Abstract
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.
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Belolipetsky, M. On 2-Systoles of Hyperbolic 3-Manifolds. Geom. Funct. Anal. 23, 813–827 (2013). https://doi.org/10.1007/s00039-013-0223-x
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DOI: https://doi.org/10.1007/s00039-013-0223-x