Action dimension of a group
Let be a group. Assume has a finite . The geometric dimension of is the minimum dimension of a complex homotopic to . The action dimension of is the minimum dimension of a manifold with boundary homotopic to . Embedding complexes in manifolds gives
1.1. Obstructor dimension
Bestvina-Kapovich-Kleiner introduced obstructor dimension. Say a finite CW complex is an -obstructor if van Kampen’s classical for embedding of in does not vanish.
Example: a non-planar graph is a 2-obstructor.
They say that if there exists a coarse embedding of the Euclidean cone over to . This holds for instance if has a boundary (e.g. hyperbolic or ). Therefore they define obstructor dimension as max of such that .
Theorem 1 (Bestvina-Kapovich-Kleiner 2001)
Equality often holds.
Example. ( factors) has . If , , .
Example. a non-uniform lattice.
Example. mapping class group.
2. Results on action dimension
Le-Davis-Huang: general Artin groups.
Le-Schreve: simple complexes of groups. This means a functor from a poset to the category of groups.
Today, will be the poset of simplices (including the empty simplex) of a simplicial complex .
Form the disjoint union of products and identify according to inclusions. Denote result by .
Example: the Artin complex of a Coxeter system is constructed from the nerve whose simplices correspond to subsets of vertices generating finite Coxeter subgroups. Call the corresponding Artin groups spherical Artin subgroups.
Example: the graph product complex. Let be a flag complex. Then is a simplicial graph. The graph procduct is the free product of vertex groups modded out by each time . This has a finite iff is a flag complex.
I compute the action dimension of the families of examples above.
Theorem 2 Let be the nerve of a Coxeter system, let be its dimension. Suppose the corresponding Artin group has a finite . Then
- implies .
- (Le’s thesis) If embeds in a contractible complex of the same dimension (EDCE), then . (Due to C. Gordon for ).
- For RAAG’s, if , then .
Theorem 3 Let be a simplicial complex, its grpah product. Let be an aspherical manifold of minimal dimension which is a . Let if has non-empty boundary, if is closed. Set . Then
- . Furthermore, if no is closed, equality holds.
- If all are closed, and if is EDCE), then .
First perform suitable gluings, and then find obstructions.
1. Gluing. Glue together manifolds along codim 0 subsets of their boundaries. Eventually multiply smaller dimensional manifolds with disks in order to raise all pieces to the same dimension. For instance, glue surfaces to a 2-disk along intervals of their boundaries. Call this disk a dual disk.
2. Obstructors. For RAAGs, the is a join of tori, contains a union of Euclidean spaces. Its visual boundary is called the octahedralization of (vertices are doubled). We show that this is a -obstructor if its top homology does not vanish.
For graph products, in a similar way we produce a join of spheres which is an obstructor.
Swenson: what about these exotic contractible manifolds whose boundaries are not spheres? We are thinking of this. Ultimately, I think we shall have to exclude them.