[Paper Review] Affine Coxeter Extensions of the Two-Holed Projective Plane

William M. Goldman, Gregory Laun|arXiv (Cornell University)|Nov 17, 2015

Geometric Analysis and Curvature Flows7 references3 citations

TL;DR

This paper classifies proper affine actions of the double extension of the fundamental group of a two-holed projective plane (a cross surface), showing that the space of proper affine deformations admitting crooked fundamental domains forms a hexagon inscribed in a quadrilateral within projective space. It proves the existence of proper actions without crooked fundamental domains, extending results from orientable to non-orientable hyperbolic surfaces.

ABSTRACT

A Margulis spacetime is a complete flat affine Lorentzian 3-manifold with free fundamental group. Associated to $M$ is a noncompact complete hyperbolic surface $Σ$. We study double extensions of $π_1 (M) \cong π_1 (Σ)$ when $Σ$ is homeomorphic to a projective plane minus two discs. We classify proper actions of this double extension on Minkowski space and show that there exist proper actions that do not admit crooked fundamental domains.

Motivation & Objective

• To classify proper affine actions of the Coxeter extension of the fundamental group of a two-holed projective plane.
• To study affine orbifolds double-covered by Margulis spacetimes with non-compact hyperbolic surfaces homeomorphic to a two-holed cross surface.
• To extend the theory of crooked fundamental domains to non-orientable hyperbolic surfaces.
• To determine whether proper affine deformations of such groups admit crooked fundamental domains.

Proposed method

• Use hyperideal triangulations of the two-holed projective plane to construct configurations of crooked planes.
• Apply the theory of crooked ideal triangulations to parametrize deformation spaces for a fixed ideal triangulation.
• Employ the flip graph to encode changes in deformation spaces under ideal triangulation changes.
• Use the arc complex as the dual of the flip graph to describe the full deformation space.
• Fix a geodesic representative for the homotopy class of the third side of a triangular fundamental domain, parameterized by θ.
• Projectivize the deformation space to analyze the geometry of crooked fundamental domains in RP².

Experimental results

Research questions

• RQ1 Does every proper affine deformation of the Coxeter extension of a two-holed projective plane fundamental group admit a crooked fundamental domain?
• RQ2 How does the space of proper affine deformations with crooked fundamental domains relate to the space of hyperbolic structures on the orbifold quotient?
• RQ3 What is the geometric shape of the projectivized space of crooked fundamental domains for such deformations?
• RQ4 Are there proper affine deformations that do not admit crooked fundamental domains, and if so, how are they characterized?
• RQ5 How does varying the geodesic representative θ affect the structure of the deformation space?

Key findings

• The space of proper affine deformations admitting crooked fundamental domains for fixed θ is a six-sided cone over the moduli space of hyperbolic structures on the orbifold quotient.
• This cone projectivizes to a hexagon H inscribed in the quadrilateral Q, which parametrizes the projectivized space of crooked fundamental domains for the index-two free group.
• Varying the geodesic representative θ results in an octagon instead of a hexagon, as two vertices are replaced by intervals.
• There exist proper affine deformations of the group that do not admit crooked fundamental domains, corresponding to points in Q ∖ H.
• The image of the positive orthant under the deformation matrix projects to a pentagon P₁ inscribed in Q, with a vertex at the top corresponding to the rank-1 matrix M₃.
• The construction yields a second pentagon P₂ from a different configuration, intersecting P₁ in a smaller quadrilateral Q_small, confirming the geometric structure of the deformation space.