  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 Teichm\"uller space and its
symmetry group ${\rm PPSL}_2({\mathbb R})$, 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.
