Notes of Michael Davis’ Cambridge lecture 16-05-2017

Action dimension of a group

1. Dimensions

Let ${G}$ be a group. Assume ${G}$ has a finite ${K(G,1)}$. The geometric dimension of ${G}$ is the minimum dimension of a complex homotopic to ${K(G,1)}$. The action dimension of ${G}$ is the minimum dimension of a manifold with boundary homotopic to ${K(G,1)}$. Embedding complexes in manifolds gives

$\displaystyle \begin{array}{rcl} gdim\leq actdim\leq 2 gdim. \end{array}$

1.1. Obstructor dimension

Bestvina-Kapovich-Kleiner introduced obstructor dimension. Say a finite CW complex ${K}$ is an ${m}$-obstructor if van Kampen’s classical for embedding of ${K}$ in ${{\mathbb R}^m}$ does not vanish.

Example: a non-planar graph is a 2-obstructor.

They say that ${K\subset\partial G}$ if there exists a coarse embedding of the Euclidean cone over ${K}$ to ${EG}$. This holds for instance if ${G}$ has a boundary (e.g. hyperbolic or ${CAT(0)}$). Therefore they define obstructor dimension as ${2+}$ max of ${m}$ such that ${K\subset\partial G}$.

Theorem 1 (Bestvina-Kapovich-Kleiner 2001)

$\displaystyle \begin{array}{rcl} actdim\geq obdim. \end{array}$

Equality often holds.

Example. ${G=F_2\times\cdots\times F_2}$ (${d+1}$ factors) has ${actdim=2d+2}$. If ${d=1}$, ${gdim=2}$, ${obdim=4}$.

Example. ${G}$ a non-uniform lattice.

Example. ${G=}$ 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, ${Q}$ will be the poset of simplices (including the empty simplex) of a simplicial complex ${L}$.

2.1. Gluing

Form the disjoint union of products ${K(G_\sigma,1)\times Cone(link(\sigma))}$ and identify according to inclusions. Denote result by ${BG(L)}$.

Example: the Artin complex of a Coxeter system is constructed from the nerve ${L}$ whose simplices correspond to subsets of vertices generating finite Coxeter subgroups. Call the corresponding Artin groups ${A_\sigma}$ spherical Artin subgroups.

Example: the graph product complex. Let ${L}$ be a flag complex. Then ${L^1}$ is a simplicial graph. The graph procduct is the free product of vertex groups ${G_v}$ modded out by ${[G_s,G_A]=1}$ each time ${\{s,A\}\in L^1}$. This has a finite ${K(G,1)}$ iff ${L}$ is a flag complex.

2.2. Results

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

Theorem 2 Let ${L}$ be the nerve of a Coxeter system, let ${d}$ be its dimension. Suppose the corresponding Artin group ${A}$ has a finite ${K(A,1)}$. Then

1. ${H_d(L,{\mathbb Z}/2{\mathbb Z})\not=0}$ implies ${actdim(A)=obdim(A)=2gdim(A)}$.
2. (Le’s thesis) If ${L}$ embeds in a contractible complex of the same dimension (EDCE), then ${actdim(A)\leq 2d+1}$. (Due to C. Gordon for ${d=1}$).
3. For RAAG’s, if ${H_d(L,{\mathbb Z}/2{\mathbb Z})=0}$, then ${actdim(A)\leq 2d+1}$.

Theorem 3 Let ${L}$ be a simplicial complex, ${G}$ its grpah product. Let ${M_v}$ be an aspherical manifold of minimal dimension which is a ${K(G_v,1)}$. Let ${m_v=dim(M_v)}$ if ${M_v}$ has non-empty boundary, ${=dim(M_v)+1}$ if ${M_v}$ is closed. Set ${m_\sigma=\sum_{v\in\sigma}m_v}$. Then

1. ${actdim(G)\leq\max\{m_\sigma\}}$. Furthermore, if no ${M_v}$ is closed, equality holds.
2. If all ${M_v}$ are closed, and if ${L}$ is EDCE), then ${actdim(G)\leq\max\{m_\sigma\}-1}$.

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 ${K(A,1)}$ is a join of tori, ${EA}$ contains a union of Euclidean spaces. Its visual boundary ${OL}$ is called the octahedralization of ${L}$ (vertices are doubled). We show that this is a ${2d+1}$-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.