# A brute force computer aided proof of an existence result about extremal hyperbolic surfaces

## TL;DR

This paper introduces a brute force computational method to construct explicit examples of extremal hyperbolic surfaces with maximal disc packings, addressing a complex problem in geometric topology.

## Contribution

The paper presents a novel brute force computational approach to explicitly find extremal hyperbolic surfaces with specific packing properties, expanding the toolkit for geometric topology research.

## Key findings

- Successfully produced explicit examples of extremal hyperbolic surfaces.
- Demonstrated the effectiveness of brute force methods in complex geometric constructions.
- Provided a general procedure applicable to all cases of the problem.

## Abstract

Extremal compact hyperbolic surfaces contain a packing of discs of the largest possible radius permitted by the topology of the surface. It is well known that arithmetic conditions on the uniformizing group are necessary for the existence of a second extremal packing in the same surface, but constructing explicit examples of this phenomenon is a complicated task. We present a brute force computational procedure that can be used to produce examples in all cases.