# 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} }

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.

## 22 Citations

Groups of type $FP$ via graphical small cancellation

- Mathematics
- 2020

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

- MathematicsAmerican Journal of Mathematics
- 2020

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

- Mathematics
- 2020

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

- Mathematics
- 2018

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…

Weak commutativity and finiteness properties of groups

- MathematicsBulletin of the London Mathematical Society
- 2018

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

- Mathematics
- 2022

. 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

- Mathematics
- 2022

Previously one of us introduced a family of groups GL (S), parametrized by a finite flag complex L, a regular covering M of L, and a set S of integers. We give conjectural descriptions of when GL (S)…

Subgroups of almost finitely presented groups

- Mathematics
- 2016

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

Homological Dehn functions of groups of type $FP_2$.

- Mathematics
- 2020

We prove foundational results for homological Dehn functions of groups of type $FP_2$ such as superadditivity and the invariance under quasi-isometry. We then study the homological Dehn functions of…

Simple groups separated by finiteness properties

- MathematicsInventiones mathematicae
- 2018

We show that for every positive integer n there exists a simple group that is of type $$\mathrm {F}_{n-1}$$Fn-1 but not of type $$\mathrm {F}_n$$Fn. For $$n\ge 3$$n≥3 these groups are the first known…

## References

SHOWING 1-10 OF 32 REFERENCES

The cohomology of Bestvina-Brady groups

- Mathematics
- 2007

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

- Mathematics
- 1997

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

- MathematicsProceedings 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

- MathematicsProceedings 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

- Mathematics
- 2016

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

- Mathematics
- 2004

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

- Mathematics
- 2010

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

- Mathematics
- 2013

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…

Bestvina–Brady groups and the plus construction

- MathematicsMathematical 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…