2017/10/20 by Grzegorz Gromadzki, Gromadzki, Grzegorz, Rubén A. Hidalgo +2 · 1 citation
Mathematics · #30F10 #30F40 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Point processes and geometric inequalities #math.GT #msc:30F10 #msc:30F40
paper · pdf · doi:10.48550/arxiv.1710.07518
openalex publication_date 2017/10/20 · arxiv created 2022/02/25 · arxiv updated 2022/02/28 · openalex created_date 2022/08/19 · openalex updated_date 2026/07/28
Given a symmetry τ of a closed Riemann surface S, there exists an extended Kleinian group K, whose orientation-preserving half is a Schottky group Γ uniformizing S, such that K/Γ induces ⟨ τ⟩; the group K is called an extended Schottky group. A geometrical structural description, in terms of the Klein-Maskit combination theorems, of both Schottky and extended Schottky groups is well known. A dihedral extended Schottky group is a group generated by the elements of two different extended Schottky groups, both with the same orientation-preserving half. Such configuration of groups corresponds to closed Riemann surfaces together with two different symmetries and the aim of this paper is to provide a geometrical structure of them. This result can be used in study of three dimensional manifolds and as an illustration we give the sharp upper bounds for the total number of connected components of the locus of fixed points of two and three different symmetries of a handlebody with a Schottky structure.