One-relator groups and proper 3-realizability

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 π 1 (K) ∼ = G whose universal coverK 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.

Relating the Freiheitssatz to the asymptotic behavior of a group

“…As a consequence of Theorem 1.10 (together with Remark 1.7), we answer Conjecture 1.5 in [8] in the affirmative; namely: Corollary 1.12. All finitely generated one-relator groups are properly 3-realizable.…”

“…Given Y ⊂ X and the subgraph J ⊂ K From now on, all CW-complexes will be PL CW-complexes, in the sense of §1.4 of [19]. Let us recall the notion of strong proper homotopy equivalence between 2-dimensional (PL) CW-complexes introduced in [8]. Note that this notion is a (proper) generalization of the notion of simple homotopy equivalence within the category of 2-dimensional (PL) CW-complexes.…”

“…Those groups which admit a one-relator group presentation P have a nice fundamental pro-group at each end (see [8]), and in this case P is special. Observe that the results in [8] make use of an additional CW-complex L, thus motivating Definitions 1.5 and 1.6. Remark 1.8.…”

A topological equivalence relation for finitely presented groups

“…In fact, any extension of an infinite finitely presented group by another infinite finitely presented group is of telescopic type at infinity [16]. One-relator groups are also of telescopic type at each end [15] (see also [36]).…”

Some Hyperbolic Space Forms With Few Generated Fundamental Groups

“…This manifold is obtained by identifying the faces of a single ideal tetrahedron; it has the first integral homology group and the isometry group both isomorphic to Z 2 . Now applying Agol's result [2], we have: Finally, we observe that the one-relator group G in Theorem 4.1 is properly 3-realizable in the sense of [5], i.e., there exists a compact 2-polyhedron K with π 1 (K) ∼ = G whose universal cover K is proper homotopy equivalent to a 3-manifold. In our case, K is a spine of the space form M (for example, the spine which corresponds to the finite presentation of G) and K is proper homotopy equivalent to H 3 .…”