Sara Fernandes, Clara Gracio, Carlos Correia Ramos, Systoles in discrete dynamical systems, Journal of Geometry and Physics, Volume 63, January 2013, Pages 129-139. See http://www.sciencedirect.com/science/article/pii/S0393044012001854
1. Liokumovich, Yevgeny; Spheres of small diameter with long sweep-outs. Proc. Amer. Math. Soc. 141 (2013), no. 1, 309-312. http://www.ams.org/mathscinet-getitem?mr=2988732
2. Filippo Cerocchi: Margulis Lemma, entropy and free products. See http://arxiv.org/abs/1204.1619
3. Chady El Mir, Zeina Yassine: Conformal Geometric Inequalities on the Klein Bottle. See http://arxiv.org/abs/1209.6202
http://arxiv.org/abs/1209.1783 Exotic arithmetic structure on the first Hurwitz triplet, by Lei Yang
1. Nabutovsky, Alexander; Rotman, Regina: Linear bounds for lengths of geodesic loops on Riemannian 2-spheres. J. Differential Geom. 89 (2011), no. 2, 217-232.
2. Hwang, Jun-Muk; To, Wing-Keung: Injectivity radius and gonality of a compact Riemann surface. Amer. J. Math. 134 (2012), no. 1, 259--283.
3. Gournay, Antoine: Widths of lp balls. Houston J. Math. 37 (2011), no. 4, 1227-1248.
4. De Pauw, Thierry; Hardt, Robert: Rectifiable and flat G chains in a metric space. Amer. J. Math. 134 (2012), no. 1, 1-69.
A Note on Riemann Surfaces of Large Systole
We examine the large systole problem, which concerns compact hyperbolic Riemannian surfaces whose systole, the length of the shortest noncontractible loops, grows logarithmically in genus. The generalization of a construction of Buser and Sarnak by Katz, Schaps, and Vishne, which uses principal "congruence" subgroups of a fixed cocompact arithmetic Fuchsian, achieves the current maximum known growth constant of \gamma = 4/3. We prove that this is the best possible value of \gamma for this construction using arithmetic Fuchsians in the congruence case. The final section compares the large systole problem with the analogous large girth problem for regular graphs.
1. Philippe, Emmanuel: Détermination géométrique de la systole des groupes de triangles. (French) [Geometric determination of the systole of triangle groups] C. R. Math. Acad. Sci. Paris 349 (2011), no. 21-22, 1183--1186.
2. On 2-systoles of hyperbolic 3-manifolds, by Mikhail Belolipetsky, arXiv:1205.5198
3. Dyck's surfaces, systoles, and capacities. By Mikhail G. Katz and Stephane Sabourau. See http://arxiv.org/abs/1205.0188
1. Hyperellipticity and Systoles of Klein Surfaces. By Mikhail G. Katz and Stephane Sabourau. See http://www.ams.org/mathscinet-getitem?mr=2944532 and http://arxiv.org/abs/1201.0361
2. Makover, Eran; McGowan, Jeffrey: The length of closed geodesics on random Riemann surfaces. Geom. Dedicata 151 (2011), 207--220. See mathscinet at http://www.ams.org/mathscinet-getitem?mr=2780746
3. Belolipetsky, Mikhail V.; Thomson, Scott A. Systoles of hyperbolic manifolds. Algebr. Geom. Topol. 11 (2011), no. 3, 1455--1469 at http://dx.doi.org/10.2140/agt.2011.11.1455 and mathscinet at http://www.ams.org/mathscinet-getitem?mr=2821431
Title: Homological Error Correcting Codes and Systolic Geometry
Authors: Ethan Fetaya
Geometry & Topology 15 (2011) 1477-1508
Isosystolic genus three surfaces critical for slow metric variations
by Stephane Sabourau
Title: Systole et rayon maximal des varietes hyperboliques non compactes
Authors: Matthieu Gendulphe
2. Makover, Eran; McGowan, Jeffrey: The length of closed geodesics on random Riemann surfaces. Geom. Dedicata 151 (2011), 207--220.
3. Langer, Joel C.; Singer, David A.:
When is a curve an octahedron?
Amer. Math. Monthly 117 (2010), no. 10, 889--902.
4. Erickson, Jeff; Worah, Pratik: Computing the shortest essential
Discrete Comput. Geom. 44 (2010), no. 4, 912--930.
1. If you are in Germany, you may be interested in Pape's course, see http://www.uni-math.gwdg.de/pape/teaching.html
Title: Geodesics, volumes and Lehmer's conjecture
Author: Mikhail Belolipetsky
3. Algebraic & Geometric Topology 11 (2011) 1455-1469
Systoles of hyperbolic manifolds
by Mikhail V Belolipetsky and Scott A Thomson
Title: Liouville's equation for curvature and systolic defect
Author: Mikhail Katz
Algebraic & Geometric Topology 11 (2011) 983-999
Stable systolic category of the product of spheres
by Hoil Ryu
1. Balacheff ; Sabourau : Diastolic and isoperimetric inequalities on surfaces. Ann. Sci. Ec. Norm. Super. (4) 43 (2010), no. 4, 579--605. See mathscinet entry
2. Guth, Larry : Volumes of balls in large Riemannian manifolds. Annals of Mathematics 173 (2011), no. 1, 51--76. See arXiv:math.DG/0610212 and mathscinet entry
Taylor, Laurence R. : Controlling indeterminacy in Massey triple products. Geom. Dedicata 148 (2010), 371--389. Taylor constructs interesting manifolds possessing nontrivial Massey triple products, leading to new examples of systolic inequalities based on: Systolic inequalities and Massey products in simply-connected manifolds. Israel J. Math. 164 ('08), 381-395. See arXiv:math.DG/0604012.
Title: Short loop decompositions of surfaces and the geometry of Jacobians
Authors: Florent Balacheff, Hugo Parlier, Stephane Sabourau
Title: The homology systole of hyperbolic Riemann surfaces
Authors: Hugo Parlier
Title: Distribution of the systolic volume of homology classes
Authors: Ivan K. Babenko, Florent Balacheff,
Title: Systoles of Hyperbolic Manifolds
Authors: Mikhail Belolipetsky, Scott A. Thomson
1. arXiv:1007.2913 [pdf, ps, other]
Title: Stable systolic category of the product of spheres
Authors: Hoil Ryu
Title: Conformal isosystolic inequality of Bieberbach 3-manifolds
Authors: Chady El Mir
Title: Flat currents modulo p in metric spaces and filling radius inequalities
Authors: Luigi Ambrosio, Mikhail G. Katz
Title: Metaphors in systolic geometry
Author: Larry Guth
1. Gendulphe, Matthieu: D ecoupages et in egalit es systoliques pour les surfaces hyperboliques a bord. (French. French summary) [Systolic cuttings and inequalities for surfaces with boundary] Geom. Dedicata 142 (2009), 23--35.
Title: The systolic constant of orientable Bieberbach 3-manifolds
Authors: Chady Elmir
Title: Infinitesimal Systolic Rigidity of Metrics all of whose Geodesics are Closed and of the same Length
Authors: J.-C. Alvarez Paiva, F. Balacheff
4. Dranishnikov, A.; Rudyak, Y.: Stable systolic category of manifolds and the cup-length. Journal of Fixed Point Theory and Applications 6 (2009), no. 1, 165-177.
1. Gromov, M.: Singularities, Expanders and Topology of Maps. Part 1: Homology Versus Volume in the Spaces of Cycles. Journal Geometric And Functional Analysis (GAFA). Online SpringerLink November 03, 2009. (Systolic matters are dealt with on pages 92-94.)
Title: Relative systoles of relative-essential 2-complexes
Authors: Karin U. Katz, Mikhail G. Katz, Stephane Sabourau, Steven Shnider, Shmuel Weinberger
Title: Filling minimality of Finslerian 2-discs
Author: Sergei Ivanov
1. Mikhail Gromov. Bull. Lond. Math. Soc. 41 (2009), no. 3, 573--575.
In '08, the London Mathematical Society has elected Professor Mikhail Gromov to Honorary Membership of the Society, noting in particular: "His bound on the length of the shortest non-contractible loop of a Riemannian manifold, the systole, together with his new invariant, the filling radius, created systolic geometry in its modern form."
Title: Small filling sets of curves on a surface
Authors: James W. Anderson, Hugo Parlier, Alexandra Pettet
Title: Area-minimizing projective planes in three-manifolds
Authors: H. Bray, S. Brendle, M. Eichmair, A. Neves
Title: Scattering at low energies on manifolds with cylindrical ends and stable systoles
Authors: Werner Muller, Alexander Strohmaier
Title: Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic
Author: Stephane Sabourau
Babenko, Ivan: Addenda a l'article intitule ``Topologie des systoles unidmensionnelles'' [Addenda to the article titled ``Topology of one-dimensional systoles''] Enseign. Math. (2) 54 (2008), no. 3-4, 397--398.
arXiv:0903.5299, Title: Systolic inequalities and minimal hypersurfaces, by Larry Guth
1. Katz, Karin Usadi; Katz, M.: Bi-Lipschitz approximation by finite-dimensional imbeddings. See arXiv:0902.3126 More details may be found at hyperreals
2. Brunnbauer, Michael: Filling inequalities do not depend on topology. J. Reine Angew. Math. 624 (2008), 217--231. See Brunnbauer
3. Pettet, Alexandra; Souto, Juan: Minimality of the well-rounded retract. Geom. Topol. 12 (2008), no. 3, 1543--1556.
4. Brunnbauer, Michael: Homological invariance for asymptotic invariants and systolic inequalities. Geom. Funct. Anal. 18 (2008), no. 4, 1087--1117. See Brunnbauer
Bangert, V; Katz, M.; Shnider, S.; Weinberger, S.: E7, Wirtinger inequalities, Cayley 4-form, and homotopy. Duke Math. J. 146 ('09), no. 1, 35-70. See
arXiv:0812.4637: Stable Systolic Category of Manifolds and the Cup-length. Authors: Alexander N. Dranishnikov, Yuli B. Rudyak
1. Balacheff, F.: A local optimal diastolic inequality on the
The author applies Loewner's torus inequality to the ramified triple cover of the sphere, so as to prove a local minimality of Calabi's "triangular pillow" metric for the least length of a geodesic loop.
2. Katz, Karin Usadi; Katz, M.: Hyperellipticity and Klein bottle companionship in systolic geometry. See arXiv:0811.1717
3. Parlier, Hugo: Fixed-point free involutions on Riemann surfaces. Israel J. Math. 166 ('08), 297-311. arXiv:math.DG/0504109
1. Martelli, Bruno: Complexity of PL-manifolds. See arXiv:0810.5478
1. Croke, C.: Small volume on big n-spheres, Proc. Amer. Math. Soc. 136 (2008), no. 2, 715-717
2. Rudyak, Yuli B.; Sabourau, Stéphane: Systolic invariants of groups and 2-complexes via Grushko decomposition. Ann. Inst. Fourier (Grenoble) 58 (2008), no. 3, 777--800
1. arXiv:0807.5040 Cohomological dimension, self-linking, and systolic geometry, by Dranishnikov, A.; Katz, M.; Rudyak, Y.
1. Bounding volume by systoles of 3-manifolds, Mikhail G. Katz; Yuli B. Rudyak in Journal of the London Mathematical Society 2008; doi: 10.1112/jlms/jdm105
2. Asymptotic properties of coverings in negative curvature, Andrea Sambusetti in Geometry & Topology 12 (2008) 617-637.
1. Frequently Asked Questions about Journal of the London Mathematical Society
2. Dranishnikov, A.; Katz, M.; Rudyak, Y.: Small values of the Lusternik-Schnirelmann category for manifolds. See
Elmir, C.; Lafontaine, J.: Sur la géométrie systolique des variétés de Bieberbach. See
1. Horowitz, C.; Katz, Karin Usadi; Katz, M.: Loewner's torus inequality with isosystolic defect. Journal of Geometric Analysis, to appear. See
2. Berger, M.: What is... a Systole? Notices of the AMS 55 (2008), no. 3, 374-376.
3. Sabourau, S.: Asymptotic bounds for separating systoles on surfaces. Comment. Math. Helv. 83 (2008), no. 1, 35--54.
Cayley 4-form comass and triality isomorphisms, by M. Katz and S. Shnider, see
Systolic volume of hyperbolic manifolds and connected sums of manifolds, by S. Sabourau, Geom. Dedicata 127 (2007), 7-18.
A study of a d-systolic upper bound in terms of the log of the total
volume of a d-dimensional complex, by
Lubotzky, A.; Meshulam, R.: A Moore bound for simplicial complexes. Bull. Lond. Math. Soc. 39 (2007), no. 3, 353--358.
On manifolds satisfying stable systolic inequalities,
A systolic lower bound for the area of CAT(0) surfaces,
1. Notes on Gromov's systolic estimate,
2. Gromov's book was out of print, but no more! Order the
3. Unlike optimal systolic constants, optimal filling constants are
independent of the topology of the manifold,
4. A study of small values of Lusternik-Schnirelmann and systolic
categories for manifolds,
5. Spines and systoles for Teichmuller space of flat tori
6. Visit the page on
1. Systolic groups acting on complexes with no flats are
2. An approach to understanding the mapping class group via the
systole function on the moduli space is proposed
1. A study of filling invariants in systolic complexes and groups,
2. An effective algorithm to determine the systolic loops of a
1. A short proof of Gromov's filling inequality,
1. The systolic constant, the minimal entropy, and the spherical
volume of a manifold depend only on the image of the fundamental class
in the Eilenberg-MacLane space,
3. An essay by Gábor Elek on the mathematics of Mikhael Gromov, has appeared at Acta Math. Hungarica . Systoles are discussed on pages 174-175.
1. A study of the global geometry of Teichmuller space by lengths of
simple closed geodesics,
Return to main SGT menu
Return to homepage