Dynamics and quasiconformal geometry
1. Motivation : Cannon’s conjecture
Gromov hyperbolic group. When is the ideal boundary a topological 2-sphere ? It is the case when is virtually the fundamental group of a compact hyperbolic 3-manifold.
Version 1: Are there other examples ?
Version 2: Show that the ideal boundary , equipped with a visual metric, is quasi-symmetric to the standard 2-sphere.
1.1. Visual metrics
Recall that the visual metric , where the Gromov product is (up to a bounded additive error) the distance of base point to the geodesic joing to . Changing the parameter changes to a snowflake equivalent metric (i.e. a power of it). The word comes from the fact that the von Koch snowflake curve is snowflake equivalent to the real line.
Quasi-symmetric means that ratios in one metric are controlled by similar ratios in the other. Equivalently, balls in one metric are pinched between concentric balls in the other, with a bounded ratio of radii. This has to do with quasi-conformality (an infinitesimal version of quasi-symmetry): for Euclidean domains, quasi-symmetry is equivalent to qausi-conformality.
Proposition 1 Under mild assumptions, Gromov hyperbolic metric spaces are quasi-isometric iff their ideal boundaries are quasi-symmetric.
A Gromov hyperbolic group acts on its ideal boundary by quasi-symmetries, in a uniform manner. To make this quantitative, one must inteoduce the notion of quasi-Möbius map (replace ratios with cross-ratios of 4-tuples of points).
Version 2 implies Version 1. Indeed, if is quasi-symmetric to the round sphere, acts by uniformly quasi-Möbius on the round sphere. A result of Sullivan and Tukia implies that the action is by Möbius transformations.
2. The quasi-symmetric uniformization problem
When is a metric space is quasi-symmetric to a standard space ?
This is relevant for the
2.1. Kapovich-Kleiner conjecture
Let be a Gromov hyperbolic group whose ideal boundary is homeomorphic to a Sierpinsky carpet (start with a square, cut in 9 pieces, remove central square, iterate in each of the 8 remaining squares). Does arise from a standard situation in hyperbolic geometry ?
This is equivalent to showing that is quassymmetric to a round carpet (remove disjoint geometric circle from a circle, until no interior is left).
Cannon conjecture implies Kapovich-Kleiner conjecture.
This is also relevant to
2.2. Other problems in semi-group dynamics
From a branched covering , one defines a Gromov hyperbolic graph (this a rather long story, I willnot give details).
Theorem 2 (Bonk-Meyer, Haissinsky-Pilgrim) Let be a postcritically finite expanding branched covering. Then is conjugate to a rational map iff is quasi-symmetric to the round sphere.
Sometimes it is true, sometimes it is not. The visual metrics arising from branched covering maybe non quasi-symmetric to the standard sphere.
2.3. Example: the snow sphere
Start with the boundary of the cube. Subdivide each face in 9 squares, build a small cube on the middle square, and iterate. This produces a metric which is not a snowflake of the standard sphere (it has rectifiable curves).
Question. Is it quasi-symmetric to the standard sphere ?
At first sight, one would bet that the answer is no. If it were, all squares in the construction should remain uniformly round. Flattening the first stage of the construction to a cube is easy, but doing this at all stages accumulates distorsion. Nevertheless,
Theorem 3 (Meyer) The snow sphere is quasi-symmetric to the round sphere.
3. What is known ?
A lot of positive results for dimensions 0 and 1. No positive results in higher dimensions . Semmes writes that all the naive facts one could think of turn out to be wrong. So interesting things happen in dimension 2.
3.1. Low dimensional results
Theorem 4 (Tukia-Väisälä) A metric on the circle is quasi-symmetri to the standard metric iff
- It is doubling.
- It has bounded turning: every arc with endpoints and has diameter.
Similar result for Cantor sets.
3.2. Results in dimension 2
Theorem 5 (Bonk-Kleiner) Let be a metric sphere homeomorphic to the 2-sphere. Assume that
- is linearly locally connected (this is a necessary condition).
- is Ahlfors 2-regular (this is not at all necessary).
Then is quasi-symmetric to the standard sphere.
Note that visual metrics of hyperbolic groups are Ahlfors-regular (Coornaert).
Theorem 6 (Bonk-Kleiner) Let be Gromov hyperbolic group whose ideal boundary is homeomorphic to the 2-sphere. Assume that the conformal dimension is attained as a minimum. Then is quasi-symmetric to the standard sphere.
Recall that the conformal dimension is the infimal dimension of Alhfors regular metric spaces quasi-symmetric to .
We strongly use 2 dimensions, but an intermediate step applies in all dimensions: conformal dimension attained implies metric is Löwner. Note that there are examples (due to Bourdon and Pajot) of groups whose conformal dimension is not attained. These examples have boundaries which are not 2-spheres.