# nLab hyperbolic 3-manifold

Contents

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## Idea

A Riemannian manifold which is both a 3-manifold and a hyperbolic manifold is a hyperbolic 3-manifold.

Equivalently this is a Riemannian manifold which is isometric to the quotient space $\mathbb{H}^3/ \Gamma$ of hyperbolic 3-space by the action of a torsion-free discrete group $\Gamma$ acting by isometries.

## Properties

### Volume conjecture

There is a curious relation of volumes of hyperbolic 3-manifolds to the action functional of Chern-Simons theory/Dijkgraaf-Witten theory (volume conjecture).

Let $G$ be a Lie group and $\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^3 U(1)$ a cocycle in degree-3 (generalized) Lie group cohomology. Write $\flat G$ for the underlying discrete group and $\flat \mathbf{c} \colon \mathbf{B} \flat G \to \mathbf{B}^3 \flat U(1)$ for the induced cocycle in ordinary (discrete) group cohomology, $[\flat \mathbf{c}] \in H^3_{Grp}(G_{disc},U(1)_{disc})$.

Then for $\Sigma$ a closed manifold of dimension 3, a map (of smooth infinity-groupoids) $\Sigma \to \mathbf{B}\flat G$ is a flat $G$-principal connection on $\Sigma$ and the composite

$[\Sigma, \mathbf{B}\flat G] \stackrel{[\Sigma, \flat \mathbf{c}]}{\to} [\Sigma, \mathbf{B}^3 \flat U(1)] \stackrel{\int_{\Sigma}}{\to} U(1)$

is the action functional for $G$-Chern-Simons theory on $\Sigma$ restricted to $G$-flat connections, or equivalently is the action functional of $\flat G$-Dijkgraaf-Witten theory.

Now for $G = SL(n,\mathbb{C})$ the complex special linear group and hence for Chern-Simons theory with complex gauge group, it turns out that the imaginary part of this flat Chern-Simons/Dijkgraaf-Witten invariant of 3-manifolds always has an expression as a combination of volumes of hyperbolic 3-manifolds.