Groups which are not properly 3-realizable

TL;DR

This paper investigates the properties of groups that are not properly 3-realizable, establishing conditions under which such groups have certain topological features, and providing the first examples of finitely presented groups lacking proper 3-realizability.

Contribution

It proves that properly 3-realizable groups with quasi-simply filtered property have specific topological features and presents the first examples of finitely presented groups that are not properly 3-realizable.

Findings

Properly 3-realizable groups with quasi-simply filtered property have semi-stable ends.

Such groups have pro-(finitely generated free) fundamental group at infinity.

Examples include large families of Coxeter groups that are not properly 3-realizable.

Abstract

A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it has {\em pro-(finitely generated free) fundamental group at infinity} and {\em semi-stable ends}. Conjecturally the quasi-simply filtration assumption is superfluous. Using these restrictions we provide the first examples of finitely presented groups which are not properly 3-realizable, for instance large families of Coxeter groups.