Schottky groups cannot act on $\mathbb{P}^{2n}_{\mathbb{C}}$ as subgroups of $PSL(2n+1,\Bbb{C})$

TL;DR

This paper proves that Schottky groups, a special class of discrete subgroups of complex projective linear groups, only exist in odd dimensions and cannot be embedded in even-dimensional complex projective spaces.

Contribution

It establishes a fundamental restriction on the dimensions in which Schottky groups can act as subgroups of complex projective linear groups.

Findings

Schottky groups only occur in odd dimensions

They cannot be realized as subgroups of $PSL(2n+1,\mathbb{C})$ for even dimensions

Basic properties of these groups and their limit sets are developed

Abstract

In this paper we look at a special type of discrete subgroups of $PSL_{n+1}(\Bbb{C})$ called Schottky groups. We develop some basic properties of these groups and their limit set when $n > 1$, and we prove that Schottky groups only occur in odd dimensions, {\it i.e.}, they cannot be realized as subgroups of $PSL_{2n+1}(\Bbb{C})$.