Sketch of a Program for Universal Automorphic Functions to Capture Monstrous Moonshine

We review and reformulate old and prove new results about the triad $ {\rm PPSL}_2({\mathbb Z})\subseteq{\rm PPSL}_2({\mathbb R})\circlearrowright ppsl_2({\mathbb R}) $, which provides a universal generalization of the classical automorphic triad ${\rm PSL}_2({\mathbb Z})\subseteq{\rm PSL}_2({\mathbb R})\circlearrowright psl_2({\mathbb R})$\mathrm{.}TheleadingPor$p$intheuniversalsettingstandsfor$piecewise$,andthegroup${\rm PPSL}_2({\mathbb Z})$ plays at once the role of universal modular group, universal mapping class group, Thompson group $T$ and Ptolemy group. We produce a new basis of the Lie algebra $pps{l}_2(\mathbb{R})$, compute its structure constants, define a central extension which is compared with the Weil-Petersson 2-form, and discuss its representation theory. We construct and study new framed holographic coordinates on the universal Teichm"uller space and its symmetry group ${\mathrm{P}\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{R})$, and construct an invariant 1-form as its Maurer-Cartan form analogous to the invariant Eisenstein 1-form $E_2(z)dz$, which gives rise to the spin 1 representation of $psl_2({\mathbb R})$ extended by the trivial representation. This suggests the full program for developing the theory of universal automorphic functions conjectured to yield the bosonic CFT${}_2$. Relaxing the automorphic condition to the commutant leads to our ultimate conjecture on realizing the Monster CFT${}_2$ via the automorphic representation for the universal triad. This conjecture is also bolstered by the links of both the universal Teichm"uller and the Monster CFT${}_2$ theories to the three-dimensional quantum gravity.