# 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} }

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.

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## References

SHOWING 1-10 OF 39 REFERENCES

Embedding obstructions and 4-dimensional thickenings of 2-complexes

- Mathematics
- 2000

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

- MathematicsBulletin 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

- Mathematics
- 1982

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

- Mathematics
- 1934

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

- EngineeringOxford 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…