  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.
