Exotic relation modules and homotopy types for certain 1-relator groups

TL;DR

This paper constructs new 2-dimensional homotopy types with trefoil fundamental groups using exotic relation modules, answering a longstanding question and providing novel examples of such modules.

Contribution

It introduces an infinite collection of homotopy types with specific properties and presents new exotic relation modules, advancing understanding of relation modules in group theory.

Findings

Constructed infinite 2-dimensional homotopy types with Euler characteristic one

Provided examples of stably free non-free relation modules

Answered a question by Berridge and Dunwoody regarding homotopy types

Abstract

Using stably free non-free relation modules we construct an infinite collection of 2-dimensional homotopy types, each of Euler-characteristic one and with trefoil fundamental group. This provides an affirmative answer to a question asked by Berridge and Dunwoody [J. London Math. Soc. 19 (1979) 433-436]. We also give new examples of exotic relation modules. We show that the relation module associated with the generating set x, y^4 for the Baumslag-Solitar group <x, y | xy^2x^{-1}=y^3> is stably free non-free of rank one.