Notes of Cornelia Drutu’s lecture

Ranks of mapping class groups

{S} orientable genus {g} surface, {p} boundary components. Define

\displaystyle  \begin{array}{rcl}  \mathrm{Complexity}(S):=\xi(S)=3g+p-3. \end{array}

The mapping class group {MCG(S)} is the group of isotopy classes of orientation preserving homeomorphisms of {S}.

Assume complexity is at least 2 (otherwise, MCG is virtually free).

We shall use various notions of rank.

1. Version 1

Let {X} be a Hadamard manifold. Assume {X} has a discrete cocompact isometry group {G}. The quasiflat rank is the maximal dimension of quasiflats, i.e. quasi-isometrically embedded Euclidean spaces. Replacing quasi-isometrically by isometrically does not change the definition (this is a theorem of Bruce Kleiner).

Example 1 Symmetric spaces. There, quasi-flat rank is the usual rank, even for non uniform lattices.

Indeed, Lubotzky-Moses-Raghunathan show that non uniform lattices of higher rank are quasi-isometrically embedded.

This definition does not quite serve our purposes, since

Theorem 1 (Kapovich-Leeb) No geometric action of {MCG(S)} on Hadamard manifolds.

Nevertheless, MCG’s behave like {CAT(0)} groups in many respects,

Theorem 2 (Farb-Lubotzky-Minsky) {MCG(S)} has {\xi(G)}-dimensional quasi-flats.

Quasi-flat rank conjecture (Hamenstädt, Behrstock-Minsky). QFlatRank {=\xi(S)} ?

2. Version 2

Let {X} be a Hadamard manifold. A {k}-sphere is {X} in a Lipschitz map {f} of round {S^k} to {X}. So is a {k+1}-ball. Define

\displaystyle  \begin{array}{rcl}  \mathrm{Fill}(f)&=&\inf\{\mathrm{volume}(g)\,;\,g\textrm{ fills }f\},\\ Iso_k(x)&=&\sup\{\mathrm{Fill}(f)\,;\,\mathrm{volume}(f)\leq A\,x^k\}. \end{array}

Definition 3 (Filling rank)

\displaystyle  \begin{array}{rcl}  \mathrm{FillRank}=\max\{k\,;\,Iso_k(x)\leq x^{k+1}\}. \end{array}

Possible alternate definition: {\min\{k\,;\,Iso_k(x)\sim x^{k}\}}.

Proposition 4 For uniform lattices in symmetric spaces, FillRank=QFlatRank.

However, for non-uniform lattices, the story is still open. Robert Young solved the case of {Sl(n,{\mathbb Z})}.

Theorem 5 (Behrstock-Drutu) For {k\leq\xi(S)-1}, {Iso_k(x)\sim x^{k+1}}.

For {k\geq\xi(S)}, {Iso_k(x)=o(x^{k+1})}.

In small genera, we can improve the second estimate.

Difficulties: cutting spheres in pieces.

3. Version 3

Let {X} be a Hadamard manifold. Let us define higher dimensional divergence: it measures how geodesics diverge, and simultaneously, filling properties.

Fix a base point {p_0}. Let {f:S^k\rightarrow X\setminus B(p_0,x)} be a sphere of volume volume{(f)\leq A\,x^k}, outside a large ball. Define

\displaystyle  \begin{array}{rcl}  \mathrm{divergence}_k(f)&=&\inf\{\mathrm{volume}(g)\,;\,g\textrm{ fills }f\textrm{ outside }B(p_0,\lambda x)\}.\\ Div_k(x)&=&\sup\{\mathrm{divergence}_k(f)\,;\,\mathrm{volume}(f)\leq A\,x^k \textrm{ outside }B(p_0,x)\}. \end{array}

Example 2 {X} symmetric space. Then

  1. If {k<\mathrm{QFlatRank}(X)-1}, {Div_k(x)\sim x^{k+1}}.
  2. If {k=\mathrm{QFlatRank}(X)-1}, {Div_k(x)\sim e^x} (Leuzinger).
  3. If {k\geq\mathrm{QFlatRank}(X)}, {Div_k(x)\leq x^{k+1}} (Hindawi), improved by Wenger into {\sim x^{k+1}}.

Theorem 6 (Behrstock-Drutu)

  1. If {k<\xi(S)-1}, {Div_k(x)\geq x^{k+2}}.
  2. If {k\geq \xi(S)}, {Div_k(x)=o(x^{k+1})}.

Again, in low genera, one can improve a bit. Also, 1-dimensional divergence is better understood.

Theorem 7 (Abrams-Brady-Duchin-Young)

\displaystyle  \begin{array}{rcl}  Div_1(x)\leq x^4. \end{array}

4. Intermediate results

Here are two facts that hold in a larger generality.

If a group {G} is {\mathcal{F}_{\infty}} and has a combing, then

\displaystyle  \begin{array}{rcl}  Iso_k(x)\leq x^{k+1}. \end{array}

If {G} is hyperbolic, then all {Iso_k} are linear (for Lipschitz maps, this is due to Urs Lang).

5. Proof

I explain why MCG’s do not behave like Hadamard spaces.

For all {k\leq} QFlatRank {=\xi(S)}, there exists a {k}-quasflat that is maximal, i.e. not contained in a quasi-flat of higher dimension. This is different from the case of symmetric spaces. These lower dimensional quasiflats arise from multicurve splittings of the surface.

For instance, split {S=U_1\cup U_2} where {U_1} is an annulus. Then Dehn twists in {U_1} provide undistorted subgroups {{\mathbb Z}\times MCG(U_2)} in MCG{(S)}.

Our main tool: let {W\subset S} be a subsurface. Then there is a Lipschitz map of {MCG(S)} to {<pA>\times MCG(W^c)} which is locally constant outside {<pA>\times MCG(W)}.

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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