## Notes of Kevin Shreve’s Cambridge lecture 23-06-2017

Action dimension and ${L^2}$ Cohomology

Joint work with Giang Le and Mike Davis.

1. Action dimension

This is the minimal dimension ${actdim(G)}$ of contractible manifolds which admit a proper ${G}$-action. The geometric dimension ${gdim(G)}$ replaces manifolds with complexes.

1.1. Examples

If ${G}$ is of type ${F}$, then ${actdim(D)\leq 2 gdim(G)}$. This comes from embedding complexes ${BG}$ into ${{\mathbb R}^N}$. ${N=2n+1}$ would be easy. ${N=2n}$ is Stallings’ theorem, using a suitable model of ${BG}$.

Bestvina-Feighn: For lattices in semi-simle Lie groups, ${actdim(G)}$ is the dimension of the symmetric space.

Desputovic: ${actdim(MCG)=dim(}$Teichmuller space${)}$.

1.2. Our favourite examples

Today, we focus on graph products of fundamental groups of closed aspherical manifolds and complements of hyperplane arrangements. We are concerned with lower bounds: when can one reduce from the obvious dimension?

The first class (circles) includes RAAG, covered by Avramidi-Davis-Okun-Shreve.

1.3. Motivation from ${L^2}$-cohomology

Let ${b_i(\tilde M)}$ denote the ${L^2}$-Betti numbers of the universal covering.

Singer conjecture: If ${M}$ is a closed aspherical manifold of dimension ${n}$, then ${b_i(\tilde M)}$ vanish if ${i\not=n/2}$.

This suggests

Action dimension conjecture. If ${b_i(G)\not=0}$, then ${actdim(G)\geq 2i}$.

Okun and I have shown that both conjectures are in fact equivalent.

2. Graph products

Let ${L}$ be a flag complex with vertex set ${S}$. The graph product of a family ${\{G_s\,;\,s\in S\}}$ of groups over ${L}$ is the quotient of the free product of ${G_s}$ by the normal subgroup generated by ${[g_s,g_t]}$, when ${st}$ is an edge of ${L}$.

Examples. If all ${G_s={\mathbb Z}}$, we get RAAG. If all ${G_s}$ are finite cyclic, we get RACG.

Theorem 1 Let ${L}$ be a ${d-1}$-dimensional flag complex, let ${G_L}$ be the corresponding graph product of fundamental groups of closed aspherical ${m}$-manifolds. Then

1. If ${H_{d-1}(L,{\mathbb Z}_2)\not=0}$, then ${actdim(G_L)=md+d}$.
2. If ${H_{d-1}(L,{\mathbb Z}_2)=0}$, then ${actdim(G_L).

2.1. Constructing aspherical manifolds

The only way to make new aspherical manifolds is to glue aspherical manifolds with boundary along codimension 0 submanifolds of their boundaries. For instance, Salvetti complexes, made of tori, do not work. We replace tori with tori ${\times}$ interval.

In general, we glue together products of ${M_v\times I}$, which is ${md+d}$ dimensional, which is sharp in some cases, as we show next. The fact that ${L}$ has vanishing homology allows to decrease dimension.

2.2. Obstructions to actions

Bestvina-Kapovitch-Kleiner coarsify van Kampen’s obstruction to embedding complexes ${K}$ into ${{\mathbb R}^N}$. This lives in ${H^n(Conf_2(K),{\mathbb Z}_2)}$ (configuration of pairs of points).

Theorem 2 (Bestvina-Kapovitch-Kleiner) Let ${G}$ be ${CAt(0)}$ or hyperbolic, let ${K\subset\partial G}$ with ${vK^n(K)\not=0}$. Then

$\displaystyle \begin{array}{rcl} actdim(G)\geq n+2. \end{array}$

Example. If ${G=F_2\times F_2}$, ${\partial G}$ contains ${K_{3,3}}$, hence ${actdim(G)\geq 4}$ (in fact, ${=2}$).

For graph products of closed aspherical manifolds, we construct a complex, denoted by ${\hat O L}$, in ${\partial G_L}$. It is a join of ${m-1}$-spheres based on ${L}$.