An Invariant Between Hyperbolic Surfaces and Lattice Spin Models

Abstract

In this succinct note, it is showed that a partition function of equivalent classes of hyperbolic surfaces can be connected to an Ising model located on the boundary of the Poincare disc, as hinted by Poincare's Uniformization theorem and Patterson-Sullivan's Theorem. Keywords: Hyperbolic spaces, Schottky groups, Ising models, locations of Lee-Yang Zeros, non-trivial zeros of Riemann zeta function, phase transition, and quantum chaos.