The Surface Group Conjectures for groups with two generators

TL;DR

This paper proves the Surface Group Conjectures for two-generator groups, showing that certain one-relator groups with free infinite-index subgroups are either free or surface groups, advancing understanding of their subgroup structures.

Contribution

It resolves the Surface Group Conjectures in the two-generator case and characterizes two-generator one-relator groups with all infinite-index subgroups free.

Findings

Every two-generator one-relator group with all infinite-index subgroups free is either free or a surface group.

The Surface Group Conjectures are confirmed for the two-generator case.

Provides a classification of two-generator one-relator groups based on subgroup properties.

Abstract

The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.