# Rigidity at infinity for the Borel function of the tetrahedral reflection lattice

## TL;DR

This paper proves Guilloux's conjecture on the rigidity of the Borel function at infinity for a specific hyperbolic reflection group related to a regular ideal tetrahedron, confirming boundedness near ideal points.

## Contribution

It establishes Guilloux's conjecture for the reflection group of a regular ideal tetrahedron in hyperbolic 3-space, a special case previously unproven.

## Key findings

- The Borel function remains bounded away from its maximum at ideal points for the tetrahedral reflection group.
- Guilloux's conjecture holds in this specific geometric setting.
- The result advances understanding of character varieties in hyperbolic geometry.

## Abstract

If $\Gamma$ is the fundamental group of a complete finite volume hyperbolic $3$-manifold, Guilloux conjectured that the Borel function on the $\text{PSL}(n,\mathbb{C})$-character variety of $\Gamma$ should be rigid at infinity, that is it should stay bounded away from its maximum at ideal points. In this paper we prove Guilloux's conjecture in the particular case of the reflection group associated to a regular ideal tetrahedron of $\mathbb{H}^3$.