Uncountably many groups of type FP

  title={Uncountably many groups of type FP},
  author={Ian J. Leary},
  journal={Proceedings of the London Mathematical Society},
  • 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. 

Groups of type $FP$ via graphical small cancellation

We construct an uncountable family of groups of type $FP$. In contrast to every previous construction of non-finitely presented groups of type $FP$ we do not use Morse theory on cubical complexes;

Uncountably many quasi-isometry classes of groups of type FP

Abstract:In an earlier paper, one of the authors constructed uncountable families of groups of type $FP$ and of $n$-dimensional Poincar\\'e duality groups for each $n\\geq 4$. We show that those

Finitely generated groups acting uniformly properly on hyperbolic space

We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. We show that only countably many of these groups can be virtually torsion-free. This gives new examples

Almost Hyperbolic Groups with Almost Finitely Presented Subgroups

We construct new examples of CAT(0) groups containing non finitely presented subgroups that are of type $FP_2$, these CAT(0) groups do not contain copies of $\mathbb{Z}^3$. We also give a

Hyperbolic groups with almost finitely presented subgroups

In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type

Virtually special non-finitely presented groups via linear characters

We present a new method for showing that groups are virtually special. This is done by considering finite quotients and linear characters. We use this to show that an infinite family of groups,

Weak commutativity and finiteness properties of groups

We consider the group X(G) obtained from G*G by forcing each element g in the first free factor to commute with the copy of g in the second free factor. Deceptively complicated finitely presented

Profinite rigidity of fibring

. We introduce the classes of TAP groups, in which various types of algebraic fibring are detected by the non-vanishing of twisted Alexander polynomials. We show that finitely presented LERF groups lie

On the virtual and residual properties of a generalization of Bestvina-Brady groups

<jats:p>Previously one of us introduced a family of groups <jats:inline-formula><jats:alternatives><jats:tex-math>$$G^M_L(S)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">

Subgroups of almost finitely presented groups

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



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


  • 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.


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


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