From local to global conjugacy of subgroups of relatively hyperbolic groups

2016/05/06 by Oleg Bogopolski, Bogopolski, Oleg, Kai-Uwe Bux +1
Mathematics · #20E45 #20F65 #20F67 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20E45 #msc:20F65 #msc:20F67

paper · pdf · doi:10.48550/arxiv.1605.01795

14 pages, 1 Figure. The proof in this version is shorter

arxiv created 2016/09/15 · arxiv updated 2016/09/19

Abstract

Suppose that a finitely generated group G is hyperbolic relative to a collection of subgroups ℙ=\P₁,…,P_m\. Let H₁,H₂ be subgroups of G such that H₁ is relatively quasiconvex with respect to ℙ and H₂ is not parabolic. Suppose that H₂ is elementwise conjugate into H₁. Then there exists a finite index subgroup of H₂ which is conjugate into H₁. The minimal length of the conjugator can be estimated. In the case where G is a limit group, it is sufficient to assume only that H₁ is a finitely generated and H₂ is an arbitrary subgroup of G.

Related

• Quasiconvexity in the Relatively Hyperbolic Groups
• Notes on relatively hyperbolic groups and relatively quasiconvex subgroups
• Detecting Relatively Quasiconvex Subgroups and Their Induced Peripheral Structure
• Conjugacy Search Problem for Relatively Hyperbolic Groups
• The universal relatively hyperbolic structure on a group and relative quasiconvexity for subgroups