Properly 3-realizable groups

Abstract

A finitely presented group G G is said to be properly 3 3 -realizable if there exists a compact 2 2 -polyhedron K K with π 1 ( K ) ≅ G \pi _1(K) \cong G and whose universal cover K ~ \tilde {K} has the proper homotopy type of a (p.l.) 3 3 -manifold with boundary. In this paper we show that, after taking wedge with a 2 2 -sphere, this property does not depend on the choice of the compact 2 2 -polyhedron K K with π 1 ( K ) ≅ G \pi _1(K) \cong G . We also show that (i) all 0 0 -ended and 2 2 -ended groups are properly 3 3 -realizable, and (ii) the class of properly 3 3 -realizable groups is closed under amalgamated free products (HNN-extensions) over a finite cyclic group (as a step towards proving that ∞ \infty -ended groups are properly 3 3 -realizable, assuming 1 1 -ended groups are).