In this paper denotes a surface homeomorphic to a punctured torus or a pair of pants. Our interest is the study of combinatorial -systoles, that is closed curves with self-intersection numbers greater than and with least combinatorial length. We show that the maximal intersection number of combinatorial -systoles of grows like and .
This result, in case of a pair of pants and a punctured torus, is a positive response to the combinatorial version of the Erlandsson-Parlier conjecture, originally formulated for the geometric length.
Revised:
Accepted:
Published online:
Keywords: closed geodesics, self-intersection, $k$-systole
ElHadji Abdou Aziz Diop 1; Masseye Gaye 1; Abdoul Karim Sane 1
@article{CML_2021__13_2_29_0, author = {ElHadji Abdou Aziz Diop and Masseye Gaye and Abdoul Karim Sane}, title = {Combinatorial $k$-systoles on a punctured torus and a pair of pants}, journal = {Confluentes Mathematici}, pages = {29--38}, publisher = {Institut Camille Jordan}, volume = {13}, number = {2}, year = {2021}, doi = {10.5802/cml.76}, language = {en}, url = {https://cml.centre-mersenne.org/articles/10.5802/cml.76/} }
TY - JOUR AU - ElHadji Abdou Aziz Diop AU - Masseye Gaye AU - Abdoul Karim Sane TI - Combinatorial $k$-systoles on a punctured torus and a pair of pants JO - Confluentes Mathematici PY - 2021 SP - 29 EP - 38 VL - 13 IS - 2 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.76/ DO - 10.5802/cml.76 LA - en ID - CML_2021__13_2_29_0 ER -
%0 Journal Article %A ElHadji Abdou Aziz Diop %A Masseye Gaye %A Abdoul Karim Sane %T Combinatorial $k$-systoles on a punctured torus and a pair of pants %J Confluentes Mathematici %D 2021 %P 29-38 %V 13 %N 2 %I Institut Camille Jordan %U https://cml.centre-mersenne.org/articles/10.5802/cml.76/ %R 10.5802/cml.76 %G en %F CML_2021__13_2_29_0
ElHadji Abdou Aziz Diop; Masseye Gaye; Abdoul Karim Sane. Combinatorial $k$-systoles on a punctured torus and a pair of pants. Confluentes Mathematici, Volume 13 (2021) no. 2, pp. 29-38. doi : 10.5802/cml.76. https://cml.centre-mersenne.org/articles/10.5802/cml.76/
[1] Universal length bounds for non-simple closed geodesics on hyperbolic surfaces, J. Topol., Volume 6 (2013) no. 2, pp. 513-524 | DOI | MR
[2] The geometry of discrete groups, Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1995, xii+337 pages (Corrected reprint of the 1983 original) | MR
[3] An algorithm for simple curves on surfaces, J. London Math. Soc. (2), Volume 29 (1984) no. 2, pp. 331-342 | DOI | MR
[4] Markov maps associated with Fuchsian groups, Inst. Hautes Études Sci. Publ. Math. (1979) no. 50, pp. 153-170 | MR
[5] Geometry and spectra of compact Riemann surfaces, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2010, xvi+454 pages (Reprint of the 1992 edition) | DOI | MR
[6] Almost simple geodesics on the triply-punctured sphere, Math. Z., Volume 291 (2019) no. 3-4, pp. 1175-1196 | DOI | MR
[7] Self-intersection numbers of curves on the punctured torus, Experimental Mathematics, Volume 19 (2010) no. 2, pp. 129-148
[8] Self-intersection numbers of curves in the doubly punctured plane, Exp. Math., Volume 21 (2012) no. 1, pp. 26-37 | DOI | MR
[9] Geodesic and horocyclic trajectories, Universitext, Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011, xii+176 pages (Translated from the 2007 French original) | DOI | MR
[10] Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000, vi+310 pages | MR
[11] Self intersections on a pair of pants, arXiv: 2108.06796
[12] Short closed geodesics with self-intersections, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 169 (2020) no. 3, pp. 623-638
[13] Short closed geodesics on cusped hyperbolic surfaces, arXiv:1912.09967
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