On 2-systoles of hyperbolic 3-manifolds

Abstract

We investigate the geometry of $π_1$-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 $π_1$-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 $π_1(M_i)$ 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.