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
5
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 32 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…
Morse theory and finiteness properties of groups
Abstract. We examine the finiteness properties of certain subgroups of “right angled” Artin groups. In particular, we find an example of a group that is of type FP(Z) but is not finitely presented.
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…
The topology of discrete groups
Bestvina–Brady groups and the plus construction
  • J. Howie
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1999
A recent result of Bestvina and Brady [1, theorem 8·7], shows that one of two outstanding questions has a negative answer; either there exists a group of cohomological dimension 2 and geometric…
...
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