Minoration du spectre des vari\'et\'es hyperboliques de dimension 3

Abstract

Let $M$ be a compact hyperbolic 3-manifold of diameter $d$ and volume $\leq V$. If $\mu_i(M)$ denotes the $i$-th egenvalue of the Hodge laplacian acting on coexact 1-forms of $M$, we prove that $\mu_1(M)\geq \frac c{d^3e^{2kd}}$ and $\mu_{k+1}(M)\geq \frac c{d^2}$, where $c>0$ depends only on $V$, and $k$ is the number of connected component of the thin part of $M$. Moreover, we prove that for any finite volume hyperbolic 3-manifold $M_\infty$ with cusps, there is a sequence $M_i$ of compact fillings of $M_\infty$ of diameter $d_i\to+\infty$ such that $\mu_1(M_i)\geq \frac c{d_i^2}$.