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Let $m$ be a Riemannian metric on $S_g$ the surface of genus $g$, and $sys(m)$ be the length of the shortest non contractible cycle with respect to $m$.

The systolic inequality claims that for any metric on $S_g$, $$\frac{sys(m)^2}{area(m)}\leq c\frac{g}{log^2 g}.$$

As far as I know the best known constant is $c=1/\pi$ due to Sabourau and Katz. On the other hand Buser and Sarnak constructed arithmetic surfaces showing that $c$ cannot be smaller than $\frac{4}{9\pi}$.

Is there an upper bound on $c_{hyp}$ that refines the value of the constant $c$ if we assume that $m$ has constant curvature $-1$, (is hyperbolic)? i.e. what is known about the upper bound on: $$c_{hyp}(g):=sup_{\textrm{m hyperbolic}} \frac{log^2 g}{g}\frac{sys(m)^2}{area(m)} .$$

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There's this paper which shows that this is the best constant one could hope for on congruence arithmetic surfaces:arxiv.org/abs/1206.2965v1 –  Ian Agol Jul 11 '13 at 17:39

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