Francisco F. Lasheras

Universal covers and 3-manifolds

In this paper, we show that if a finitely presented group G is the fundamental group of a finite fake surface in which the link of any vertex is not homeomorphic to the 1-skeleton of a tetrahedron, then there is a finite 2-complex K with π1(K)≅G and whose universal cover K has the proper homotopy type of a 3-manifold. As a consequence, the cohomology group H2(G;ZG) is free abelian.

One-relator groups and proper

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

3

G

-realizability

is said to be properly 3-realizable if there exists a compact 2-polyhedron

Direct products and properly 3-realisable groups

K

Amalgamated products and properly 3-realizable groups

with

ASCENDING HNN-EXTENSIONS AND PROPERLY 3-REALISABLE GROUPS

\pi_1(K) \cong G

Detecting cohomology classes for the proper LS category. The case of semistable 3-manifolds

whose universal cover

Relating the Freiheitssatz to the asymptotic behavior of a group.

\tilde{K}

On manifolds with nonhomogeneous factors

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.