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
${\rm PSL}_2({\mathbb Z})\subseteq{\rm PSL}_2({\mathbb R})\circlearrowright psl_2({\mathbb R})$. The leading P or $p$ in the universal setting stands for $piecewise$, and
the group ${\rm PPSL}_2({\mathbb Z})$ 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 $ppsl_2({\mathbb R})$, 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 Teichmrm
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 als