Spectral gap for random Schottky surfaces
We establish a spectral gap for resonances of the Laplacian of random Schottky surfaces, which is optimal according to a conjecture of Jakobson and Naud.
math.SP↗
arXiv subjects
Irving Calderón
Publications and source records attributed to Irving Calderón.
$S$-integral quadratic forms and homogeneous dynamics
Let $S = \{ \infty \} \cup S_f$ be a finite set of places of $\mathbb{Q}$. Using homogeneous dynamics, we establish two new quantitative and explicit results about integral quadratic forms in three or more variables: The first is a criterion of $S$-integral equivalence. The second determines a finite generating set of any $S$-integral orthogonal group. Both theorems--which extend results of H. Li and G. Margulis for $S = \{ \infty\}$--are given by polynomial bounds on the size of the coefficients of the quadratic forms.
math.NT↗
Explicit spectral gap for Schottky subgroups of $\mathrm{SL} (2,\mathbb{Z})$
Let $Γ$ be a Schottky subgroup of $\mathrm{SL} (2,\mathbb{Z})$. We establish a uniform and explicit lower bound of the second eigenvalue of the Laplace-Beltrami operator of congruence coverings of the hyperbolic surface $Γ\backslash \mathbb{H}^2$ provided the limit set of $Γ$ is thick enough.
math.SP↗