Logarithmic systolic growth for hyperbolic surfaces in every genus (Q6654044)

The systole \({\mathrm{sys}}(S)\) of a closed hyperbolic surface \(S\) is the length of a shortest non-contractible closed geodesic, and a consequence of Mumford compactness is that this achieves a maximum on the closed hyperbolic surfaces of a given genus. It is well-known that \({\mathrm{sys}}(S)\leqslant 2\log(4g-2)\) for surfaces \(S\) of genus \(g\). Using arithmetic constructions \textit{R. Brooks} [J. Reine Angew. Math. 390, 117--129 (1988; Zbl 0641.53041)] and \textit{P. Buser} and \textit{P. Sarnak} [Invent. Math. 117, No. 1, 27--56 (1994; Zbl 0814.14033)] found a sequence a sequence of closed hyperbolic surfaces exhibiting a logarithmic lower bound of the form \(\frac43\log(g)-C\) along the sequence giving the range \(\frac43\leqslant\limsup_{g\to\infty}\max_{S\text{ of genus }g}\frac{{\mathrm{sys}}(S)}{\log g}\leqslant2\). This construction has been generalized in several ways for certain genera. Liu and Petri [Preprint, arXiv:2312.11428 [math.GT] (2023).] used random surfaces to show that for \(c\leqslant\frac29\) there is a constant \(C>0\) such that for any genus \(g\geqslant2\) there is a closed hyperbolic surface \(S\) of genus \(g\) with \({\mathrm{sys}}(S_g)\geqslant c\log(g)-C\), giving a lower bound of the form \(\liminf_{g\to\infty}\max_{S\text{ of genus }g}\frac{{\mathrm{sys}}(S)}{\log g}\geqslant\frac29\). Here a similar result is found using non-random methods closer to those of Brooks and Buser and Sarnak. The main result is that for any \(c<\frac{19}{120}\) there is a constant \(C>0\) such that for every genus \(g\geqslant2\) there is a closed hyperbolic surface \(S_g\) of genus \(g\) with \({\mathrm{sys}}(S_g)\geqslant c\log(g)-C\).