Largest systoles of closed hyperbolic surfaces by genus
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How large can the systole, the shortest closed geodesic length, be on a closed hyperbolic surface of genus g? The target construction has systole at least log g−12 log log g for every sufficiently large g.
Why it matters and what counts as progress
Why it matters: It asks for large shortest geodesics uniformly across genera, improving the lower-limit coefficient relative to log g rather than only a subsequence.
What counts as progress: Larger systoles uniformly at every sufficiently large genus, or upper bounds narrowing the asymptotic log-genus coefficient.
How the problem developed
- 1994 · Sequence lower bound [1]
Peter Buser and Peter Sarnak Constructed a sequence of closed hyperbolic surfaces with systole at least (4/3) log g up to an additive constant.
- 18 Dec 2023 · Every genus lower bound [2]
Mingkun Liu and Bram Petri Improved the explicit every-genus lower-bound coefficient for the maximal systole to 2/9.
- 27 Aug 2026 · Every large genus lower bound [3]
Yifei Cai; GPT-5.6 Sol developed the proof through extended discussion For every sufficiently large genus g, constructed a surface with systole at least log g−12 log log g, improving the liminf coefficient to 1.
AI contribution
A note on surfaces with large systoles
For every sufficiently large genus g, Cai constructs a closed hyperbolic surface with systole at least log g−12 log log g. The lower limit of the largest possible systole divided by log g is therefore at least one.
Sources
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