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.


About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
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