On the topology of character varieties of once-punctured torus bundles

TL;DR

This paper develops a method to analyze the character varieties of hyperbolic once-punctured torus bundles by restricting characters to the fiber, extending previous results, and demonstrating an infinite family with unbounded genus of character curves.

Contribution

It introduces a natural approach to study character varieties of these bundles, extending prior work and providing explicit examples with unbounded genus of character curves.

Findings

Extended results of Boyer, Luft, and Zhang on character varieties.

Constructed an infinite family of bundles with unbounded genus of PSL(2,C) character curves.

Analyzed both SL(2,C) and PSL(2,C) character varieties.

Abstract

This paper presents, for the special case of once-punctured torus bundles, a natural method to study the character varieties of hyperbolic 3-manifolds that are bundles over the circle. The main strategy is to restrict characters to the fibre of the bundle, and to analyse the resulting branched covering map. This allows us to extend results of Steven Boyer, Erhard Luft and Xingru Zhang. Both $SL(2, \mathbb{C})$-character varieties and $PSL(2, \mathbb{C})$-character varieties are considered. As an explicit application of these methods, we build on work of Baker and Petersen to show that there is an infinite family of hyperbolic once-punctured bundles with canonical curves of $PSL(2, \mathbb{C})$-characters of unbounded genus.