An overview on complex Kleinian groups

TL;DR

This paper provides an overview of complex Kleinian groups, extending classical Kleinian group theory from $PSL(2, ext{C})$ to higher dimensions, highlighting its emerging status and potential in mathematics and physics.

Contribution

It introduces the study of discrete subgroups of $PSL(n, ext{C})$ for $n > 2$, an area still in early development, and summarizes current knowledge and open problems.

Findings

Classical Kleinian groups are well-studied and act on the Riemann sphere.

Higher-dimensional analogs of Kleinian groups are less developed and represent a new research frontier.

The paper highlights the importance of these groups in mathematics and physics.

Abstract

Classical Kleinian groups are discrete subgroups of $PSL(2,\C)$ acting on the complex projective line $\P^1$, which actually coincides with the Riemann sphere, with non-empty region of discontinuity. These can also be regarded as the monodromy groups of certain differential equations. These groups have played a major role in many aspects of mathematics for decades, and also in physics. It is thus natural to study discrete subgroups of the projective group $PSL(n,\C)$, $ n > 2$. Surprisingly, this is a branch of mathematics which is in its childhood, and in this article we give an overview of it.