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Riemannian limits of the product geometries $\mathbb{H}^2\times\mathbb{R}$ and $S^2\times\mathbb{R}$

Geometry/Topology

Speaker: Steve Trettel, Stanford University
Location: 3106 MSB
Start time: Tue, Feb 25 2020, 3:10PM

As $k$ goes to zero, the 1-parameter family of hyperbolic spaces with curvature $k$ converges to Euclidean space. This fact was already noted by Klein at the end of the 19th century, who produced an explicit model of this transition as a conjugacy limit in projective space. As understanding the limiting behaviors of a geometry informs the study of degenerations of geometric structures, the classification of such limits is of interest to geometric topology. And in dimension three, geometrization suggests a natural project: classifying the limits of the Thurston geometries. More precisely - when can a Thurston geometry $X$ be deformed in projective space so that it degenerates to some other Thurston geometry $Y$? Work of Cooper, Danciger, and Wienhard nearly completed this classification, resolving 62 of the 64 possible cases. In this talk I will discuss the remaining two, and show the product geometries $\mathbb{H}^2\times\mathbb{R}$ and $S^2\times\mathbb{R}$ do not limit to $\mathsf{Nil}$.