Heegaard genera in congruence towers of hyperbolic 3-manifolds (Q450510)

The author constructs a tower of covers of a 3-manifold with Heegaard genus tending to infinity and bounds the Heegaard genera from below in terms of the degree of the cover. Two of the main theorems of the paper follow:NEWLINENEWLINE{ Theorem:} Let \(M\) be a closed hyperbolic 3-manifold and \(\epsilon > 0\) any (small) number. Then there exists a tower of finite congruence covers NEWLINE\[NEWLINE\cdots \rightarrow M_i \rightarrow \cdots M_2 \rightarrow MNEWLINE\]NEWLINE such that the Heegaard genus of each \(M_i\) is at least \([\pi_1(M): \pi_1(M_1)]^{\frac{1}{8} - \epsilon}\). If \(M\) is arithmetic, we can improve the exponent \({\frac{1}{8} - \epsilon}\) to \({\frac{1}{4} - \epsilon}\).NEWLINENEWLINE{ Theorem:} For \(M\) a given arithmetic non-compact hyperbolic 3-manifold and \(\epsilon > 0\), there exists a tower of finite congruence covers NEWLINE\[NEWLINE\cdots \rightarrow M_i \rightarrow \cdots M_2 \rightarrow MNEWLINE\]NEWLINE such that the Heegaard genus of each \(M_i\) is at least \([\pi_1(M): \pi_1(M_1)]^{\frac{1}{4} - \epsilon}\).