On uniformly continuous endomorphisms of hyperbolic groups

2021/02/16 by Carvalho, André · 1 citation
#20E36 #20F67 #51M10 #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2102.08294

Abstract

We prove a generalization of the fellow traveller property for a certain type of quasi-geodesics and use it to present three equivalent geometric formulations of the bounded reduction property and prove that it is equivalent to preservation of a coarse median. We then provide an affirmative answer to a question from Araújo and Silva as to whether every nontrivial uniformly continuous endomorphism of a hyperbolic group with respect to a visual metric satisfies a Hölder condition. We remark that these results combined with the work done by Paulin prove that every endomorphism admitting a continuous extension to the completion has a finitely generated fixed point subgroup.

Cited by

• Conjugacy languages and conjugacy growth relative to subsets of groups

Related

• Maps between Boundaries of Relatively Hyperbolic Groups
• On endomorphisms of automatic groups
• Fixed points of endomorphisms of virtually free groups
• Martin boundary covers Floyd boundary
• On the quasiconvexity of the fixed subgroup of endomorphisms of relatively hyperbolic groups