Heegaard genera in congruence towers of hyperbolic 3-manifolds

Details Heegaard genera in congruence towers of hyperbolic 3-manifolds Heegaard genera in congruence towers of hyperbolic 3-manifolds

2011-05-12

arxiv.org/abs/1105.2372v1

Mathematics

Geometric Topology

Scientific paper

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.

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