Construction of Fuchsian Schottky group with conformal boundary at infinity

TL;DR

This paper constructs generalized Fuchsian Schottky groups of arbitrary finite rank, explores their hyperbolic ends, and describes their Teichmüller space coordinates, extending classical Schottky group theory to include orientation-reversing isometries.

Contribution

It introduces a new class of Fuchsian Schottky groups with arbitrary finite rank, including orientation-reversing isometries, and analyzes their hyperbolic and conformal structures.

Findings

Construction of Fuchsian Schottky groups of arbitrary finite rank.

Analysis of hyperbolic ends via Euler characteristic.

Derivation of Fenchel-Nielsen coordinates for the associated Teichmüller space.

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 Teichm\"uller space corresponding to any finite rank Fuchsian Schottky groups are also obtained.