2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/34723We review some of our recent work on the conformal geometry corresponding to the triangulated surfaces used in 2-dimensional simplicial quantum gravity. In particular, we discuss the regularized Liouville action associated with random Regge triangulations and its connection with Hodge-Deligne theory.42 pages, 12 figures, published on "Advances in General Relativity and Cosmology" - Edited by Giorgio Ferrarese - Pitagora EditriceGeneral Relativity and Quantum CosmologyHigh Energy Physics - TheoryMathematical Physicstext