Connected components of Isom($\mathbb{H}^3$)-representations of non-orientable surfaces

TL;DR

This paper investigates the structure of representations of non-orientable surface groups into hyperbolic 3-space isometries, revealing a specific number of connected components distinguished by topological invariants.

Contribution

It establishes the exact count of connected components of these representations and characterizes them via Stiefel-Whitney classes.

Findings

There are 2^{k+1} connected components of representations.

Connected components are distinguished by Stiefel-Whitney classes.

The 'square map' behavior is analyzed in the context of hyperbolic isometries.

Abstract

Let $N_k$ denote the closed non-orientable surface of genus $k$. In this paper we study the behaviour of the `square map' from the group of isometries of hyperbolic 3-space to the subgroup of orientation preserving isometries. We show that there are $2^{k+1}$ connected components of representations of $\pi_1(N_k)$ in Isom$(\mathbb{H}^3)$, which are distinguished by the Stiefel-Whitney classes of the associated flat bundle.