On 2-Systoles of Hyperbolic 3-Manifolds

We investigate the geometry of π ₁-injective surfaces in closed hyperbolic 3-manifolds. First we prove that for any ${epsilon > 0}$ , if the manifold M has sufficiently large systole sys₁(M), the genus of any such surface in M is bounded below by ${{rm exp}((frac{1}{2} - epsilon){rm sys}_1(M))}$ . Using this result we show, in particular, that for congruence covers M _i → M of a compact arithmetic hyperbolic 3-manifold we have: (a) the minimal genus of π ₁-injective surfaces satisfies ${{rm log} , {rm sysg}(M_i) gtrsim frac{1}{3} {rm log} , {rm vol}(M_i) ; (b)}$ there exist such sequences with the ratio Heegard ${{rm genus}(M_i)/{rm sysg}(M_i) gtrsim {rm vol}(M_i)^{1/2}}$ ; and (c) under some additional assumptions π ₁(M _i ) is k-free with ${{rm log} , k gtrsim frac{1}{3}{rm sys}_1(M_i)}$ . The latter resolves a special case of a conjecture of Gromov.