Residual properties of 1-relator groups

TL;DR

This paper surveys results showing that most 1-relator groups with three or more generators are virtually residually finite p-groups and coherent, using methods from combinatorial group theory, algebraic geometry, and probability theory.

Contribution

It proves that almost all 1-relator groups with at least three generators are virtually residually finite p-groups and coherent, combining diverse mathematical techniques.

Findings

Most 1-relator groups with ≥3 generators are virtually residually finite p-groups.

Such groups are also coherent.

The proof employs combinatorial, algebraic geometric, and probabilistic methods.

Abstract

This is a survey of two papers joint with A. Borisov and a paper joint with I. Spakulova. It is based on my lectures at the conference "Groups St. Andrews 2009", Bath (August 2009). We prove that almost all 1-related groups with at least 3 generators are virtually residually (finite p-)groups for almost all primes p, and coherent. The proof involves methods from combinatorial group theory (the congruence extension property of certain subgroups of free groups) algebraic geometry (dynamics of polynomial maps over finite and p-adic fields) and probability theory (convex hulls of Brownian bridges).