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In Buser and Sarnak's "On the period matrix of a Riemann surface of large genus", we get $$\frac4{3}\le\limsup_{g\rightarrow\infty}\frac{\max\{\operatorname{sys}(S)|S\in\mathcal{M}_g\}}{\log g}\le 2$$ Here $\operatorname{sys}(S)$ is the shortest length of closed noncontractible geodesics in $S$, and $\mathcal{M}_g$ is the moduli space. However, I want to ask if there exists a constant $C>0$ such that $$\liminf_{g\rightarrow\infty}\frac{\max\{\operatorname{sys}(S)|S\in\mathcal{M}_g\}}{\log g}\ge C\quad ?$$ Any help will be appreciated.

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  • $\begingroup$ Yes, there is such $C$: It boils down to constructing for each $b$ a connected 3-valent graph of girth $\ge C \log(b)$ and with the 1st Betti number $b$ (the number $b$ is your genus $g$). The existence of such graphs is usually proven by a simple counting argument (a probabilistic argument). I will write a proof when I have more time. $\endgroup$ – Moishe Kohan Jun 19 '20 at 10:00
  • $\begingroup$ @MoisheKohan Thanks! I known the latter result, but why It boils down to constructing such trivalent graph? Does it related to gluing pair of pants along trivalent graph? $\endgroup$ – Jugendtraum Jun 20 '20 at 14:14
  • $\begingroup$ Not quite: Using pairs of pants would give a surface of small systole (cuffs of the pants are uniformly bounded). Instead, one glues together ribbon graphs which leads to a hyperbolic surface with boundary. Then one "doubles" such a surface "with a half-twist" along each boundary component. $\endgroup$ – Moishe Kohan Jun 20 '20 at 14:47
  • $\begingroup$ @MoisheKohan Sorry, I'm a little confused. Why It boils down to constructing such trivalent graph? $\endgroup$ – Jugendtraum Jun 20 '20 at 15:28
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    $\begingroup$ Sorry, currently I just do not have time. I will come back to this in about a week, unless somebody writes an answer which would be just fine as well. $\endgroup$ – Moishe Kohan Jun 20 '20 at 18:16

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