Heegaard genera in congruence towers of hyperbolic 3-manifolds

TL;DR

This paper constructs towers of hyperbolic 3-manifold covers with increasing Heegaard genus and establishes explicit lower bounds, also demonstrating the existence of congruence covers with large embedded balls relative to their volume.

Contribution

It provides explicit lower bounds on Heegaard genus in congruence towers and constructs covers with large embedded balls, advancing understanding of geometric properties of hyperbolic 3-manifolds.

Findings

Heegaard genus grows with the degree of covers.

Existence of congruence covers with large embedded balls.

Results extend to arithmetic non-compact 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 $\text{vol}(B_x) > (\text{vol}(M_i))^{\tfrac{1}{4}-\epsilon}$. We get similar results in the arithmetic non-compact case.