One-relator hierarchies
Abstract
We prove that one-relator groups with negative immersions are hyperbolic and virtually special; this resolves a recent conjecture of Louder and Wilton. As a consequence, one-relator groups with negative immersions are residually finite, linear and have isomorphism problem decidable among one-relator groups. Using the fact that parafree one-relator groups have negative immersions, we answer a question of Baumslag's from 1986. The main new tool we develop is a refinement of the classic Magnus--Moldavanskii hierarchy for one-relator groups. We introduce the notions of Z-stable HNN-extensions and Z-stable hierarchies. We then show that a one-relator group is hyperbolic and has a quasi-convex one-relator hierarchy if and only if it does not contain a Baumslag--Solitar subgroup and has a Z-stable one-relator hierarchy.
Keywords
Cite
@article{arxiv.2202.11324,
title = {One-relator hierarchies},
author = {Marco Linton},
journal= {arXiv preprint arXiv:2202.11324},
year = {2024}
}
Comments
35 pages, 6 figures. Incorporated referees' comments and corrections. Most of section 5 has been removed, but still appears in the author's thesis. This is the final version to appear in Duke Math. J