** Action dimension of a group **

**1. Dimensions **

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 **

Avramidi-Davis-Okun-Schreve: RAAG’s.

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 .

** 2.1. Gluing **

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.

** 2.2. Results **

I compute the action dimension of the families of examples above.

Theorem 2Let 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 3Let 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 .

**3. Proofs **

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.

**4. Questions **

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.