Finite-sided deformation spaces of complete affine 3-manifolds. (English) Zbl 1297.30070

Charette, Virginie; Drumm, Todd A.; Goldman, William M.

J. Topol. 7, No. 1, 225-246 (2014).

Let \(E\) denote the real affine \(3\)-dimensional space endowed with the Minkowski inner product. A Margulis spacetime is a manifold obtained as the quotient of \(E\) by the proper action of a discrete free group \(\Gamma\) of affine Minkowski isometries. The natural linear representation of \(\Gamma\) into the group of linear Minkowski isometries \(\mathrm{SO}(2,1)\) is faithful and discrete. Its image \(\Gamma_0\) is thus a Fuchsian group, and it follows that one can naturally associate to each Margulis spacetime a hyperbolic surface with free fundamental group.
The paper is devoted to the study of Margulis spacetimes whose associated hyperbolic surfaces are topologically real projective planes with two punctures. The authors prove that in this case the Margulis spacetime is homeomorphic to the interior of handlebody of genus \(2\). Moreover, these Margulis spacetimes are “tame” in the sense that \(\Gamma\) admits a fundamental domain bounded by “crooked planes”. This latter result proves the “Crooked Plane Conjecture” for proper affine deformations of Fuchsian groups uniformising the twice-punctured real projective plane.
Some remarks on Margulis spacetimes whose associated hyperbolic surfaces have free fundamental group of rank \(2\) (sphere with four punctures, once-punctured torus and Klein bottle) are also provided.

Reviewer: Luisa Paoluzzi (Marseille)

MSC:

30F60	Teichmüller theory for Riemann surfaces
20H10	Fuchsian groups and their generalizations (group-theoretic aspects)
57M05	Fundamental group, presentations, free differential calculus

Keywords:

Margulis spacetime; affine deformations; Margulis invariant; Fricke space; crooked planes