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We develop the Lorentzian geometry of a crooked halfspace in 2 + 1-dimensional Minkowski space. We calculate the affine, conformal and isometric automorphism groups of a crooked halfspace, and discuss its stratification into orbit types, giving an explicit slice for the action of the automorphism group. The set of parallelism classes of timelike lines, or particles, in a crooked halfspace is a geodesic halfplane in the hyperbolic plane. Every point in an open crooked halfspace lies on a particle. The correspondence between crooked halfspaces and halfplanes in hyperbolic 2-space preserves the partial order defined by inclusion, and the involution defined by complementarity. We find conditions for when a particle lies completely in a crooked half space. We revisit the disjointness criterion for crooked planes developed by Drumm and Goldman in terms of the semigroup of translations preserving a crooked halfspace. These ideas are then applied to describe foliations of Minkowski space by crooked planes.Date: November 7, 2018. 2000 Mathematics Subject Classification. 53B30 (Lorentz metrics, indefinite metrics), 53C50 (Lorentz manifolds, manifolds with indefinite metrics).

Associated to every complete affine 3-manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface Σ. We classify such complete affine structures when Σ is homeomorphic to a three-holed sphere. In particular, for every such complete hyperbolic surface Σ, the deformation space identifies with two opposite octants in R 3 . Furthermore every M admits a fundamental polyhedron bounded by crooked planes. Therefore M is homeomorphic to an open solid handlebody of genus two. As an explicit application of this theory, we construct proper affine deformations of an arithmetic Fuchsian group inside Sp(4, Z).

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The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on dimension 2 + 1, in which there is a rich interplay with symplectic geometry.