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
- Hyperbolic dynamics: acting on the 2-torus, negatively curved geodesic flow.
- Isometric dynamics: action on compact Lie group of dense subgroup , action of group on its profinite (resp. pro-) completion.
2. Topological entropy
If , compact metrizable, pick arbitrary compatible background metric on . Fot , define as the space of length orbits, i.e. metrized by
Fix . Let be the number of -balls needed to cover . Then topological entropy is
Result does not depend on choice of , neither on choice of background metric .
Definition 1 (Gromov ?) Let be the space of geodesic segments, i.e. metrized by
Since, coarsely, this is a half-line, replace with the corresponding -geodesic metric
Question. Show that coarse space does not depend on nor on choice of background metric.
If so, every large scale invariant of become a dynamical invariant. For instance, topological entropy equals the exponential volume growth of .
Question. If is the time 1 geodesic flow on unit tangent bundle of a compact negatively curved manifold , then is quasi-isometric to universal covering space ?
3. John Roe’s warped cone construction
Riemannian manifold, with metric . Roe warps it with the action on arising from -action on .
Definition 2 metric space. acts by homeos of with finite generating set . Define warped metric largest metric on with is and such that for all , . Alternatively,
Define warped cone as warped by action on arising from -action on .
As a coarse space, does not depend on choices of background metric on and generating set of . If acts by bi-Lipschitz homeos, has bounded geometry.
Example. Let and cyclic group generated by translation . Then Euclidean plane, is generated by the rotation of angle . The diameter of level seems to be governed by
If , then . Thus if , there are infinitely many values of (Hurwitz) for which the diameter of level collapses to something pretty small (not smaller than , though).
Question. Describe this example in more detail.
Remark. A smooth -action on can be suspended into a -bundle over with a foliation transverse to fibers. Pick a complement to in ambient tangent bundle. Warp metric on as follows:
Resulting space is coarse equivalent to .
Example. Let and cyclic group generated by translation . Then suspension is torus with linear foliation of slope .
Question. Let acting by matrix on the 2-torus. Compare with warping along linear foliation defined by one eigenspace.
Question. Let act on its profinite completion (or on its pro- completion). Show 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.
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 has property A if there exists a sequence of families of probability measures on such that
- Uniformly bounded support: .
- Spreading: uniformly as .
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 action on compact space is amenable if there exists a family of finite support probability measures on such that ,
Note that , as a metric space, has property A iff it admits an amenable action on some compact space.
Roe 2005. If action of on is amenable, then has Yu’s property A. In particular, it has a coarse embedding into a Hilbert space.
Roe 2005. Let be a compact Lie group and a dense subgroup. Then
These results indicate that warped cones can be monstruous (if 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 be a generating set for . Say action is expanding in measure if for all subsets with .
Vigolo 2016. Let preserve a probability measure on . Then levels of constitute a coarse expander iff action on 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 ( acting on ) 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 is equivalent to spectral gap on .
Definition 5 Let act linearly isometrically on some Banach space . Say action has a spectral gap if if there exists and a generating set such that , , such that .
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 . Nevertheless, even free groups can have spectral gaps in specific representations.
Therefore we want to understand a few more examples of dense subgroups with a spectral gap.
The adequate context is that of a symmetric probability measure on (think of as generated by the support of ). One always assumes that the support of generates a dense subgroup. A spectral gap on means that the spectral radius of the averaging operator
Benoist-de Saxcé 2016 (generalizing Bourgain-Gamburd). Let be a compact simple Lie group. has a spectral gap iff is almost Diophantine in the following sense: does not put much mass on tubular neighborhoods of connected subgroups. Specifically, and such that , for every proper closed subgroup ,
Benoist-de Saxcé 2016. Assume that , has algebraic entries (in a fixed basis of ). Then is almost Diophantine.
We see that the structure of does not show up (is free, most of the time ?), the essential role is played by the structure of .
The main tool is de Saxce’s Product Theorem (elaboration on results of Bourgain-Gamburd): if a subset near identity does not grow much under tripling , then is Hausdorff close to a proper closed subgroup. Size is measured by covering number .