Heegaard genera in congruence towers of hyperbolic 3-manifolds

Abstract

Given a closed hyperbolic 3-manifold $M$, we construct a tower of covers with increasing Heegaard genus, and give an explicit lower bound on the Heegaard genus of such covers as a function of their degree. Using similar methods we prove that for any $\epsilon >0$ there exist infinitely many congruence covers $\{M_i\}$ such that, for any $x\in M$, $M_i$ contains an embbeded ball $B_x$ (with center $x$) satisfying $\text{vol}(B_x)>(\text{vol}(M_i){)}^{\frac{1}{4}-\epsilon }$. We get similar results in the arithmetic non-compact case.