scholarly journals Outer automorphism groups of right-angled Coxeter groups are either large or virtually abelian

2019 ◽  
Vol 372 (11) ◽  
pp. 7785-7803
Author(s):  
Andrew Sale ◽  
Tim Susse
Keyword(s):  
Coxeter Groups ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Outer Automorphism Groups

On outer automorphism groups of coxeter groups

1997 ◽  
Vol 93 (1) ◽  
pp. 499-513 ◽  
Author(s):  
R. B. Howlett ◽  
P. J. Rowley ◽  
D. E. Taylor
Keyword(s):  
Coxeter Groups ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Outer Automorphism Groups

scholarly journals OUTER AUTOMORPHISM GROUPS OF CERTAIN TREE PRODUCTS OF ABELIAN GROUPS

2008 ◽  
Vol 77 (1) ◽  
pp. 9-20 ◽  
Author(s):  
Y. D. CHAI ◽  
YOUNGGI CHOI ◽  
GOANSU KIM ◽  
C. Y. TANG
Keyword(s):  
Automorphism Groups ◽  
Abelian Groups ◽  
Outer Automorphism ◽  
Finitely Generated ◽  
Residually Finite ◽  
Outer Automorphism Groups

AbstractWe prove that certain tree products of finitely generated Abelian groups have Property E. Using this fact, we show that the outer automorphism groups of those tree products of Abelian groups and Brauner’s groups are residually finite.

scholarly journals Cannon–Thurston maps for $${{\,\mathrm{CAT}\,}}(0)$$ groups with isolated flats

2021 ◽  
Author(s):  
Beeker Benjamin ◽  
Matthew Cordes ◽  
Giles Gardam ◽  
Radhika Gupta ◽  
Emily Stark
Keyword(s):  
Automorphism Groups ◽  
Structure Theorem ◽  
Hyperbolic Group ◽  
Outer Automorphism ◽  
Hyperbolic Groups ◽  
Normal Subgroups ◽  
Infinite Index ◽  
Outer Automorphism Groups ◽  
Visual Boundaries

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 ) 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.

The Dehn-Nielsen-Baer Theorem

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Automorphism Groups ◽  
Harmonic Maps ◽  
Outer Automorphism ◽  
Mapping Class ◽  
Original Proof ◽  
Outer Automorphism Groups ◽  
And Algebra ◽  
Manifold Theory

This chapter deals with the Dehn–Nielsen–Baer theorem, one of the most beautiful connections between topology and algebra in the mapping class group. It begins by defining the objects in the statement of the Dehn–Nielsen–Baer theorem, including the extended mapping class group and outer automorphism groups. It then considers the use of the notion of quasi-isometry in Dehn's original proof of the Dehn–Nielsen–Baer theorem. In particular, it discusses a theorem on the fundamental observation of geometric group theory, along with the property of being linked at infinity. It also presents the proof of the Dehn–Nielsen–Baer theorem and an analysis of the induced homeomorphism at infinity before concluding with two other proofs of the Dehn–Nielsen–Baer theorem, one inspired by 3-manifold theory and one using harmonic maps.

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