Notes of Ian Leary’s Cambridge lecture 26-01-2017

Generalizing Bestvina-Brady groups using branched covers

Joint with Ignat Soroko and Robert Kropholler.

Initial motivation: prove that there exist uncountably many groups of type {FP}.

1. Finiteness properties

Recall that {G} is type {F} if it has a finite {K(\pi,1)}, i.e. it acts freely, cellularly and cocompactly on a contractible CW complex.

Say {G} is type {FH} if it acts freely, cellularly and cocompactly on an acyclic CW complex (same homology as a point).

Say {G} is type {FH} if it acts freely, cellularly and cocompactly on an acyclic CW complex (same integral homology as a point).

Say {G} is type {FL} if {{\mathbb Z}} has a resolution of finite length by finitely generated free {{\mathbb Z} G}-modules.

Say {G} is type {FP} if {{\mathbb Z}} has a resolution of finite length by finitely generated projective {{\mathbb Z} G}-modules.

All these properties imply that {G} is torsion free.

There is no known group that is {FP} but not {FL} or not {FH} over {{\mathbb Z}}. If one replaces {{\mathbb Z}} with {{\mathbb Q}}, one finds examples showing that {FP({\mathbb Q})\not= FL({\mathbb Q})\not=FH({\mathbb Q})}.

2. Bestvina-Brady’s groups

Bestvina-Brady associate a group {BB_L} to a flag complex {L}, the kernel of the obvious map of the Artin group {A_L} to {{\mathbb Z}}. Then

{BB_L} is finitely generated iff {L} is connected.

{BB_L} is finitely presented iff {L} is 1-connected.

{BB_L} is type {F} iff {L} is a point.

{BB_L} is {FH} iff {BB_L} is {FP} iff {L} is acyclic.

There are countably such groups. All embed in groups of type {F}.

3. New examples

3.1. Properties

Start with a connected finite flag complex {L}, with universal cover {\tilde L}, and a subset {S\subset{\mathbb Z}}. I build a finitely generated group {G_L(S)} with following properties.

If {S\subset T}, there is an epimorphism {G_L(S)\rightarrow G_L(T)\rightarrow 1}.

For fixed {L}, the following are equivalent:

  1. {\forall S\subset{\mathbb Z}}, {G_L(S)} is {FP}.
  2. {\forall S\subset{\mathbb Z}}, {G_L(S)} is {FH}.
  3. {L} and {\tilde L} are both acyclic.

If {L} is not simply connected,

  • There exist uncountably many isomorphism types of {G_L(S)} as {S} varies. With Ignat Soroko and Robert Kropholler, we even show that there are uncountably many quasi-isometry types.
  • {G_L(S)} is finitely presented iff {S} is finite,
  • {G_L(S)} embeds in a finitely prresented group iff {S} is recursively enumerable.

3.2. Corollaries

Using a trick of Mike Davies, we get

Theorem 1 For all {n\geq 4}, there exist an aspherical {n}-manifold {M^n} with uncountably many regular acyclic covers with non-isomorphic (and propably non quasi-isometric) groups of deck transformations.

Theorem 2 Every countable group embeds in a group of type {FP_2}.

Question. If you believe in Gromov’s assertion (that a statement balid for all countable groups is either false or obvious), find a straightforward proof of this.

3.3. Construction

Let {T_L} be the Salvetti complex. View it as a subcomplex of the torus (product of circles, one for each vertex). {T_L} is a {K(A_L,1)}.

Addition defines a map {T_L\rightarrow T_p} which, in homotopy, is Bestvina-Brady’s morphism. Lift it to induced {{\mathbb Z}}-cover, get a real function {h} on {\hat T_L}. {BB_L} is the fundamental group of {\hat T_L}. Each vertex link in this complex is the sphericalisation {S(L)}: replace vertices by simplices and edges by joins of such simplices.

Theorem 3 Fer every subset {S} of integers, there is a {CAT(0)} cube complex {X_L^{(S)}} which is a regular branched covering of {\hat T_L}, branched only at vertices, whose links are isomorphic to

  1. {S(L)} for vertices in {h^{-1}(S)},
  2. {S(\tilde L)} for vertices not in {h^{-1}(S)},

We define {G_L(S)} as the fundamental group of {X_L^{(S)}}.

Recall that in a {CAT(0)} space, the link of a vertex is a strong deformation retract of its complement, hense fundamental groups of links. If {L} has no local cut points, {S(\tilde L)=\widetilde{S(L)}}. If {L} has cut-points, embed {L} into {L\times I} to get rid of them.

To distinguish these groups, we use Bowditch’s length spectrum.


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