## Notes of Pierre Pansu’s informal Cambridge lecture 07-02-2017

Coarse spaces associated to dynamics and spectral gaps

As an incentive for a working seminar, I informally survey papers by Roe, Drutu-Nowak, Vigolo, Benoist-De Saxcé (+ Sawicki).

Goal: understand ways of encoding dynamical properties of group actions in a coarse metric space.

1. Examples of dynamics

1. Hyperbolic dynamics: ${\begin{pmatrix} 2 &1 \\ 1 & 1 \end{pmatrix}}$ acting on the 2-torus, negatively curved geodesic flow.
2. Isometric dynamics: action on ${G}$ compact Lie group of dense subgroup ${\Gamma< G}$, action of group on its profinite (resp. pro-${p}$) completion.

2. Topological entropy

If ${A\in\mathrm{Homeo}(Y)}$, ${Y}$ compact metrizable, pick arbitrary compatible background metric ${d_Y}$ on ${Y}$. Fot ${t\in{\mathbb N}}$, define ${Y_t}$ as the space of length ${t}$ orbits, i.e. ${Y}$ metrized by

$\displaystyle \begin{array}{rcl} d_t(y,y')=\sup_{0\leq s\leq t}d_Y(A^s(y),A^s(y')). \end{array}$

Fix ${\epsilon>0}$. Let ${N(Y_t,\epsilon)}$ be the number of ${\epsilon}$-balls needed to cover ${Y_t}$. Then topological entropy is

$\displaystyle \begin{array}{rcl} h(A):=\lim_{t\rightarrow\infty}\frac{1}{t}\log N(Y_t,\epsilon). \end{array}$

Result does not depend on choice of ${\epsilon}$, neither on choice of background metric ${d}$.

Definition 1 (Gromov ?) Let ${X}$ be the space of geodesic segments, i.e. ${X=Y\times{\mathbb N}}$ metrized by

$\displaystyle \begin{array}{rcl} \tilde d((y,t),(y',t'))=|t-t'|+\sup_{0\leq s\leq t}d_Y(A^s(y),A^s(y')). \end{array}$

Since, coarsely, this is a half-line, replace ${\tilde d}$ with the corresponding ${\epsilon}$-geodesic metric

$\displaystyle \begin{array}{rcl} d_\epsilon(x,x')=\inf\{\sum_{i=0}^{k-1}\tilde d(x_i,x_{i+1})\,;\,x_0=x,\,x_k=x',\,\sup_i \tilde d(x_i,x_{i+1})<\epsilon\}. \end{array}$

Question. Show that coarse space ${X_\epsilon}$ does not depend on ${\epsilon}$ nor on choice of background metric.

If so, every large scale invariant of ${X_\epsilon}$ become a dynamical invariant. For instance, topological entropy equals the exponential volume growth of ${X_\epsilon}$.

Question. If ${A}$ is the time 1 geodesic flow on unit tangent bundle ${Y=T_1 M}$ of a compact negatively curved manifold ${M}$, then ${X_\epsilon}$ is quasi-isometric to universal covering space ${\tilde M}$ ?

3. John Roe’s warped cone construction

${Y}$ Riemannian manifold, ${\mathrm{cone}(Y)=Y\times{\mathbb R}_+}$ with metric ${t^2g_Y+dt^2}$. Roe warps it with the ${\Gamma}$ action on ${Y\times{\mathbb R}_+}$ arising from ${\Gamma}$-action on ${Y}$.

Definition 2 ${X}$ metric space. ${\Gamma}$ acts by homeos of ${X}$ with finite generating set ${S}$. Define warped metric ${d_\gamma=}$ largest metric on ${X}$ with is ${\leq d_X}$ and such that for all ${s\in S}$, ${d_\Gamma(x,sx)\leq 1}$. Alternatively,

$\displaystyle \begin{array}{rcl} d_\Gamma(x,x')=\inf\{\sum_{i=0}^{k-1}d_X(x_i,x_{i+1})+|\gamma_i|\,;\,x_0=x,\,x_k=x',\,\gamma_i\in\Gamma\}. \end{array}$

Define warped cone ${O_\Gamma(Y)}$ as ${\mathrm{cone}(Y)}$ warped by ${\Gamma}$ action on ${Y\times{\mathbb R}_+}$ arising from ${\Gamma}$-action on ${Y}$.

As a coarse space, ${O_\Gamma (Y)}$ does not depend on choices of background metric on ${Y}$ and generating set of ${\Gamma}$. If ${\Gamma}$ acts by bi-Lipschitz homeos, ${O_\Gamma (Y)}$ has bounded geometry.

Example. Let ${Y={\mathbb R}/{\mathbb Z}}$ and ${\Gamma}$ cyclic group generated by translation ${y\mapsto y+\alpha}$. Then ${X=}$ Euclidean plane, ${\Gamma}$ is generated by the rotation of angle ${2\pi\alpha}$. The diameter of level ${t}$ seems to be governed by

$\displaystyle \begin{array}{rcl} \min_{0\leq s\leq t} 2\pi t\{s\alpha\}. \end{array}$

If ${|\alpha-\frac{p}{q}|\leq \frac{\epsilon}{q^2}}$, then ${q\{q\alpha\}\leq\epsilon}$. Thus if ${\alpha\notin{\mathbb Q}}$, there are infinitely many values of ${t}$ (Hurwitz) for which the diameter of ${t}$ level collapses to something pretty small (not smaller than ${\log t}$, though).

Question. Describe this example in more detail.

Remark. A smooth ${\Gamma}$-action on ${Y}$ can be suspended into a ${Y}$-bundle over ${B\Gamma}$ with a foliation ${\mathcal{F}}$ transverse to fibers. Pick a complement ${\nu}$ to ${T\mathcal{F}}$ in ambient tangent bundle. Warp metric on ${\mathrm{cone}(Y)}$ as follows:

$\displaystyle \begin{array}{rcl} g_{\mathcal{F}}+t^2 g_{\nu}+dt^2. \end{array}$

Resulting space is coarse equivalent to ${O_{\Gamma}(Y)}$.

Example. Let ${Y={\mathbb R}/{\mathbb Z}}$ and ${\Gamma}$ cyclic group generated by translation ${y\mapsto y+\alpha}$. Then suspension is torus ${({\mathbb R}/{\mathbb Z})^2}$ with linear foliation of slope ${\alpha}$.

Question. Let ${\Gamma={\mathbb Z}}$ acting by matrix ${\begin{pmatrix} 2 &1 \\ 1 & 1 \end{pmatrix}}$ on the 2-torus. Compare ${O_\Gamma(T^2)}$ with warping along linear foliation defined by one eigenspace.

Question. Let ${\Gamma}$ act on its profinite completion ${\hat\Gamma}$ (or on its pro-${p}$ completion). Show ${O_{\Gamma}\hat\Gamma}$ is coarsely equivalent to the box space.

G. Arzhantseva points out Damian Sawicki’s paper on warped cones and profinite completions. It contains many examples and results parallel to those of Roe below.

4. Theorems

4.1. Monstruosity

Yu introduced property A as a coarse metric generalization of Haagerup property. It is designed to produce coarse Hilbert embeddings.

Definition 3 (Yu) Say space ${X}$ has property A if there exists a sequence of families of probability measures ${x\mapsto f_n(x)}$ on ${X}$ such that

1. Uniformly bounded support: ${\mathrm{supp}(f_n(x))\subset B(x,r_n)}$.
2. Spreading: ${\|f_{n}(x)-f_n(x')\|_1 \leq\epsilon_n(s)\rightarrow 0}$ uniformly as ${d(x,x')\leq s}$.

Next is one of a hundred equivalent definitions of amenability of actions. All actions of amenable groups are amenable, but there are more. For instance, the action of a hyperbolic group on its ideal boundary is amenable. Morally (this is exactly true only for rank one Lie groups), this means that stabilizers are amenable.

Definition 4 Say ${\Gamma}$ action on compact space ${Y}$ is amenable if there exists a family of finite support probability measures ${y\mapsto\mu_{n}(y)}$ on ${\Gamma}$ such that ${\forall \gamma\in\Gamma}$,

$\displaystyle \begin{array}{rcl} \sup_{y\in Y}\|\gamma\cdot\mu_{n}(y)-\mu_{n}(\gamma y)\|_1 \rightarrow 0 \quad \textrm{as }n\rightarrow\infty. \end{array}$

Note that ${\Gamma}$, as a metric space, has property A iff it admits an amenable action on some compact space.

Roe 2005. If action of ${\Gamma}$ on ${Y}$ is amenable, then ${O_\Gamma(Y)}$ has Yu’s property A. In particular, it has a coarse embedding into a Hilbert space.

Roe 2005. Let ${G}$ be a compact Lie group and ${\Gamma a dense subgroup. Then

$\displaystyle \begin{array}{rcl} O_\Gamma (G) \textrm{ has property }A \quad &\Rightarrow&\quad \Gamma \textrm{ is amenable}.\\ O_\Gamma (G) \textrm{ has coarse Hilbert embedding} \quad &\Rightarrow&\quad \Gamma \textrm{ has Haagerup property}.\\ \end{array}$

These results indicate that warped cones can be monstruous (if ${\Gamma}$ is Kazhdan, for instance). Roe even announced a warped cone counterexample to coarse Baum-Connes conjecture, but nothing appeared until

Drutu-Nowak 2015. Certain warped cones have ghost projections in their Roe algebras.

4.2. A new source of expanders

Let ${S}$ be a generating set for ${\Gamma}$. Say ${\Gamma}$ action is expanding in measure if ${\mu(SA)\geq (1+\epsilon)\mu(A)}$ for all subsets ${A}$ with ${\mu(A)\leq 1/2}$.

Vigolo 2016. Let ${\Gamma}$ preserve a probability measure on ${Y}$. Then levels of ${O_\Gamma (Y)}$ constitute a coarse expander iff ${\Gamma}$ action on ${L^2(Y)}$ is expanding in measure.

Question. Investigate further properties of these new expanders.

Question. Compare Vigolo’s construction to classical constructions of expanders, like Margulis type (${\Gamma}$ acting on ${\hat\Gamma}$) or zig-zag products (perhaps requires generalizing warped cones to finitely generated equivalence relations).

5. Actions with spectral gaps

It turns out that measure expansion on ${Y}$ is equivalent to spectral gap on ${L^2(Y)}$.

Definition 5 Let ${\Gamma}$ act linearly isometrically on some Banach space ${B}$. Say action has a spectral gap if if there exists ${\epsilon>0}$ and a generating set ${S}$ such that ${\forall v\in B}$, ${|v|=1}$, ${\exists s\in S}$ such that ${|sv-v|\geq\epsilon}$.

For instance, Kazhdan groups have a uniform spectral gap in any unitary representation. On the other hand, irrational rotations of the circle never have spectral gaps on ${L^2(\textrm{circle})}$. Nevertheless, even free groups can have spectral gaps in specific representations.

Therefore we want to understand a few more examples of dense subgroups ${\Gamma with a spectral gap.

The adequate context is that of a symmetric probability measure ${\mu}$ on ${G}$ (think of ${\Gamma}$ as generated by the support of ${\mu}$). One always assumes that the support of ${\mu}$ generates a dense subgroup. A spectral gap on ${L^2(G)}$ means that the spectral radius of the averaging operator

$\displaystyle \begin{array}{rcl} \int_{G}L_g \,d\mu(g) \end{array}$

is ${<1}$.

Benoist-de Saxcé 2016 (generalizing Bourgain-Gamburd). Let ${G}$ be a compact simple Lie group. ${\mu}$ has a spectral gap iff ${\mu}$ is almost Diophantine in the following sense: ${\mu}$ does not put much mass on tubular neighborhoods of connected subgroups. Specifically, ${\exists c}$ and ${C}$ such that ${\forall n}$, for every proper closed subgroup ${H,

$\displaystyle \begin{array}{rcl} \mu\{HB(e^{-Cn})\}\leq e^{-cn}. \end{array}$

Benoist-de Saxcé 2016. Assume that ${\forall g\in\mathrm{supp}(\mu)}$, ${Ad_g}$ has algebraic entries (in a fixed basis of ${\mathfrak{g}}$). Then ${\mu}$ is almost Diophantine.

We see that the structure of ${\Gamma}$ does not show up (is ${\Gamma}$ free, most of the time ?), the essential role is played by the structure of ${G}$.

The main tool is de Saxce’s Product Theorem (elaboration on results of Bourgain-Gamburd): if a subset ${A\subset G}$ near identity does not grow much under tripling ${A\mapsto AAA}$, then ${A}$ is Hausdorff close to a proper closed subgroup. Size is measured by covering number ${N(A,\delta)}$.