  For bordered surfaces S, we develop a complete parallel between the geometry
of the combinatorial Teichm\"uller space $T_S^{comb}$ equipped with Kontsevich
symplectic form $\omega_K$, and then the usual Weil-Petersson geometry of
Teichm\"uller space $T_S$. The basis for this is an identification of
$T_S^{comb}$ with a space of measured foliations with transverse boundary
conditions. We equip $T_S^{comb}$ with an analog of the Fenchel-Nielsen
coordinates (defined similarly as Dehn-Thurston coordinates) and show they are
Darboux for $\omega_K$ (analog of Wolpert formula). We then set up the
geometric recursion of Andersen-Borot-Orantin to produce mapping class group
invariants functions on $T_S^{comb}$ whose integration with respect to
Kontsevich volume form satisfy topological recursion. Further we establish an
analog of Mirzakhani-McShane identities, and provide applications to the study
of the enumeration of multicurves with respect to combinatorial lengths and
Masur-Veech volumes. The formalism allows us to provide uniform and completely
geometric proofs of Witten's conjecture/Kontsevich theorem and Norbury's
topological recursion for lattice point count in the combinatorial moduli
space, parallel to Mirzakhani's proof of her recursion for Weil-Petersson
volumes. We strengthen results of Mondello and Do on the convergence of
hyperbolic geometry to combinatorial geometry along the rescaling flow,
allowing us to flow systematically natural constructions on the usual
Teichm\"uller space to their combinatorial analogue, such as a new derivation
of the piecewise linear structure of $T_S^{comb}$ originally obtained in the
work of Penner, as the limit under the flow of the smooth structure of $T_S$.
