On a compact hyperbolic surface, the Laplacian has a spectral gap between 0 and the next smallest eigenvalue if and only if the surface is connected. The size of the spectral gap measures how "highly connected" the surface is. We study the spectral gap of a random covering space of a fixed surface, and show that for every ε>0 , with high probability as the degree of the cover tends to ∞, the smallest new eigenvalue is at least 3/16-ε.
Our main tool is a new method to analyze random permutations "sampled by surface groups". I intend to give some background to the result and discuss some ideas from the proof.
This is based on joint works with Michael Magee and Frederic Naud.