## Notes of Alain Valette’s lecture nr 2

Compression of embeddings

Goal of talk is to explain why geometric group theorists are interested in metric embeddings.

1. Coarse embeddings

Definition 1 Let ${X}$ be a metric space, ${B}$ a Banach space. Say a map ${f:X\rightarrow B}$ is a coarse embedding if

${x}$, ${y\in X}$ are far apart iff ${f(x)}$, ${f(y)\in B}$ are far apart.

More precisely, if there exist two functions ${\rho_+}$, ${\rho_- :{\mathbb R}_+ \rightarrow {\mathbb R}_+}$, tending to ${+\infty}$, such that for all ${x}$, ${y\in X}$,

$\displaystyle \begin{array}{rcl} \rho_- (d(x,y))\leq |f(x)-f(y)|\leq \rho_+ (d(x,y)). \end{array}$

Remark 1 If ${X}$ is a connected graph, one can take ${\rho_+}$ linear.

We are mainly interested in finitely generated groups. These are metric spaces, in a unique way up to coarse equivalence.

Example 1 Baumslag-Solitar’s ${BS(2,1)=\langle a,\,b\,|\,abba^{-1}b^{-1}\rangle}$.

Represent it by matrices ${a=\begin{pmatrix} 1/2 &0 \\ 0 & 1 \end{pmatrix}}$ and ${b=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}}$. One can view these matrices as plane hyperbolic motions. The Cayley graph looks like the product of a degree ${3}$ tree and an interval.

Definition 2 If ${G}$ is a group acting isometrically on ${B}$, we say that ${f:X\rightarrow B}$ is an equivariant coarse embedding if there exists an (affine) isometric action of ${G}$ on ${B}$ such that for all ${x\in X}$ and ${g\in G}$, ${f(g.x)=g.f(x)}$.

Example 2 The case of trees.

Let ${X=(V,E)}$ be a tree. Fix a base point ${x_0 :in V}$. Define ${f:V\rightarrow \ell^{p}(E)}$ by

$\displaystyle \begin{array}{rcl} f(x)=\textrm{ characteristic function of }\{\textrm{set of edges on path }[x_0, x]\}. \end{array}$

Then ${|f(x)-f(y)|_{p}=d(x,y)^{1/p}}$. So ${f}$ is a coarse embedding with ${\rho_- (t)=t^{1/p}}$.

Modify ${f}$, to make it equivariant. Let ${\mathop{\mathbb E}}$ denote the set of oriented edges. Define ${c:V\times V\rightarrow\ell^{p}(E)}$ by

$\displaystyle \begin{array}{rcl} c(x,y)=e\mapsto &1& \textrm{if }e\textrm{ is traversed positively when travelling from }x\textrm{ to }y\\ &-1& \textrm{if }e\textrm{ is traversed negatively when travelling from }x\textrm{ to }y\\ &0& \textrm{otherwise}. \end{array}$

This is a cocycle: ${c(x,y)+c(y,z)=c(x,y)}$. Furthermore, ${c}$ is equivariant with respect to the natural representation of ${G=Aut(X)}$ on ${\ell^{p}(\mathop{\mathbb E})}$. Fix vertex ${x_0}$. Define an isometric action ${\alpha}$ of ${G}$ on ${\ell^{p}(\mathop{\mathbb E})}$ as follows. For ${\xi\in\ell^{p}(\mathop{\mathbb E})}$, ${g\in G}$,

$\displaystyle \begin{array}{rcl} \alpha(g)\xi=g\cdot \xi + c(gx_0,x_0). \end{array}$

Now define ${f(x)=c(x_0 ,x)}$. Then ${f}$ is equivariant.

2. Why do we care ?

Guoliang’s Yu ‘s 2000 theorem cast topology into the coarse world.

Theorem 3 Let ${G}$ be a finitely generated group. Assume that ${G}$ admits a coarse embedding into a Hilbert space. Then Novikov conjecture holds for ${G}$.

Kasparov-Yu 2004: one can replace Hilbert space by uniformly convex Banach space.

Novikov conjecture is hard to formulate, but here a closely related conjecture, the Borel stable conjecture: Let ${M}$, ${N}$ be closed manifolds with fundamental group ${G}$. Assume that their universal covers are contractible. If ${M}$ and ${N}$ are homotopy equivalent, ${M\times{\mathbb R}^k}$ and ${N\times{\mathbb R}^k}$ are homeomorphic for ${k}$ large enough.

Question: can one remove ${k}$ ? Answer: yes, then it is called Borel conjecture, it is much more ambitious.

Higson and Kasparov later proved a much stronger result.

Theorem 4 Let ${G}$ be a finitely generated group. Assume that ${G}$ admits a coarse embedding into a Hilbert space. Then Baum-Connes conjecture holds for ${G}$.

Remark 2 Some metric spaces do not admit any coarse embedding into Hilbert spaces.

E.g. spaces containing an expander family. This follows from Linial, London and Rabinovich’s paper mentioned this morning by Sanjeev Arora. Gromov was able to coarsely embed an expander family into the Cayley graph of a finitely generated group. This produces groups which do not admit any coarse embedding into Hilbert spaces.

Linial: can one use expanders arising from ${PSL(3,{\mathbb Q}_p)}$ ? Answer; yes, since, for well chosen generators, these graphs are not only expanders, but have large girth, a fact which is required in Gromov’s construction.

3. Compression

The following terminology has been introduced by Kaminker and Guentner.

Definition 5 The compression exponent of a map ${f:X\rightarrow B}$ is the supremum of ${\alpha}$ such that ${\rho_+}$ is linear and ${\rho_- (t)\geq C\,t^{\alpha}-D}$.

The ${L^{p}}$ compression of ${X}$, ${\alpha_{p}(X)}$, is the supremum of compressions of maps of ${X}$ to ${L^p}$.

Similarly, define the equivariant compression ${\alpha_{p}^{\#}(X)}$ for spaces with an isometric group action.

Example 3 ${\alpha_2 (\textrm{tree})=1}$. ${\alpha_{p}^{\#}(\textrm{Free group})=1/2}$(Kaminker and Guentner).

Proposition 6 ${\alpha_{2}(X)>\frac{1}{2}\Rightarrow}$ Yu’s property (A) ${\Rightarrow X}$ embeds coarsely into Hilbert space.

Think of Yu’s property (A) as a weak form of amenability.

Proposition 7 ${\alpha_{p}^{\#}(G)>\max\{\frac{1}{2},\frac{1}{p}\}\Rightarrow G}$ is amenable ${\Rightarrow X}$ embeds coarsely into ${\ell^p}$.

This is due to Naor and Peres, Bekka, Chérix and Valette.

4. Permanence properties

4.1. Wreath products

S. Li, after Naor and Peres.

Theorem 8 ${\alpha_p (G\wr H)\geq\max\{\frac{1}{2},\frac{1}{p}\}\min\{\alpha_1 (G),\frac{\alpha_1 (H)}{1+\alpha_1 (H)}\}}$

and a similar statement for equivariant compression.

4.2. Free and amalgamated products

Dreesen and Valette 2010.

Theorem 9 Let ${G=A\star_C B}$ where ${C}$ is finite, and has index at least ${3}$ in both ${A}$ and ${B}$. Then

1. ${\alpha_{2}^{\#}(G)=\min\{\alpha_{2}^{\#}(A),\alpha_{2}^{\#}(B),\frac{1}{2}\}}$
2. ${\min\{\alpha_{2}(A),\alpha_{2}(B),\frac{1}{2}\} \leq\alpha_{2}(G)\leq\min\{\alpha_{2}(A),\alpha_{2}(B)\}}$.

4.3. Extensions

Kazhdan’s relative property (T) restricts lower bounds on equivariant compression of extensions in terms of equivariant compression of the factors. For instance, the natural semi-direct product ${SL(2,{\mathbb Z})\times{\mathbb Z}^2}$ has equivariant compression ${0}$ (in every Hilbert action, ${{\mathbb Z}^2}$ fixes a point) whereas both factors have positive equivariant compression.

In the other hand, Dreesen and Valette 2010 show

Theorem 10 Let ${1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1}$. Equip ${N}$ with the distance induced from ${G}$.

1. If ${Q}$ has polynomial growth, then ${\alpha_2 (G)\geq\frac{1}{3}\alpha_2 (N)}$.
2. If ${Q}$ is hyperbolic, then ${\alpha_2 (G)\geq\frac{1}{5}\alpha_2 (N)}$.

References