Uncountably many groups of type FP

@article{Leary2015UncountablyMG,
  title={Uncountably many groups of type FP},
  author={Ian J. Leary},
  journal={Proceedings of the London Mathematical Society},
  year={2015},
  volume={117}
}
  • I. Leary
  • Published 21 December 2015
  • Mathematics
  • Proceedings of the London Mathematical Society
We construct uncountably many discrete groups of type FP; in particular we construct groups of type FP that do not embed in any finitely presented group. We compute the ordinary, ℓ2 , and compactly supported cohomology of these groups. For each n⩾4 we construct a closed aspherical n ‐manifold that admits an uncountable family of acyclic regular coverings with non‐isomorphic covering groups. 
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22 Citations
Highly Influential Citations
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Background Citations
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Methods Citations
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Results Citations
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22 Citations

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References

SHOWING 1-10 OF 29 REFERENCES

The cohomology of Bestvina-Brady groups

For each subcomplex of the standard CW-structure on any torus, we compute the homology of a certain infinite cyclic regular covering space. In all cases when the homology is finitely generated, we…

Presentations for subgroups of Artin groups

Recently, M. Bestvina and N. Brady have exhibited groups that are of type FP but not finitely presented. We give explicit presentations for groups of the type considered by Bestvina-Brady. This leads…

THE EULER CLASS OF A POINCARÉ DUALITY GROUP

  • I. Leary
  • Mathematics
    Proceedings of the Edinburgh Mathematical Society
  • 2002
Abstract Under an extra hypothesis satisfied in every known case, we show that the Euler class of an orientable odd-dimensional Poincaré duality group over any ring has order at most two. We…

Subgroups of finitely presented groups

  • G. Higman
  • Mathematics
    Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
  • 1961
The main theorem of this paper states that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. It…

Subgroups of almost finitely presented groups

We show that every countable group embeds in a group of type $$FP_2$$FP2.

GROTHENDIECK'S PROBLEMS CONCERNING PROFINITE COMPLETIONS AND REPRESENTATIONS OF GROUPS

In 1970 Alexander Grothendieck [6] posed the following problem: let Γ1 and Γ2 be finitely presented, residually finite groups, and let u :Γ 1 → Γ2 be a homomorphism such that the induced map of…

Cohomology computations for Artin groups, Bestvina-Brady groups, and graph products

We compute: * the cohomology with group ring coefficients of Artin groups (or actually, of their associated Salvetti complexes), Bestvina-Brady groups, and graph products of groups, * the L^2-Betti…

ON THE FINITENESS PROPERTIES OF GROUPS

For an automorphism ' of the group G, the connection between the centralizer CG(') and the commutator (G,') is investigated and as a con- sequence of the Schur theorem it is shown that if G/CG(') and…

A metric Kan–Thurston theorem

For every simplicial complex X, we construct a locally CAT(0) cubical complex TX, a cellular isometric involution τ on TX and a map tX:TX→X with the following properties: tXτ=tX; tX is a homology…

Examples of Poincaré duality groups

A closed aspherical manifold may have a fundamental group which is not residually finite and contains infinitely divisible elements. The purpose of this note is to exhibit examples of Poincare…