On Character Varieties with Non-Connected Structure Groups

TL;DR

This paper explores character varieties with non-connected structure groups, analyzing their properties, classifying related semi-direct products, and relating these to fixed points in connected group character varieties.

Contribution

It introduces a framework for character varieties with non-connected groups and classifies semi-direct product isomorphism classes, extending understanding of their structure.

Findings

Classification of semi-direct products $G\rtimes\Gamma$

Analysis of generic conjugacy classes

Relation between non-connected and connected structure group varieties

Abstract

We define character varieties with non-connected structure groups of finitely presented discrete groups and study some basic aspects, such as generic conjugacy classes and relation with fixed points in character varieties with connected structure groups. In the process, we give the classification of the isomorphism classes of the semi-direct products $G\rtimes\Gamma$, with fixed $G$ and $\Gamma$.