1. The existence of a uniform lower bound for systoles of arithmetic hyperbolic manifolds is apparently equivalent to the Mahler conjecture. See paper by Agol on hyperbolic 4-manifolds. A volunteer is hereby sought to provide a summary.
2. Lengths of suitable closed geodesics are widely known as the Fenchel-Nielsen coordinates on Teichmuller space of Riemann surfaces. The (inverse of the) systole is an exhaustion function on Teichmuller space. See the work of S. Wolpert and the recent paper by Z. Huang. A volunteer is hereby sought to provide a summary.
3. Local extrema of the systole on the moduli space of Riemann surfaces have been studied by C. Bavard, H. Akrout, P. Schmutz, J. Quine, and others. There are elegant connections to lattice theory. A volunteer is hereby sought to provide a summary.
4. C. Adams and A. Reid have several works on systoles of hyperbolic 3-manifolds. A volunteer is hereby sought to provide a summary.
5. The precise asymptotic constant has been identified for principal congruence subgroups of Fuchsian groups. Namely, the corresponding Riemann surfaces Sg satisfy the following precise genus asymptotics:
In other words, 4/3 is the sharp constant asymptotically, for principal congruence subgroups. See the work by Katz, Schaps, and Vishne, as well as the follow-up paper by Vishne (encourage him to place it on the arxiv!).
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Systoles in hyperbolic geometry and Riemann surfaces
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