Properly 3-realizable groups

Abstract

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

One-relator groups and proper 3-realizability

“…In the torsion-free case, the above translates into a decomposition of G into a free product of a free group with a one-relator group, the latter having at most one end. The conclusion now follows from Theorem 1.1 and the fact that free groups are properly 3-realizable, and free products of properly 3-realizable groups are again properly 3-realizable (see ( [1], Thm. 1.4)).…”

“…. , T is be those trees of the collection which intersect C n (1) , and take Z n(1),m ⊂ T im to be either the connected subtree satisfying…”

“…Finally, repeating this process (starting off with the successive C n,m ⊂ K m and going each time one step further inside K P ) we get a filtration C 1 ⊂ C 2 ⊂ · · · ⊂ K P of compact simply connected subcomplexes. Moreover, choosing base points on any given base ray, one can easily check that rank(π 1 …”

“…where α is the lift of the inverse path of the (oriented) generating circle corresponding to A whose initial endpoint α(0) coincides with β (1). Note that as A generates an infinite cyclic subgroup in both group presentations P and P , the support of each of the above edge paths is homeomorphic to a closed interval.…”

“…Note that we are already done in the 0-ended or 2-ended case as the 0-ended case corresponds to finite cyclic groups, and in the 2-ended case we only have to deal with the group Z of integers, which is the only one-relator group having Z as a subgroup of finite index. See also ( [1], Cor. 1.2).…”