Limits of the Thurston Geometries
The analysis of collapsing hyperbolic cone-manifolds plays an important role in both Thurston's hyperbolic Dehn surgery and the proof of the Orbifold Theorem, as
other geometric structures (such as Euclidean, Nil, and Sol) can be produced by carefully rescaling the hyperbolic metric along a collapsing family of singular hyperbolic structures. Understanding which structures may occur in this way amounts to classifying the rescaling limits of hyperbolic space. In 2014 Cooper, Danciger and Wienhard completed this classification, and raised the natural extension to the geometries of Geometrization: when does some rescaled limit of Thurston geometry X contain the Thurston geometry Y? Their further work on this problem resolved 62 of the 64 possible cases. In this talk I will discuss the remaining two, and show the product geometries H2xR and S2xR do not limit to Nil.