Endomorphisms of relatively hyperbolic groups

2005/01/20 by Igor Belegradek, Belegradek, Igor, Andrzej Szczepański +5
Computer Science · Mathematics · #20F65 #Combinatorics #Endomorphism #FOS: Mathematics #Finite Group Theory Research #Finitely-generated abelian group #Free group #Geometric and Algebraic Topology #Group (periodic table) #Group Theory (math.GR) #Hyperbolic function #Hyperbolic group #Hyperbolic manifold #Mathematical analysis #Mathematics #Metric Geometry (math.MG) #Normal subgroup #Pure mathematics #Quotient #Relatively hyperbolic group #math.GR #math.MG #msc:20F65 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.math/0501321

final version

openalex publication_date 2005/01/20 · arxiv created 2007/08/11 · arxiv updated 2011/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. (1) If G is a finitely generated non-elementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group. (2) If a finitely generated non-parabolic subgroup H of a non-elementary relatively hyperbolic group is not Hopfian, then H acts non-trivially on an R-tree. (3) Every non-elementary relatively hyperbolic group has a non-elementary relatively hyperbolic quotient that is Hopfian. (4) Any finitely presented group is isomorphic to a finite index subgroup of Out(H) for some Kazhdan group H. (This sharpens a result of Ollivier-Wise).

Citations

• An Introduction to the Theory of Groups
• Tree-graded spaces and asymptotic cones of groups
• The Isomorphism Problem for Hyperbolic Groups I
• A topological characterisation of relatively hyperbolic groups
• Stable actions of groups on real trees
• Relatively Hyperbolic Groups
• Accidental parabolics and relatively hyperbolic groups
• Small cancellations over relatively hyperbolic groups and embedding theorems

Related

• A non-Hopfian relatively hyperbolic group with respect to a Hopfian subgroup
• On the quasiconvexity of the fixed subgroup of endomorphisms of relatively hyperbolic groups
• Splittings and automorphisms of relatively hyperbolic groups
• A recipe for constructing non-Hopfian relatively hyperbolic groups with Hopfian peripheral subgroups
• Notes on relatively hyperbolic groups and relatively quasiconvex subgroups