## 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 ${FP({\mathbb Q})\not= FL({\mathbb Q})\not=FH({\mathbb Q})}$.

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.