Limits of geometries

TL;DR

This paper develops a framework for understanding geometric transitions between different structures, classifies limits of hyperbolic geometry within projective geometry, and identifies which geometries can arise as limits.

Contribution

It introduces a general method to analyze geometric limits via sub-geometries in ambient geometries and classifies all such limits for three-dimensional hyperbolic geometry.

Findings

Classified all limits of hyperbolic geometry in projective geometry.

Identified Euclidean, Nil, and Sol geometries as limits.

Showed that certain Thurston geometries do not appear as limits.

Abstract

A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spaces, describing the basic process by which one homogeneous geometry may transform into another. We develop a general framework to describe transitions in the context that both geometries involved are represented as sub-geometries of a larger ambient geometry. Specializing to the setting of real projective geometry, we classify the geometric limits of any sub-geometry whose structure group is a symmetric subgroup of the projective general linear group. As an application, we classify all limits of three-dimensional hyperbolic geometry inside of projective geometry, finding Euclidean, Nil, and Sol geometry among the limits. We prove, however, that the other Thurston geometries, in particular $\mathbb{H}^2 \times \mathbb{R}$ and $\widetilde{\operatorname{SL}_2 \mathbb{R}}$, do not embed in any limit of hyperbolic geometry in this sense.