A sufficiently complicated noded Schottky group of rank three

TL;DR

This paper constructs an explicit example of a sufficiently complicated noded Schottky group of rank three, advancing the understanding of non-classical Schottky groups beyond rank two.

Contribution

It provides the first explicit construction of a rank three sufficiently complicated noded Schottky group, enabling the creation of non-classical Schottky groups of the same rank.

Findings

Explicit example of a rank three noded Schottky group provided

Method to construct non-classical Schottky groups of rank three explained

Advances the understanding of higher-rank non-classical Schottky groups

Abstract

The theoretical existence of non-classical Schottky groups is due to Marden. Explicit examples of such kind of groups are only known in rank two, the first one by by Yamamoto in 1991 and later by Williams in 2009. In 2006, Maskit and the author provided a theoretical method to obtain examples of non-classical Schottky groups in any rank. The method assumes the knowledge of some algebraic limits of Schottky groups, called sufficiently complicated noded Schottky groups, whose existence was stated. In this paper we provide an explicit construction of a sufficiently complicated noded Schottky group of rank three and it is explained how to construct explicit non-classical Schottky groups of rank three.