## 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.

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

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 ${\times MCG(W^c)}$ which is locally constant outside ${\times MCG(W)}$.