If the systole is defined as the length of the shortest essential simple closed curve are there any known upper bounds for hyperbolic surfaces with punctures?
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Yes, since the injectivity radius (defined as the max of injectivity radii over all points of the surface) is bounded by roughly the log of the area (think "embedded disk"). For interesting papers on this subject, check out:
@article {MR1269424, AUTHOR = {Buser, P. and Sarnak, P.}, TITLE = {On the period matrix of a {R}iemann surface of large genus}, NOTE = {With an appendix by J. H. Conway and N. J. A. Sloane}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {117}, YEAR = {1994}, NUMBER = {1}, PAGES = {27--56}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {22E40 (14H15 14H42 32G20)}, MRNUMBER = {1269424 (95i:22018)}, MRREVIEWER = {Jos{\'e} M. Mu{\~n}oz Porras}, DOI = {10.1007/BF01232233}, URL = {http://dx.doi.org/10.1007/BF01232233}, }
@article {MR3065183, AUTHOR = {Basmajian, Ara}, TITLE = {Universal length bounds for non-simple closed geodesics on hyperbolic surfaces}, JOURNAL = {J. Topol.}, FJOURNAL = {Journal of Topology}, VOLUME = {6}, YEAR = {2013}, NUMBER = {2}, PAGES = {513--524}, ISSN = {1753-8416}, MRCLASS = {30F40 (30F45 58E10)}, MRNUMBER = {3065183}, MRREVIEWER = {Makoto Masumoto}, DOI = {10.1112/jtopol/jtt005}, URL = {http://dx.doi.org/10.1112/jtopol/jtt005}, }
answered May 14, 2014 at 18:09
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$\begingroup$ The argument with largest embedded disk does not work in the presence of punctures: the disk might overlap itself around a puncture, without creating a closed geodesic. The paper by Basmajian has a bound but with a constant $d_S$ which a priori depends on the metric... $\endgroup$ Commented May 31, 2014 at 23:50