Uncountably many quasi-isometry classes of groups of type FP

  title={Uncountably many quasi-isometry classes of groups of type FP},
  author={Robert P. Kropholler and Ian J. Leary and Ignat Soroko},
  journal={American Journal of Mathematics},
  pages={1931 - 1944}
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 groups comprise uncountably many quasi-isometry classes. We deduce that for each $n\\geq 4$ there are uncountably many quasi-isometry classes of acyclic $n$-manifolds admitting free cocompact properly discontinuous discrete group actions. 

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;

Homological Dehn functions of groups of type $FP_2$.

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

Quasi‐isometric diversity of marked groups

We use basic tools of descriptive set theory to prove that a closed set S of marked groups has 2ℵ0 quasi‐isometry classes, provided that every non‐empty open subset of S contains at least two

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">

Superexponential Dehn functions inside CAT(0) groups

We construct 4–dimensional CAT(0) groups containing finitely presented subgroups whose Dehn functions are exp(x) for integers n,m ≥ 1 and 6–dimensional CAT(0) groups containing finitely presented

Bowditch Taut Spectrum and dimensions of groups

For a finitely generated group G, let H(G) denote Bowditch’s taut loop length spectrum. We prove that if G = (A ∗B)/〈〈R〉〉 is a C′(1/12) small cancellation quotient of a the free product of finitely

Constructing groups of type $FP_2$ over fields but not over the integers

We construct examples of groups that are FP2(Q) and FP2(Z/pZ) for all primes p but not of type FP2(Z).



Uncountably many groups of type FP

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

On Degrees of Growth of Finitely Generated Groups

We prove that for an arbitrary function ρ of subexponential growth there exists a group G of intermediate growth whose growth function satisfies the inequality vG,S(n) ⩾ ρ(n) for all n. For every

Degrees of Growth of Finitely Generated Groups, and the Theory of Invariant Means

This paper gives a negative solution to the problem of Milnor concerning the degrees of growth of groups. The construction also answers a question of Day concerning amenable groups. A number of other

The geometry and topology of Coxeter groups

These notes are intended as an introduction to the theory of Coxeter groups. They closely follow my talk in the Lectures on Modern Mathematics Series at the Mathematical Sciences Center in Tsinghua

Metric Spaces of Non-Positive Curvature

This book describes the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by

Continuously many quasiisometry classes of 2-generator groups

Abstract. We construct continuously many quasiisometry classes of torsion-free 2-generator small cancellation groups.

Cohomology of groups

  • L. Evens
  • Engineering
    Oxford 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

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 cohomology of a Coxeter group with group ring coefficients