Structure of quasiconvex virtual joins

2023/09/30 by Lawk Mineh, Mineh, Lawk
Mathematics · #20F67 #Combinatorics #Computer science #Discrete mathematics #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Geometry #Group (periodic table) #Group Theory (math.GR) #Joins #Mathematical Dynamics and Fractals #Mathematics #Normal subgroup #Physics #Pure mathematics #Quasiconvex function

paper · pdf · doi:10.48550/arxiv.2310.00512

openalex publication_date 2023/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a relatively hyperbolic group and let Q and R be relatively quasiconvex subgroups. It is known that there are many pairs of finite index subgroups Q' \leqslant_f Q and R' \leqslant_f R such that the subgroup join ⟨ Q', R' ⟩ is also relatively quasiconvex, given suitable assumptions on the profinite topology of G. We show that the intersections of such joins with maximal parabolic subgroups of G are themselves joins of intersections of the factor subgroups Q' and R' with maximal parabolic subgroups of G. As a consequence, we show that quasiconvex subgroups whose parabolic subgroups are almost compatible have finite index subgroups whose parabolic subgroups are compatible, and provide a combination theorem for such subgroups.

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