Almost all one-relator groups with at least three generators are residually finite

TL;DR

This paper demonstrates that most one-relator groups with three or more generators are residually finite, virtually residually (finite p)-group, and coherent, using a combination of combinatorial group theory and probabilistic methods.

Contribution

It establishes that almost all such groups are residually finite and coherent, employing novel probabilistic techniques in group theory.

Findings

Most one-relator groups with ≥3 generators are residually finite

These groups are virtually residually (finite p)-groups for large p

They are also coherent

Abstract

We prove that with probability tending to 1, a 1-relator group with at least 3 generators and relator of length n is residually finite, virtually residually (finite p)-group for all sufficiently large p, and coherent. The proof uses both combinatorial group theory and non-trivial results about Brownian motions, bridges and excursions in R^k.