Counting cycles in labeled graphs: The nonpositive immersion property for one-relator groups

TL;DR

This paper proves that one-relator 2-complexes without torsion possess the nonpositive immersion property, extending the Hanna Neumann Theorem to a broader class of complexes and providing new insights into their structure.

Contribution

It establishes a rank 1 version of the Hanna Neumann Theorem for one-relator 2-complexes without torsion, generalizing to staggered and reducible complexes.

Findings

Every one-relator 2-complex without torsion has the nonpositive immersion property

The proof extends to staggered and reducible 2-complexes

Provides a new understanding of the structure of one-relator groups

Abstract

We prove a rank 1 version of the Hanna Neumann Theorem. This shows that every one-relator 2-complex without torsion has the nonpositive immersion property. The proof generalizes to staggered and reducible 2-complexes.