Projective representations of the Thurston geometries

homogeneous geometry is a pair (X, G) where X is a simply-connected Riemannian manifold and G is a maximal group of isometries of X with compact point stabilizers.

There are eight 3-dimensional homogeneous geometries, often called the Thurston geometries:

  • Euclidean 3-space \mathbb{E}^3, hyperbolic 3-space \mathbb{H}^3, and the 3-sphere S^3;
  • the product geometries S^2 \times \mathbb{R} and \mathbb{H}^2 \times \mathbb{R}; and
  • the more exotic fibered geometries \widetilde{\mathrm{SL}_2\mathbb{R}}Nil, and Sol.

The goal of this post is to show how all of these geometries can be modeled as projective geometries: more precisely, for any of these eight geometries (X, G), we can find some open domain \Omega \subset \mathbb{RP}^3 such that \Omega \cong X (possibly up to index 2), and a subgroup \Gamma \leq \mathrm{PSL}(4,\mathbb{R})= \mathrm{Aut}(\mathbb{RP}^3) such that \Gamma \cong G (again, possibly up to index 2), and \Gamma is the maximal subgroup of \mathrm{PSL}(4,\mathbb{R}) which preserves \Omega.

In other words, up to finite index and coverings, each of these geometries embeds (or, if we eliminate the preceding qualification, locally embeds) into 3-dimensional projective geometry.

I will follow the approach of Emil Molnár’s 1997 paper, which, for some reason, I seem to have great difficulty reading. Hopefully the process of writing this post / reading the article to reprocess the details for this purpose will help.

To describe the goal of this post somewhat more precisely: I wish to exhibit examples of such projective representations, or rather sketch semi-impressionistic outlines for how to obtain them. Readers in need of more (concrete, computational) details should consult Molnár paper; this post will hopefully provide a useful broad guide to the paper, but is not meant to be a substitute.

The general setup: projective space and symmetric bilinear forms

We recall the definition of \mathbb{RP}^3 as the projectivization of a 4-dimensional vector space V_4, i.e. \mathbb{RP}^3 := V_4 / \sim where we identify vectors in V_4 which are related by (real) scalar multiplication.

We also recall that \mathrm{PSL}(4,\mathbb{R}) = \mathrm{PGL}(4,\mathbb{R}) is in fact the full group of collineations in this case, since the Galois group \mathrm{Gal}(\mathbb{R} / \mathbb{Q}) is trivial.

The general game now is to consider symmetric bilinear forms \langle \cdot , \cdot \rangle on V_4, of varying signatures, and possibly preserving certain fiberings in the case of the product / fibered geometries; then we take \Omega to be some sort of invariant set w.r.t. \langle \cdot , \cdot \rangle, and \Gamma to be the orthogonal (sub)group w.r.t. \langle \cdot , \cdot \rangle (in \mathrm{PGL}(4,\mathbb{R}).)

We note that Molnár’s paper sets up a bunch of additional notation, as well as the notion of a polarity (which is equivalent to the bilinear form, via the dual vector space.) It appears that this is used to perform more explicit computations and more generally characterize projective representations of the various Thurston geometries, but are not strictly necessary for our purposes.

The non-fibered geometries

Spherical geometry can be represented by taking a (positive- or negative-) definite bilinear form. Then the isotropy group \Gamma to be [a projectivization of] the orthogonal group \mathrm{O}(4), and we may take \Omega to be all of \mathbb{RP}^3, or even better all of its better cover S^3.

Hyperbolic geometry can be represented by considering the hyperboloid model: take a Lorentz form, i.e. a form \langle \cdot, \cdot \rangle with signature (-+++), let \Omega be the projectivization of \{p \in V_4 : \langle p, p \rangle < 0\}; then the isotropy group is isomorphic to (the appropriate finite-index subgroup of) \mathrm{SO}(1,3).

(3-dimensional) Euclidean geometry may be represented by considering any affine chart, and the corresponding affine group sitting inside \mathrm{O}(4); the bilinear form in this case is degenerate, with signature (0+++).

The need for additional constraints

For \widetilde{\mathrm{SL}_2\mathbb{R}}, we start with a bilinear form of signature (–++), but observe (after some work) that the point stabilizers under the orthogonal group are not compact.

We would now like to impose additional structure on our bilinear form, coming from the structure of our homogeneous geometry \widetilde{\mathrm{SL}_2\mathbb{R}}, which will whittle down the orthogonal group to a more manageable size.

This additional structure comes in the form of a non-trivial line bundle structure on the unit sphere \Omega for our bilinear form, with base space a hyperboloid \mathcal{H} and fibres given the skew lines given by the S-orbits of \Omega, where S = \left\{ \left( \begin{array}{cc} R_\theta \\ & R_{-\theta} \end{array} \right) : \theta \in \mathbb{R} \right\} \cong \mathrm{SO}(2) (R_\theta being the rotation matrix \left( \begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array} \right) \in \mathrm{SO}(2)) is what Molnár calls the screw collineation group.

We observe that \tilde{S} \cong \mathbb{R} acts freely on the base space \mathcal{H}, and that we may describe the unit sphere \Omega as the universal cover of the unit tangent bundle T^1\mathcal{H}.

The collineation group \Gamma we seek is then given by elements of our orthogonal group which also preserve our line bundle; Molnár has a very concrete description of this group, in suitable (standard) bases, as

\left\{ \left( \begin{array}{cc} \cosh r R_\psi & \sinh r R_\alpha \\ \sinh r R^*_{\alpha-\omega} & \cosh r R^*_{\psi+\omega} \end{array} \right), \left( \begin{array}{cc} \cosh r R^*_\psi & \sinh r R^*_\alpha \\ \sinh r R_{\alpha-\omega} & \cosh r R_{\psi+\omega} \end{array} \right) : \psi, \omega, \alpha \in \mathbb{R},  r \geq 0 \right\}

where R_\theta is the rotation matrix \left( \begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array} \right) and R^*_\theta is the twisted rotation matrix \left( \begin{array}{cc} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{array} \right)

The product geometries

We adapt this approach, in an intuitively rather more straightforward manner, for the product geometries S^2 \times \mathbb{R} and \mathbb{H}^2 \times \mathbb{R}.

For S^2 \times \mathbb{R} we again consider a bilinear form with signature (0+++), but now take as \Omega no longer an entire affine chart on which the form is non-degenerate, but rather a punctured affine chart \mathbb{A}^3 \setminus \{O\}, where O is a fixed origin, and also equip the resulting punctured affine space with a line bundle—or really, a product structure—, with base space homeomorphic to the 2-sphere, and fibres given by open half-rays pointing outwards from O.

This gives us a distinguished family of 2-spheres (those with centre O); the group \Gamma is then, as before, the subgroup of the orthogonal group preserving this line bundle structure. It is generated by the \mathrm{SO}(2)-subgroup of isometries of the unit 2-sphere, dilatations in the \mathrm{R} direction, and inversions in our distinguished family of 2-spheres.

Analogously, for \mathbb{H}^2 \times \mathbb{R}, we consider a bilinear form with signature (0-++), and let \Omega be the subspace (in the topological, not vector space sense) of a maximal affine chart on which the form is non-degenerate given by a punctured hyperboloid cone \{p \in \mathbb{A}^3 : \langle p, p \rangle < 0\} \setminus \{O\} where again O is a fixed origin in the affine chart. Analogous to the previous case, equip \Omega with a line bundle (product structure), with base space homeomorphic to the hyperboloid, and fibres given by open half-rays in pointing outwards from O. 

We may then obtain \Gamma as the subgroup of the orthogonal group preserving the product structure; it is generated, analogously to the previous case, by the \mathrm{SO}(1,2)-subgroup of isometries of the hyperbolic 2-space, dilatations in the \mathrm{R} direction, and inversions in the distinguished family of hyperboloids coming from the line bundle structure.

Sol and Nil

For Sol, we again start with a bilinear form with signature (0-++). Now (and for Nil) we do not have any (Cartesian) product structure; we take as \Omega the entirety of an affine chart, corresponding to a maximal subspace on which our form is non-degenerate, and attempt to impose additional structure on this \Omega.

For Sol this additional structure comes in the form of a (trivial?) parallel plane fibering; the collineation group is then generated by a plane reflection, and a half-turn, and the elements \left\{ \left( \begin{array}{cccc} 1 & a & b & c \\ 0 & e^c & 0 & 0 \\ 0 & 0 & e^{-c} & 0 \\ 0 & 0 & 0 & 1 \end{array} \right): a, b, c \in \mathbb{R} \right\}.

For Nil we start with a bilinear form of signature (000+), and take as \Omega an affine chart corresponding to a maximal affine chart on which our form is degenerate. There is an additional structure on this affine 3-space which takes the form of a line bundle. To describe this line bundle, we recall the description of Nil as a Heisenberg group; the base spaces of this fibration are then the horizontal levels \left\{ \left( \begin{array}{ccc} 1 & a & c \\ & 1 & b \\ & & 1 \end{array} \right) : a, b \in \mathbb{R} \right\} of constant height c, and the fibres the “vertical” lines \left\{ \left( \begin{array}{ccc} 1 & a & c \\ & 1 & b \\ & & 1 \end{array} \right) : c \in \mathbb{R} \right\}.

\Gamma is then given by (a suitable representation of) the Heisenberg group extended by pointwise stabilizers (isomorphic to a finite extension of \mathrm{SO}(2)); Molnár describes it explicitly as

\left\{ \left( \begin{array}{cccc} 1 & x & y & z \\ & \cos\omega & \sin\omega & x \sin\omega \\ & \mp\sin\omega & \pm\cos\omega & \pm x\cos\omega \\ & & & \pm 1 \end{array}\right) : (x,y,z) \in \mathbb{R}^3 \setminus \{(0,0,0)\}, \omega \in \mathbb{R} \right\}.

(in a suitably-chosen basis.)