The Poincare disk maps the infinite hyperbolic plane into the open unit disk. Distances grow without bound near the boundary: a step that looks tiny in Euclidean terms covers the same hyperbolic distance as a step near the center. Straight lines in hyperbolic space — geodesics — appear as arcs of circles that meet the boundary at right angles, or as diameters through the center. The {7,3} tiling shown faintly on the disk demonstrates this: each tile is a regular hyperbolic heptagon, and exactly three meet at every vertex. In Euclidean geometry this is impossible. In hyperbolic geometry the angle sum of a triangle is less than π, and there is room.
Given a graph, its Laplacian matrix L = D − A (degree minus adjacency) encodes how the graph is connected. The eigenvalues 0 = λ_0 ≤ λ_1 ≤ λ_2 ≤ ... measure this: λ_1 is the spectral gap. When λ_1 is large the graph is an expander — every subset of vertices has many edges leaving it, information diffuses rapidly, and random walks mix fast. For a d-regular graph, the Alon-Boppana bound says the adjacency spectral radius of non-trivial eigenvalues is at least 2√(d−1) − o(1). Graphs meeting this bound are called Ramanujan.
Joel Friedman proved (2003/2008) that a random d-regular graph is almost surely nearly Ramanujan: its non-trivial adjacency eigenvalues are at most 2√(d−1) + ε for any ε > 0, in the large-n limit. This is the discrete analogue of a deep result in geometry.
For a compact hyperbolic surface, the Laplace-Beltrami operator has a spectral gap λ_1, and Selberg's conjecture (proven for certain arithmetic surfaces) posits λ_1 ≥ 1/4. Nalini Anantharaman and Laura Monk proved (2021–2024) that random hyperbolic surfaces — sampled by the Weil-Petersson measure on moduli space — almost surely satisfy λ_1 ≥ 1/4 − ε as the genus tends to infinity. The surfaces are nearly optimally expanding, just as random regular graphs are nearly Ramanujan.
Two worlds — combinatorial graphs and continuous Riemannian surfaces — governed by the same principle: randomness, with overwhelming probability, produces near-optimal expansion.