Virtually abelian quotients of random groups

TL;DR

This paper investigates the properties of random groups with few relators, specifically their ability to map onto infinite virtually abelian groups, extending known results about their first betti number.

Contribution

It introduces the study of homomorphisms from few-relators random groups to infinite virtually abelian groups, expanding understanding of their algebraic structure.

Findings

Random groups with few relators often admit maps to infinite virtually abelian groups.

Extension of known vanishing first betti number results to broader classes of groups.

Abstract

It is well-known that random groups with at least as many relators as generators have vanishing first betti number with high probability. In this paper we extend this question and study maps from few-relators random groups to infinite virtually abelian groups.