Notes of Kevin Shreve’s Cambridge lecture 23-06-2017

Action dimension and {L^2} Cohomology

Joint work with Giang Le and Mike Davis.

1. Action dimension

This is the minimal dimension {actdim(G)} of contractible manifolds which admit a proper {G}-action. The geometric dimension {gdim(G)} replaces manifolds with complexes.

1.1. Examples

If {G} is of type {F}, then {actdim(D)\leq 2 gdim(G)}. This comes from embedding complexes {BG} into {{\mathbb R}^N}. {N=2n+1} would be easy. {N=2n} is Stallings’ theorem, using a suitable model of {BG}.

Bestvina-Feighn: For lattices in semi-simle Lie groups, {actdim(G)} is the dimension of the symmetric space.

Desputovic: {actdim(MCG)=dim(}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 {L^2}-cohomology

Let {b_i(\tilde M)} denote the {L^2}-Betti numbers of the universal covering.

Singer conjecture: If {M} is a closed aspherical manifold of dimension {n}, then {b_i(\tilde M)} vanish if {i\not=n/2}.

This suggests

Action dimension conjecture. If {b_i(G)\not=0}, then {actdim(G)\geq 2i}.

Okun and I have shown that both conjectures are in fact equivalent.

2. Graph products

Let {L} be a flag complex with vertex set {S}. The graph product of a family {\{G_s\,;\,s\in S\}} of groups over {L} is the quotient of the free product of {G_s} by the normal subgroup generated by {[g_s,g_t]}, when {st} is an edge of {L}.

Examples. If all {G_s={\mathbb Z}}, we get RAAG. If all {G_s} are finite cyclic, we get RACG.

Theorem 1 Let {L} be a {d-1}-dimensional flag complex, let {G_L} be the corresponding graph product of fundamental groups of closed aspherical {m}-manifolds. Then

  1. If {H_{d-1}(L,{\mathbb Z}_2)\not=0}, then {actdim(G_L)=md+d}.
  2. If {H_{d-1}(L,{\mathbb Z}_2)=0}, then {actdim(G_L)<md+d}.

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 {\times} interval.

In general, we glue together products of {M_v\times I}, which is {md+d} dimensional, which is sharp in some cases, as we show next. The fact that {L} has vanishing homology allows to decrease dimension.

2.2. Obstructions to actions

Bestvina-Kapovitch-Kleiner coarsify van Kampen’s obstruction to embedding complexes {K} into {{\mathbb R}^N}. This lives in {H^n(Conf_2(K),{\mathbb Z}_2)} (configuration of pairs of points).

Theorem 2 (Bestvina-Kapovitch-Kleiner) Let {G} be {CAt(0)} or hyperbolic, let {K\subset\partial G} with {vK^n(K)\not=0}. Then

\displaystyle  \begin{array}{rcl}  actdim(G)\geq n+2. \end{array}

Example. If {G=F_2\times F_2}, {\partial G} contains {K_{3,3}}, hence {actdim(G)\geq 4} (in fact, {=2}).

For graph products of closed aspherical manifolds, we construct a complex, denoted by {\hat O L}, in {\partial G_L}. It is a join of {m-1}-spheres based on {L}.

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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 http://www.math.ens.fr/metric2011/
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