Thurston's classification of rational maps, Cannon'sconjecture, and Quasispheres

• Antragsteller: Professor Daniel Meyer, Ph.D.
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2013 bis 2016
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 237093066

Cannon's conjecture stipulates that every group that acts topologically as a Kleinian group is in fact a Kleinian group. Thus we are asking if topology implies geometry, a prominent question in Thurston's work. This conjecture was originally formulated as part of a program to prove Thurston's hyperbolization conjecture. While this conjecture (indeed, the whole geometrization conjecture) has now been proved, thanks to the groundbreaking work of Perelman, Cannon's conjecture is independent. In particular, if the conjecture is false there would be groups that in a topological sense resemble Kleinian groups, yet are distinct. According to Sullivan's dictionary, there is a close correspondence between the dynamics of Kleinian groups and of rational maps. The question corresponding to Cannon's conjecture in the rational map case has been answered by Thurston in a celebrated paper. Namely Thurston gives a topological criterion, when a (postcritically finite) map that acts topologically as a rational map acts geometrically as a rational map. More precisely Thurston gives a criterion when such a map is in fact equivalent to a rational map. Namely such a Thurston map is equivalent if and only if there are no so-called Thurston-obstructions. This is an algebraic condition on how such a map pulls back curves. Cannon's conjecture and the statement in Thurston's theorem can both be equivalently formulated as meaning that certain metric spheres are in fact quasipsheres, i.e., quasisymmetrically equivalent to the standard sphere. This offers several lines of investigation. First one may use methods from quasiconformal geometry to study when a metric sphere is in fact a quasisphere. For certain spheres arising in the setting of Thurston's theorem this can be done. Thus one obtains an independent proof of Thurston's theorem (in these cases). The ultimate aim would be to obtain an alternative proof of Thurston's theorem, in the best possible case one that would help in attacking Cannon's conjecture. The second line of invesitigation is to consider certain self-similar metric spheres and use Thurston's theorem to decide if they are quasispheres. Thurston's theorem would need to be adjusted somewhat, which would gain new insight into this important theorem. In particular it is proposed to study Thurston's theorem in a setting that is close to the setting in the group case. In fact, a description of such Thurston maps that closely resembles the geometric description of the involved groups in Cannon's conjecture has been achieved by the applicant in joint work with Mario Bonk.

• DFG-Verfahren: Sachbeihilfen