On conformally flat circle bundles over surfaces

TL;DR

This paper investigates conformally flat circle bundles over surfaces arising from surface groups in SO(4,1), establishing bounds on their Euler number and constructing new examples via a grafting path in the representation space.

Contribution

It provides new bounds on the Euler number of conformally flat circle bundles and introduces a novel grafting construction in the space of surface group representations.

Findings

Established soft bounds on the Euler number of conformally flat circle bundles.

Constructed new examples of surface group representations via grafting.

Explored the space of representations, including non-discrete regions.

Abstract

We study surface groups $\Gamma$ in $SO(4,1)$, which is the group of Mobius tranformations of $S^3$, and also the group of isometries of $\mathbb{H}^4$. We consider such $\Gamma$ so that its limit set $\Lambda_\Gamma$ is a quasi-circle in $S^3$, and so that the quotient $(S^3 - \Lambda_\Gamma) / \Gamma$ is a circle bundle over a surface. This circle bundle is said to be conformally flat, and our main goal is to discover how twisted such bundle may be by establishing a bound on its Euler number. By combinatorial approaches, we have two soft bounds in this direction on certain types of nice structures. In this article we also construct new examples, a "grafting" type path in the space of surface group representations into $SO(4,1)$: starting inside the quasi-Fuschsian locus, going through non-discrete territory and back.