# Van Kampen’s embedding obstruction for discrete groups

```@article{Bestvina2000VanKE,
title={Van Kampen’s embedding obstruction for discrete groups},
author={Mladen Bestvina and Michael Kapovich and Bruce Kleiner},
journal={Inventiones mathematicae},
year={2000},
volume={150},
pages={219-235}
}```
• Published 13 October 2000
• Mathematics
• Inventiones mathematicae
Abstract.We give a lower bound to the dimension of a contractible manifold on which a given group can act properly discontinuously. In particular, we show that the n-fold product of nonabelian free groups cannot act properly discontinuously on ℝ2n−1.
Rational manifold models for duality groups
We show that a finite type duality group of dimension d > 2 is the fundamental group of a (d + 3)-manifold with rationally acyclic universal cover. We use this to find closed manifolds with
Kleinian groups via strict hyperbolization
In this paper, we construct Kleinian groups Γ < Isom(H) from the direct product of n copies of the rank 2 free group F2 via strict hyperbolization. We give a description of the limit set and its
Topology of open nonpositively curved manifolds
This is a survey of topological properties of open, complete nonpositively curved manifolds which may have infinite volume. Topics include topology of ends, restrictions on the fundamental group, as
Towards a Sullivan dictionary in dimension two, Part I: Purely parabolic complex Kleinian groups
• Mathematics
• 2018
In this article we provide a full description of all the complex kleinian groups of \$PSL(3,\Bbb{C})\$ which contains only parabolic elements.
Action dimensions of some simple complexes of groups
• Mathematics
Journal of Topology
• 2019
The action dimension of a discrete group G is the minimum dimension of a contractible manifold that admits a proper G ‐action. We compute the action dimension of the direct limit of a simple complex
Action dimension of lattices in Euclidean buildings
The action dimension of a group G is the minimal dimension of a contractible manifold that G acts on properly discontinuously. We show that if G acts properly and cocompactly on a thick Euclidean
Proper actions of lattices on contractible manifolds
• Mathematics
• 2000
Abstract.Every lattice Γ in a connected semi-simple Lie group G acts properly discontinuously by isometries on the contractible manifold G/K (K a maximal compact subgroup of G). We prove that if Γ
The action dimension of right‐angled Artin groups
• Mathematics
• 2014
The action dimension of a discrete group Γ is the smallest dimension of a contractible manifold that admits a proper action of Γ . Associated to any flag complex L there is a right‐angled Artin
Nonhyperbolic Coxeter groups with Menger curve boundary
• Mathematics
• 2018
A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article we produce the first known examples of non-hyperbolic \$CAT(0)\$ groups
A lower bound to the action dimension of a group
The action dimension of a discrete group , actdim(), is defined to be the smallest integer m such that admits a properly dis- continuous action on a contractible m-manifold. If no such m exists, we

## References

SHOWING 1-10 OF 39 REFERENCES
Embedding obstructions and 4-dimensional thickenings of 2-complexes
The vanishing of Van Kampen’s obstruction is known to be necessary and sufficient for embeddability of a simplicial n-complex into R2n for n 6= 2, and it was recently shown to be incomplete for n =
Embedding up to homotopy type in Euclidean space
• Mathematics
Bulletin of the Australian Mathematical Society
• 1993
We give 8 short proof of the classical Stallings theorem that every finite n-dimensional cellular complex embeds up to homotopy in the 2n-dimensional Euclidean space. As an application we solve a
Proper actions of lattices on contractible manifolds
• Mathematics
• 2000
Abstract.Every lattice Γ in a connected semi-simple Lie group G acts properly discontinuously by isometries on the contractible manifold G/K (K a maximal compact subgroup of G). We prove that if Γ
Hyperbolic groups with low-dimensional boundary - eScholarship
• Mathematics
• 2000
If a torsion-free hyperbolic group G has 1-dimensional boundary ∂ ∞ G, then ∂ ∞ G is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When ∂ ∞ G is a Sierpinski
Cohomology of Groups
This advanced textbook introduces students to cohomology theory. No knowledge of homological algebra is assumed beyond what is normally taught in a first course in algebraic topology.
Van Kampen’s embedding Obstruction is incomplete for \$2\$-Complexes in \$\rz^{4}\$
• Mathematics
• 1994
In 1933, anticipating formal cohomology theory, van Kampen [5] gave a slightly rough description of an obstruction o(K) ∈ H Z/2(K , Z) which vanishes if and only if an n-dimensional simplicial
Moduli of graphs and automorphisms of free groups
• Mathematics
• 1986
This paper represents the beginning of an a t tempt to transfer, to the study of outer au tomorphisms of free groups, the powerful geometric techniques that were invented by Thurs ton to study
Topological Immersion of Peanian Continua in a Spherical Surface
In 1930 Kuratowski2 established the following result: THEOREM A: A peanian continuum,3 containing but a finite number of simple closed curves, is homneomorphic with a subset of the plane, provided
Cohomology of groups
• L. Evens
• Engineering
Oxford mathematical monographs
• 1991
A rink-type roller skate is provided with a plastic sole plate. To mount a toe stop on the skate, a novel bushing is embedded in the sole plate. The bushing has relatively small diameter ends and a