A topological equivalence relation for finitely presented groups

@article{Cardenas2020ATE,
  title={A topological equivalence relation for finitely presented groups},
  author={M. C'ardenas and Francisco F. Lasheras and Antonio Quintero and Ranja Roy},
  journal={arXiv: Geometric Topology},
  year={2020}
}
In this paper, we consider an equivalence relation within the class of finitely presented discrete groups attending to their asymptotic topology rather than their asymptotic geometry. More precisely, we say that two finitely presented groups $G$ and $H$ are "proper $2$-equivalent" if there exist (equivalently, for all) finite $2$-dimensional CW-complexes $X$ and $Y$, with $\pi_1(X) \cong G$ and $\pi_1(Y) \cong H$, so that their universal covers $\widetilde{X}$ and $\widetilde{Y}$ are proper $2… 

References

SHOWING 1-10 OF 57 REFERENCES
A Note on Group Extensions and Proper 3-Realizability
AbstractThe interaction between the study of three-dimensional manifolds and a particular stream of group theory has often been fruitful. In the realm of this, we recall that a finitely presented
One-relator groups and proper 3-realizability
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
Quasi-isometries between groups with infinitely many ends
Abstract. Let G, F be finitely generated groups with infinitely many ends and let¶ $ \pi_1(\Gamma,\mathcal A), \pi_1(\Delta ,\mathcal B) $ be graph of groups decompositions of F, G such that all edge
Topological properties of spaces admitting free group actions
In 1992, David Wright proved a remarkable theorem about which contractible open manifolds are covering spaces. He showed that if a one-ended open manifold M has pro-monomorphic fundamental group at
Free abelian cohomology of groups and ends of universal covers
Abstract Let G be a group for which there exists a K(G, 1)-complex X having finite n-skeleton (for n = 1 or 2 this is equivalent to saying that G is finitely generated or finitely presented). If X n
Semistability at the end of a group extension
A 1-ended CW-complex, Q, is semistable at oo if all proper maps r: [ 0, oo) Q are properly homotopic. If XI and X2 are finite CW-complexes with isomorphic fundamental groups, then the universal cover
On proper homotopy theory for noncompact 3-manifolds
Proper homotopy groups analogous to the usual homotopy groups are defined. They are used to prove, modulo the Poincare conjecture, that a noncompact 3-manifold having the proper homotopy type of a
Topological properties of spaces admitting a coaxial homeomorphism
Wright showed that, if a 1-ended simply connected locally compact ANR Y with pro-monomorphic fundamental group at infinity admits a proper Z-action, then that fundamental group at infinity can be
Properly 3-realizable groups
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 cover K has the proper homotopy type of a (p.l.)
Planar groups and the Seifert conjecture
We describe a number of characterisations of virtual surface groups which are based on the following result. Let G be a group and F be a field. We show that if G is FP2 over F and if H2ðG; FÞ,
...
1
2
3
4
5
...