Action dimension and Cohomology
Joint work with Giang Le and Mike Davis.
1. Action dimension
This is the minimal dimension of contractible manifolds which admit a proper -action. The geometric dimension replaces manifolds with complexes.
If is of type , then . This comes from embedding complexes into . would be easy. is Stallings’ theorem, using a suitable model of .
Bestvina-Feighn: For lattices in semi-simle Lie groups, is the dimension of the symmetric space.
Desputovic: 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 -cohomology
Let denote the -Betti numbers of the universal covering.
Singer conjecture: If is a closed aspherical manifold of dimension , then vanish if .
Action dimension conjecture. If , then .
Okun and I have shown that both conjectures are in fact equivalent.
2. Graph products
Let be a flag complex with vertex set . The graph product of a family of groups over is the quotient of the free product of by the normal subgroup generated by , when is an edge of .
Examples. If all , we get RAAG. If all are finite cyclic, we get RACG.
Theorem 1 Let be a -dimensional flag complex, let be the corresponding graph product of fundamental groups of closed aspherical -manifolds. Then
- If , then .
- If , then .
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 interval.
In general, we glue together products of , which is dimensional, which is sharp in some cases, as we show next. The fact that has vanishing homology allows to decrease dimension.
2.2. Obstructions to actions
Bestvina-Kapovitch-Kleiner coarsify van Kampen’s obstruction to embedding complexes into . This lives in (configuration of pairs of points).
Theorem 2 (Bestvina-Kapovitch-Kleiner) Let be or hyperbolic, let with . Then
Example. If , contains , hence (in fact, ).
For graph products of closed aspherical manifolds, we construct a complex, denoted by , in . It is a join of -spheres based on .