Cannon-Thurston maps for relatively hyperbolic groups