Voronoi’s Projective Cone
We begin with a question.
Let \(D=D^2\) be the two-dimensional unit disk \(D=\{z| |z| \leq 1\} \subset {\bf{R}}^2\). Can the reader readily construct continuous group actions \[(D, \partial D) \times GL({\bf{R}}^2) \to (D, \partial D)\] which satisfies the following properties:
the action is transitive on the interior;
point stabilizers are compact;
the action is elementary.
The final criteria (iii) is subjective, but important for calculations.
The reader has likely seen actions satisfying the first condition before, namely the group of “homographic mappings”, also known as Mobius transformations and their formulae. The action is conventionally presented in complex coordinates as \[{\bf{C}} \times GL({\bf{R}}^2) \to {\bf{C}}, \quad (z, \begin{pmatrix}a & b \\ c & d \end{pmatrix}) \mapsto (az+b) / (cz+d),\] after we identify \({\bf{R}}^2 \approx \bf{C}\). But does anybody really understand the above formula? Is the formula for Mobius transformations not incredible?
JHM Labs has always found the Mobious formula \(\frac{az+b}{cz+d}\) hard to understand. Indeed we argue that it’s not elementary in the sense of (iii). Is it elementary that \({\bf{R}}^2\) possesses a division operation \((u,v)\mapsto u/v\)? Thus we argue that Mobius formulas are not elementary.
We propose the following description of an elementary action of \(GL({\bf{R}}^2)\) on the disk \(D\). The description arises from representation theory. It begins with:
(a1) The matrix group \(GL({\bf{R}}^2)\) acts on \({\bf{R}}^2\) via matrix multiplication \(\rho:(v,g) \mapsto v.g\).
(a2) The above representation has natural dual representation \(\rho^*\) of \(GL({\bf{R}}^2)\) acting on the vector space of linear functionals \({{\bf{R}}^2}^*\), where we adopt left action \(g.\ell(v):=\ell(g^{-1}(v))\).
(a3) The symmetric square of the dual representation \(Sym^2(\rho^*)\) yields \(GL({\bf{R}}^2)\) acting on the vector space of homogeneous symmetric quadratic polynomials on \({\bf{R}}^2\).
(a4) We restrict \(Sym^2(\rho^*)\) to the convex cone \(Q\) of semidefinite quadratic states which satisfy \(q(v) \geq 0\) for all \(v \in {\bf{R}}^2\). The cone \(Q\) is Voronoi’s cone of quadratic states.
(a5) The affine action \(Q \times GL({\bf{R}}^2)\to Q\) commutes with the scalar \({\bf{R}}^{\times}_{>0}\)-action. Thus \(GL({\bf{R}}^2)\) acts continuously on the projectivization \(Proj(Q):=(Q-0) / {\bf{R}}^{\times}_{>0}\). But the projectivization of a cone is an affine bounded topological disk. Thus we find \(PGL({\bf{R}}^2)\) acts nontrivially on the bounded disk.
One finds the criteria (i), (ii), and (iii) are satisfied for every representation in steps (a1)–(a5). Thus we
Claim: The natural action of \(GL({\bf{R}}^2)\) on the projectivized Voronoi cone \(Proj(Q)\) is an elementary group action satisfying criteria (i), (ii), (iii) above.
Historically the diverse models of hyperbolic geometries discovered by Gauss, Bolyai, Lobachevskii, Beltrami, Poincare, Klein, Minkowski, etc.. are all unified with Voronoi’s projective model. Indeed we observe that all the historical models are noncanonical sections of the projectivization \(Q - 0 \to Proj(Q)\). For example, the hypersurfaces \(\{\text{tr}=1\}\) and \(\{\text{ disc }=1\}\) yield the Klein and Minkowski models of the hyperbolic plane. Compare (Thurston 2014), (Thurston 2022, 27:Ch.2). We remark that trace and dicriminant are defined as tensors on quadratic states.
This idea about Voronoi’s projective model has applications to Teichmueller spaces, and to arithmetic groups like \(Sp({\bf{R}}^{2g})\) and \(\Gamma=Sp({\bf{Z}}^{2g})\). From Voronoi we learn the idea that there can exist many “equivalent” models of a given moduli space, but they are different sections of a universal projective model.
For example, the symmetric space associated to the group \(Sp({\bf{R}}^{2g})\) is well-known as Siegel’s upper half space \({\bf{H}}_{2g}\) consisting of complex \(g \times g\) matrices \(Z\) which satisfy \(Im(Z)>0\). This latter condition implies the Gaussian \(e^-Z\) is absolutely convergent. Compare Folland’s treatment (Folland 1989).
For Teichmueller space, the existence of a universal projective model is an open problem. Geometers are not easily moved past the Teichmueller metric, which strikes us as simple an arbitrary canonical section of a projectivization not yet understood.
[To be continued - JHM]