On neoclassical Schottky groups

TL;DR

This paper constructs an infinite collection of non-classical Schottky groups, demonstrating their abundance and complexity, and provides explicit examples of such groups and associated noded Riemann surfaces.

Contribution

It introduces a theoretical framework for constructing infinitely many non-classical Schottky groups and provides explicit examples, expanding understanding of their structure and boundary behavior.

Findings

Existence of infinitely many non-classical noded Schottky groups.

Identification of 'sufficiently complicated' non-classical groups near boundary points.

Explicit examples of non-classical noded Schottky groups in genus 3.

Abstract

The goal of this paper is to describe a theoretical construction of an infinite collection of non-classical Schottky groups. We first show that there are infinitely many non-classical noded Schottky groups on the boundary of Schottky space, and we show that infinitely many of these are "sufficiently complicated". We then show that every Schottky group in an appropriately defined relative conical neighborhood of any sufficiently complicated noded Schottky group is necessarily non-classical. Finally, we construct two examples; the first is a noded Riemann surface of genus $3$ that cannot be uniformized by any neoclassical Schottky group (i.e., classical noded Schottky group); the second is an explicit example of a sufficiently complicated noded Schottky group in genus $3$.