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

坎农的猜想规定，每一个在拓扑上充当克莱因群的群实际上都是克莱因群。因此，我们在问拓扑学是否意味着几何学，这是瑟斯顿著作中的一个突出问题。这一猜想最初是作为证明瑟斯顿双曲猜想的程序的一部分而提出的。虽然这个猜想(实际上，整个几何化猜想)现在已经被证明，但由于佩雷尔曼的开创性工作，坎农的猜想是独立的。特别地，如果猜想是假的，则存在在拓扑意义上类似于Klein群但又是不同的群。根据Sullivan的词典，在Klein群的动力学和有理映射之间存在密切的对应。瑟斯顿在一篇著名的论文中回答了与Cannon在有理映射情况下的猜想相对应的问题。也就是说，当一个(后临界有限)映射在拓扑上起到有理映射的作用时，瑟斯顿给出了一个拓扑判据。更准确地说，瑟斯顿给出了这样一个映射实际上等价于有理映射的判据。也就是说，当且仅当不存在所谓的瑟斯顿障碍时，这样的瑟斯顿映射是等价的。这是关于这样的映射如何拉回曲线的一个代数条件。卡农猜想和瑟斯顿定理中的表述都可以等价地表述为某些度规球体实际上是准psheres，即准对称等价于标准球体。这提供了几条调查线索。首先，人们可以使用拟共形几何的方法来研究度规球体何时实际上是准球体。对于在瑟斯顿定理的背景下出现的某些球体，这是可以做到的。因此，我们得到了瑟斯顿定理的一个独立证明(在这些情况下)。最终目的将是获得瑟斯顿定理的另一种证明，在最好的情况下，这种证明将有助于反驳坎农的猜想。研究的第二条线是考虑某些自相似度规球体，并利用瑟斯顿定理来判定它们是否是准球。瑟斯顿定理需要作一些调整，这将对这一重要定理有新的认识，特别是建议在接近群情形的背景下研究瑟斯顿定理。事实上，申请者与马里奥·邦克的合作已经实现了对这种瑟斯顿映射的描述，这种描述非常类似于Cannon猜想中涉及的群的几何描述。

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 theconjecture 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.