Construction of Fuchsian Schottky group with conformal boundary at infinity
Absos Ali Shaikh, Uddhab Roy
TL;DR
The authors construct and analyze finite-rank Fuchsian Schottky groups, first in a purely hyperbolic setting and then with orientation-reversing side-pairings, to produce discrete free groups with Cantor limit sets and rich end structures. They study the associated hyperbolic surfaces via Nielsen regions, convex/compact cores, and Euler-characteristic constraints, revealing decompositions into pants and ends that yield novel surfaces such as Loch Ness Monster and Jacob's Ladder, and deriving explicit Fenchel-Nielsen coordinates for the corresponding Teichmüller spaces. A key contribution is the explicit non-tight pants decompositions with Bers constant bounds, together with a detailed account of the conformal boundary at infinity and the behavior of the limit set (including a small Hausdorff-dimension Cantor limit set and absence of cusps). Overall, the work extends Schottky- and Fuchsian-group theory by integrating orientation-reversing symmetries and providing concrete geometric and moduli-space descriptions for finite-rank constructions.
Abstract
In this article, we have constructed an interesting type of generalized Schottky group, named as Fuchsian Schottky group of arbitrary finite rank, in the context of the classical Schottky group (i.e., Schottky curves which are Euclidean circles). After that, we initiated the construction of the generalized Fuchsian Schottky group of any finite rank by including orientation-reversing isometries of the hyperbolic plane as side-pairing transformations. Further, we have investigated the hyperbolic ends for any arbitrary finite rank Fuchsian Schottky groups from the point of view of the Euler characteristic in the hyperbolic surface. Finally, we have shown that the compact core of the conformally compact Riemann surface can be decomposed into non-tight pairs of pants by using suitable twist parameters with some fixed Bers' constant. The Fenchel-Nielsen coordinates for Teichmüller space corresponding to any finite rank Fuchsian Schottky groups are also obtained.