The geometry of one-relator groups satisfying a polynomial isoperimetric inequality

TL;DR

This paper constructs specific one-relator groups with polynomial isoperimetric inequalities, exploring their geometric properties and providing counterexamples to existing conjectures about automaticity and CAT(0) actions.

Contribution

It introduces a family of one-relator groups with controlled Dehn functions, demonstrating they are not automatic nor CAT(0), and addresses open questions in geometric group theory.

Findings

Constructed groups with Dehn function ~ n^{2α}

Groups lack subgroups isomorphic to nontrivial Baumslag-Solitar groups

These groups are not automatic and do not act freely on CAT(0) cube complexes

Abstract

For every pair of positive integers $p > q$ we construct a one-relator group $R_{p,q}$ whose Dehn function is $\simeq n^{2 \alpha}$ where $\alpha = \log_2(2p / q)$. The group $R_{p,q}$ has no subgroup isomorphic to a Baumslag-Solitar group $BS(m,n)$ with $m \neq \pm n$, but is not automatic, not CAT(0), and cannot act freely on a CAT(0) cube complex. This answers a long-standing question on the automaticity of one-relator groups and gives counterexamples to a conjecture of Wise.