Gromov Conjecture on Surface Subgroups: Computational Experiments

TL;DR

This paper explores Gromov's conjecture that all one-ended word hyperbolic groups contain surface subgroups, using computational experiments on double groups with three generators to provide empirical evidence.

Contribution

It provides the first large-scale computational analysis supporting Gromov's conjecture for a broad class of hyperbolic groups.

Findings

96% of randomly selected double groups with three generators have surface subgroups

Experiments conducted using MAGMA software

Supports Gromov's conjecture with empirical data

Abstract

In this paper we investigate Gromov's question: whether every one-ended word hyperbolic group contains a surface subgroup. The case of double groups is considered by studying the associated one relator groups. We show that the majority (96%) of the randomly selected double groups with three generators have the property. The experiments are performed on MAGMA software.