Geometric Transitions

A geometry, in Klein’s sense, is a space together with its group of symmetries, and the classical geometries, spherical, Euclidean and hyperbolic, all sit inside projective space as subgeometries. Inside projective space one geometry can be deformed into another: conjugate its symmetry group by a family of projective transformations that runs off to infinity, and take the limit. A sphere of growing radius becomes the Euclidean plane this way. Cooper, Danciger and Wienhard developed the general theory of these limits, and the reason topologists care is that geometric structures on a manifold collapse along exactly such paths, as hyperbolic cone manifolds do in Thurston’s Dehn surgery theorem. My thesis at UC Santa Barbara was about these transitions, and so are two of my papers.

The Heisenberg plane (Algebraic & Geometric Topology, 2023) works out one limiting geometry in full. Its symmetries are the translations of the plane and the shears parallel to a fixed line, which is the geometry of Galilean relativity in one space and one time dimension, and it is the common limit of the sphere, the Euclidean plane, the hyperbolic plane and their Lorentzian relatives, the most degenerate two-dimensional geometry there is. It has no invariant metric, so the usual tools for classifying geometric structures do not apply. The paper classifies the closed orbifolds that carry a Heisenberg structure, nine of them, computes their deformation spaces, and shows every Heisenberg structure is complete. It then asks which Heisenberg tori arise as limits of collapsing constant-curvature cone tori.

Tori whose holonomy contains a shear cannot regenerate, because on such a torus all simple geodesics are parallel, while a cone torus always has geodesics that cross.

Degenerations of the product geometries in projective space that contain Nil (Topology and its Applications, 2024, with Tom Shifley) fills the one gap left in Cooper, Danciger and Wienhard’s account of which Thurston geometries degenerate to which: whether $\mathbb{H}^2 \times \mathbb{R}$ and $S^2 \times \mathbb{R}$ can degenerate to contain the transitive subgroup of Nil.

These are the first limits to any model of Nil, and the paper gives the paths of matrices explicitly. We found them by reducing the problem to a system of limit equations in sixteen unknowns and searching numerically, then cleaning up the solutions by hand. Whether the product geometries degenerate to all of Nil, rotations included, is open.