One-relator groups and proper 3-realizability

TL;DR

This paper investigates the proper 3-realizability of finitely generated one-relator groups, showing that those with finitely many ends are properly 3-realizable and analyzing their asymptotic properties.

Contribution

It demonstrates that one-relator groups with finitely many ends are properly 3-realizable and provides new insights into their asymptotic and homotopy properties.

Findings

One-relator groups with finitely many ends are properly 3-realizable.

Describes the fundamental pro-group structure of these groups.

Provides an alternative proof that one-relator groups are semistable at infinity.

Abstract

How different is the universal cover of a given finite 2-complex from a 3-manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group $G$ is said to be properly 3-realizable if there exists a compact 2-polyhedron $K$ with $\pi_1(K) \cong G$ whose universal cover $\tilde{K}$ has the proper homotopy type of a PL 3-manifold (with boundary). In this paper, we study the asymptotic behavior of finitely generated one-relator groups and show that those having finitely many ends are properly 3-realizable, by describing what the fundamental pro-group looks like, showing a property of one-relator groups which is stronger than the QSF property of Brick (from the proper homotopy viewpoint) and giving an alternative proof of the fact that one-relator groups are semistable at infinity.