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

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 CAT ( 0 ) groups with isolated flats with resp...

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Bibliographic Details
Published inMathematische annalen Vol. 384; no. 1-2; pp. 1 - 25
Main Authors Benjamin, Beeker, Cordes, Matthew, Gardam, Giles, Gupta, Radhika, Stark, Emily
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 2022
Springer Nature B.V
Subjects
Automorphisms
Mathematics
Mathematics and Statistics
Subgroups
Topology
20F65
20E07
57M27
20F67
57M07
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Summary: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 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 Z 2 subgroups.
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-021-02245-z