News and recent systolic developments

nov '18

Hugo Akrout, Bjoern Muetzel. Construction of surfaces with large systolic ratio. See See

jul '18

Local maxima of the systole function, by Maxime Fortier Bourque, Kasra Rafi. See

jan '18

A local systolic-diastolic inequality in contact and symplectic geometry, by Gabriele Benedetti, Jungsoo Kang. See

nov '17

On growth of systole along congruence coverings of Hilbert modular varieties, by Plinio G P Murillo, Algebraic & Geometric Topology 17:5 (2017), 2753-2762

feb '16

1. Bou-Rabee, Khalid; Cornulier, Yves; Systolic growth of linear groups. Proc. Amer. Math. Soc. 144 (2016), no. 2, 529-533.

2. Fanoni, Federica; Parlier, Hugo; Systoles and kissing numbers of finite area hyperbolic surfaces. Algebr. Geom. Topol. 15 (2015), no. 6, 3409-3433.

3. Chen, Lizhi; Li, Weiping. Systoles of surfaces and 3-manifolds. Geometry and topology of submanifolds and currents, 61-80, Contemp. Math., 646, Amer. Math. Soc., Providence, RI, 2015.

4. Grácio, Clara. Systoles on compact Riemann surfaces with symbolic dynamics. Nonlinear maps and their applications, 269-288, Springer Proc. Math. Stat., 112, Springer, Cham, 2015.

june '15

1. arXiv:1506.09051 Regular geometric cycles. Guillaume Bulteau

2. arXiv:1506.08341 Systolic Surfaces of Arithmetic Hyperbolic 3-Manifolds. Benjamin Linowitz, Jeffrey S. Meyer

3. arXiv:1506.07848 Systolic geometry and regularization technique. Guillaume Bulteau

4. arXiv:1506.07121 Combinatorial systolic inequalities. Ryan Kowalick, Jean-François Lafont, Barry Minemyer

april '15

1. Elsner, Tomasz; Januszkiewicz, Tadeusz; Homotopical systole and hyperbolicity. Bull. Lond. Math. Soc. 47 (2015), no. 2, 203-205.

2. Katz, Mikhail G.; Sabourau, Stephane; Dyck's surfaces, systoles, and capacities. Trans. Amer. Math. Soc. 367 (2015), no. 6, 4483-4504.

3. Chepoi, Victor; Osajda, Damian Dismantlability of weakly systolic complexes and applications. Trans. Amer. Math. Soc. 367 (2015), no. 2, 1247-1272.

4. Zachos, Anastasios N. Minimal systolic circles. J. Convex Anal. 21 (2014), no. 3, 619-650.

5. Mondal, Sugata Systole and \lambda_{2g-2} of closed hyperbolic surfaces of genus g. Enseign. Math. 60 (2014), no. 1-2, 3-24.

6. Chen, Lizhi; Systolic freedom of 3-manifolds. Thesis (Ph.D.) Oklahoma State University. 2014. 83 pp.

june '14

Lizhi Chen. Z_2-coefficient Homology (1,2)-systolic Freedom of RP^3#RP^3.

may '14

1. Karin Katz, Mikhail Katz, Michael Schein, Uzi Vishne. Bolza quaternion order and asymptotics of systoles along congruence subgroups. See

2. Larry Guth, Alexander Lubotzky. Quantum error-correcting codes and 4-dimensional arithmetic hyperbolic manifolds. See

apr '14

1. Systoles and Dehn surgery for hyperbolic 3-manifolds
Grant S Lakeland and Christopher J Leininger
Algebraic & Geometric Topology 14 (2014) 1441-1460 DOI: 10.2140/agt.2014.14.1441
Abstract. Given a closed hyperbolic 3-manifold M of volume V, and a link L in M such that the complement M\L is hyperbolic, we establish a bound for the systole length of M\L in terms of V. This extends a result of Adams and Reid, who showed that in the case that M is not hyperbolic, there is a universal bound of 7.35534... As part of the proof, we establish a bound for the systole length of a noncompact finite volume hyperbolic manifold which grows asymptotically like 4/3 logV.

2. Massart, Daniel; Muetzel, Bjoern. On the intersection form of surfaces. Manuscripta Math. 143 (2014), no. 1-2, 19-49. See

jun '13

Belolipetsky, Mikhail; On 2-Systoles of Hyperbolic 3-Manifolds. Geom. Funct. Anal. 23 (2013), no. 3, 813-827. See

may '13

On the finite dimensional approximation of the Kuratowski-embedding for compact manifolds
Malte Roeer

nov '12

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

oct '12

1. Liokumovich, Yevgeny; Spheres of small diameter with long sweep-outs. Proc. Amer. Math. Soc. 141 (2013), no. 1, 309-312.

2. Filippo Cerocchi: Margulis Lemma, entropy and free products. See

3. Chady El Mir, Zeina Yassine: Conformal Geometric Inequalities on the Klein Bottle. See

sep '12 Exotic arithmetic structure on the first Hurwitz triplet, by Lei Yang

july '12

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
Shotaro Makisumi
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.

may '12

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

jan '12

1. Hyperellipticity and Systoles of Klein Surfaces. By Mikhail G. Katz and Stephane Sabourau. See and

2. Makover, Eran; McGowan, Jeffrey: The length of closed geodesics on random Riemann surfaces. Geom. Dedicata 151 (2011), 207--220. See mathscinet at

3. Belolipetsky, Mikhail V.; Thomson, Scott A. Systoles of hyperbolic manifolds. Algebr. Geom. Topol. 11 (2011), no. 3, 1455--1469 at and mathscinet at

aug '11
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
DOI: 10.2140/gt.2011.15.1477

july '11

1. arXiv:1107.5975
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 cycle.
Discrete Comput. Geom. 44 (2010), no. 4, 912--930.

june '11

1. If you are in Germany, you may be interested in Pape's course, see

2. arXiv:1106.1834
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
DOI: 10.2140/agt.2011.11.1455

may '11

Title: Liouville's equation for curvature and systolic defect
Author: Mikhail Katz

apr '11

Algebraic & Geometric Topology 11 (2011) 983-999
Stable systolic category of the product of spheres
by Hoil Ryu
DOI: 10.2140/agt.2011.11.983

mar '11

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

jan '11

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.

nov '10

Title: Short loop decompositions of surfaces and the geometry of Jacobians
Authors: Florent Balacheff, Hugo Parlier, Stephane Sabourau

oct '10

Title: The homology systole of hyperbolic Riemann surfaces
Authors: Hugo Parlier

sep '10

1. arXiv:1009.2835
Title: Distribution of the systolic volume of homology classes
Authors: Ivan K. Babenko, Florent Balacheff,

aug '10

Title: Systoles of Hyperbolic Manifolds
Authors: Mikhail Belolipetsky, Scott A. Thomson

july '10

1. arXiv:1007.2913 [pdf, ps, other]
Title: Stable systolic category of the product of spheres
Authors: Hoil Ryu

2. arXiv:1007.0877
Title: Conformal isosystolic inequality of Bieberbach 3-manifolds
Authors: Chady El Mir

april '10

Title: Flat currents modulo p in metric spaces and filling radius inequalities
Authors: Luigi Ambrosio, Mikhail G. Katz

march '10

Title: Metaphors in systolic geometry
Author: Larry Guth

december '09

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.

2. arXiv:0912.3894
Title: The systolic constant of orientable Bieberbach 3-manifolds
Authors: Chady Elmir

3. arXiv:0912.3413
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.

november '09

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.)

2. arXiv:0911.4265
Title: Relative systoles of relative-essential 2-complexes
Authors: Karin U. Katz, Mikhail G. Katz, Stephane Sabourau, Steven Shnider, Shmuel Weinberger

october '09

Title: Filling minimality of Finslerian 2-discs
Author: Sergei Ivanov

september '09

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."

2. arXiv:0909.1966
Title: Small filling sets of curves on a surface
Authors: James W. Anderson, Hugo Parlier, Alexandra Pettet

3. arXiv:0909.1665
Title: Area-minimizing projective planes in three-manifolds
Authors: H. Bray, S. Brendle, M. Eichmair, A. Neves

july '09

1. arXiv:0907.3517
Title: Scattering at low energies on manifolds with cylindrical ends and stable systoles
Authors: Werner Muller, Alexander Strohmaier

2. arXiv:0907.2223
Title: Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic
Author: Stephane Sabourau

april '09

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.

march '09

arXiv:0903.5299, Title: Systolic inequalities and minimal hypersurfaces, by Larry Guth

february '09

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

january '09

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:math.DG/0608006.

december '08

arXiv:0812.4637: Stable Systolic Category of Manifolds and the Cup-length. Authors: Alexander N. Dranishnikov, Yuli B. Rudyak

november '08

1. Balacheff, F.: A local optimal diastolic inequality on the two-sphere. See arXiv:0811.0330
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

october '08

1. Martelli, Bruno: Complexity of PL-manifolds. See arXiv:0810.5478

september '08

1. Croke, C.: Small volume on big n-spheres, Proc. Amer. Math. Soc. 136 (2008), no. 2, 715-717 mathscinet review
It is known that an n-sphere with a metric invariant by the antipodal involution, admits a curvature-free volume lower bound in terms of the least distance L between a point and its antipodal image (the bound follows from Gromov's filling inequality from '83, applied to the quotient real projective n-space). The paper shows that in the absence of the condition of invariance by the antipodal map, the total volume is no longer constrained by L.

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 mathscinet

august '08

1. arXiv:0807.5040 Cohomological dimension, self-linking, and systolic geometry, by Dranishnikov, A.; Katz, M.; Rudyak, Y.

july '08

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.

june '08

1. Frequently Asked Questions about Journal of the London Mathematical Society

may '08

1. Neil's theology and systoles

2. Dranishnikov, A.; Katz, M.; Rudyak, Y.: Small values of the Lusternik-Schnirelmann category for manifolds. See arXiv:0805.1527

april '08

Elmir, C.; Lafontaine, J.: Sur la géométrie systolique des variétés de Bieberbach. See arXiv:0804.1419

march '08

1. Horowitz, C.; Katz, Karin Usadi; Katz, M.: Loewner's torus inequality with isosystolic defect. Journal of Geometric Analysis, to appear. See arXiv:0803.0690

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.

january '08

Cayley 4-form comass and triality isomorphisms, by M. Katz and S. Shnider, see arXiv:0801.0283

december '07

Systolic volume of hyperbolic manifolds and connected sums of manifolds, by S. Sabourau, Geom. Dedicata 127 (2007), 7-18.

october '07

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.

august '07

On manifolds satisfying stable systolic inequalities,
by M. Brunnbauer

july '07

A systolic lower bound for the area of CAT(0) surfaces,
by Y. Chai and D. Lee

june '07

1. Notes on Gromov's systolic estimate,
by L. Guth

2. Gromov's book was out of print, but no more! Order the new edition now

3. Unlike optimal systolic constants, optimal filling constants are independent of the topology of the manifold,
by M. Brunnbauer

4. A study of small values of Lusternik-Schnirelmann and systolic categories for manifolds,
by A. Dranishnikov, M. Katz, and Y. Rudyak

5. Spines and systoles for Teichmuller space of flat tori
by A. Pettet and J. Souto

6. Visit the page on simplicial nonpositive curvature,
maintained by T. Elsner

may '07

1. Systolic groups acting on complexes with no flats are word-hyperbolic, by
by P. Przytycki. See also systolic group theory

2. An approach to understanding the mapping class group via the systole function on the moduli space is proposed
by M. Bestvina

april '07

1. A study of filling invariants in systolic complexes and groups,
by T. Januszkiewicz and J. Świątkowski. See also systolic group theory

2. An effective algorithm to determine the systolic loops of a hyperbolic surface,
by H. Akrout

march '07

1. A short proof of Gromov's filling inequality,
by S. Wenger

2. A book entitled "Systolic geometry and topology" is published by the
AMS, Mathematical Surveys and Monographs, vol. 137.

february '07

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,
by M. Brunnbauer

2. Finding closed geodesics, as well as loops that are nearly closed geodesics, in a tight sweep-out of a 2-sphere,
by T. Colding and W. Minicozzi

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.

january '07

1. A study of the global geometry of Teichmuller space by lengths of simple closed geodesics,
by Zheng Huang

2. A study of closed geodesics for Finsler metrics on the 2-sphere, in relation to the theory of H. Hofer, K. Wysocki, and E. Zehnder,
by A. Harris and G. Paternain

3. Ever wonder how short a closed geodesic can be on a hyperbolic 4-manifold? Find the answer
by I. Agol

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