Systolic arithmetic


The classic 1994 paper by P. Buser and P. Sarnak explores systolic ramifications of the properties of congruence subgroups of Fuchsian groups, in the case of rational trace field. M. Gromov (1996) makes tantalizing remarks in section 3.C.6. Much arithmetic work was done by P. Schmutz Schaller from 1993 until 2001. The 2007 text Logarithmic growth treats both the Fuchsian and the Kleinian cases, over an arbitrary numberfield. See also related work by R. Brooks (1999) on Platonic surfaces, as well as papers by P. Sarnak and X. Xue, N. Bergeron, etc.

The general philosophy is that lower bounds for the eigenvalue of the Laplacian should yield good systolic lower bounds asymptotically. Brooks obtains a lower bound of 5/36 for the Platonic surfaces he constructs by compactifying congruence surfaces of the modular group. He did not clarify the systolic behavior of the resulting compact surfaces.






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