Non-classical generating sets in Fuchsian Schottky groups
Absos Ali Shaikh, Uddhab Roy
TL;DR
This work addresses the existence and explicit construction of non-classical generating sets for Fuchsian Schottky groups, extending the non-classical Schottky paradigm from Kleinian groups to the Fuchsian setting. By formulating a rank-2 non-classical framework in the upper-half plane and introducing four hollow half-moon Jordan curves alongside two hyperbolic Möbius generators, the authors derive a rigorous contraction-based approach to prove non-classicality for small parameters. The main result, Theorem 1, shows that the group generated by ${h^*}_{(3,2)_{\kappa}}$ and ${h^{**}}_{(4,1)_{\kappa}}$ is non-classical for $\kappa<10^{-11}$, with corollaries giving two explicit single-parameter families of examples in the upper-half plane for $\lambda=2$ and $\lambda=5/3$ and $\kappa$ below tight thresholds. This advances understanding of non-classical generating structures in Fuchsian Schottky groups and provides concrete constructions for further geometric and group-theoretic analysis.
Abstract
The goal of this article is to initiate the study of estimates of the non-classical Schottky structure in the discrete subgroups of the projective special linear group over the real numbers degree $2$. In fact, in this paper, we have investigated the non-classical generating sets in the Fuchsian Schottky groups on the hyperbolic plane with boundary. A Schottky group is usually considered non-classical if the curves used in the Schottky construction are Jordan curves (except the Euclidean circles). More precisely, in this manuscript, we have provided a structure of the rank $2$ Fuchsian Schottky groups with non-classical generating sets by utilizing two suitable hyperbolic Möbius transformations on the upper-half plane model. In particular, we have derived two non-trivial examples of Fuchsian Schottky groups with non-classical generating sets in the upper-half plane with the circle at infinity as the boundary.