On 2-systoles of hyperbolic 3-manifolds

TL;DR

This paper explores the relationship between systoles and injective surfaces in hyperbolic 3-manifolds, establishing bounds on surface genus and implications for congruence covers, advancing understanding of geometric and topological properties.

Contribution

It proves new bounds relating systole length and surface genus, and applies these to congruence covers, partially resolving Gromov's conjecture in hyperbolic 3-manifolds.

Findings

Genus of injective surfaces grows exponentially with systole length.

Minimal genus scales logarithmically with volume in congruence covers.

Constructs sequences with high Heegaard genus to systole ratio.

Abstract

We investigate the geometry of $\pi_1$-injective surfaces in closed hyperbolic 3-manifolds. First we prove that for any $e>0$, if the manifold $M$ has sufficiently large systole $\sys_1(M)$, the genus of any such surface in $M$ is bounded below by $\exp((1/2-e)\sys_1(M))$. Using this result we show, in particular, that for congruence covers $M_i\to M$ of a compact arithmetic hyperbolic 3-manifold we have: (a) the minimal genus of $\pi_1$-injective surfaces satisfies $\log \sysg(M_i) \gtrsim (1/3)\log\vol(M_i)$; (b) there exist such sequences with the ratio Heegard genus$(M_i)/\sysg(M_i) \gtrsim \vol(M_i)^{1/2}$; and (c) under some additional assumptions $\pi_1(M_i)$ is k-free with $\log k \gtrsim (1/3)\sys_1(M_i)$. The latter resolves a special case of a conjecture of M. Gromov.