Freiheitss\"{a}tze for one-relator quotients of surface groups and of limit groups

TL;DR

This paper extends Freiheitssatz results to one-relator quotients of limit groups with specific splittings, providing new proofs and broadening understanding of these algebraic structures.

Contribution

It introduces three versions of the Freiheitssatz for limit groups with 1-acylindrical splittings, including two novel proofs in the context of surface groups.

Findings

Proved three versions of the Freiheitssatz for limit groups

Extended previous results from surface groups to limit groups

Provided two new proofs even for the surface group case

Abstract

Three versions of the Freiheitssatz are proved in the context of one-relator quotients of limit groups, where the latter are equipped with 1-acylindrical splittings over cyclic subgroups. These are natural extensions of previously published corresponding statements for one-relator quotients of orientable surface groups. Two of the proofs are new even in that restricted context.