Extension of Alon's and Friedman's conjectures to Schottky surfaces

Michael Magee , Frédéric Naud (1)

1 IMJ - Institut de Mathématiques de Jussieu

Let X = Λ\H be a Schottky surface, that is, a conformally compact hyperbolic surface of infinite area. Let δ denote the Hausdorff dimension of the limit set of Λ. We prove that for any compact subset K ⊂ { s : Re(s) > δ 2 }, if one picks a random degree n cover X n of X uniformly at random, then with probability tending to one as n → ∞, there are no resonances of X n in K other than those already belonging to X (and with the same multiplicity). This result is conjectured to be the optimal one for bounded frequency resonances and is analogous to both Alon's and Friedman's conjectures for random graphs, which are now theorems due to Friedman and Bordenave-Collins, respectively.

Domaines: Mathématiques [math]

Michael Magee, Frédéric Naud. Extension of Alon's and Friedman's conjectures to Schottky surfaces. 2022. ⟨hal-03841740⟩