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 .
1. Finiteness properties
Recall that is type if it has a finite , i.e. it acts freely, cellularly and cocompactly on a contractible CW complex.
Say is type if it acts freely, cellularly and cocompactly on an acyclic CW complex (same homology as a point).
Say is type if it acts freely, cellularly and cocompactly on an acyclic CW complex (same integral homology as a point).
Say is type if has a resolution of finite length by finitely generated free -modules.
Say is type if has a resolution of finite length by finitely generated projective -modules.
All these properties imply that is torsion free.
There is no known group that is but not or not over . If one replaces with , one finds examples showing that .
2. Bestvina-Brady’s groups
Bestvina-Brady associate a group to a flag complex , the kernel of the obvious map of the Artin group to . Then
is finitely generated iff is connected.
is finitely presented iff is 1-connected.
is type iff is a point.
is iff is iff is acyclic.
There are countably such groups. All embed in groups of type .
3. New examples
Start with a connected finite flag complex , with universal cover , and a subset . I build a finitely generated group with following properties.
If , there is an epimorphism .
For fixed , the following are equivalent:
- , is .
- , is .
- and are both acyclic.
If is not simply connected,
- There exist uncountably many isomorphism types of as varies. With Ignat Soroko and Robert Kropholler, we even show that there are uncountably many quasi-isometry types.
- is finitely presented iff is finite,
- embeds in a finitely prresented group iff is recursively enumerable.
Using a trick of Mike Davies, we get
Theorem 1 For all , there exist an aspherical -manifold 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 .
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.
Let be the Salvetti complex. View it as a subcomplex of the torus (product of circles, one for each vertex). is a .
Addition defines a map which, in homotopy, is Bestvina-Brady’s morphism. Lift it to induced -cover, get a real function on . is the fundamental group of . Each vertex link in this complex is the sphericalisation : replace vertices by simplices and edges by joins of such simplices.
Theorem 3 Fer every subset of integers, there is a cube complex which is a regular branched covering of , branched only at vertices, whose links are isomorphic to
- for vertices in ,
- for vertices not in ,
We define as the fundamental group of .
Recall that in a space, the link of a vertex is a strong deformation retract of its complement, hense fundamental groups of links. If has no local cut points, . If has cut-points, embed into to get rid of them.
To distinguish these groups, we use Bowditch’s length spectrum.