Systolic invariants of groups and $2$-complexes via Grushko decomposition

Rudyak, Yuli B.; Sabourau, Stéphane

Annales de l'Institut Fourier, Tome 58 (2008) no. 3, pp. 777-800.

Nous prouvons un résultat de finitude pour l’aire systolique des groupes. Précisément, nous montrons qu’il n’existe qu’un nombre fini de facteurs non-libres dans les groupes fondamentaux des $2$-complexes d’aire systolique uniformément bornée. Nous montrons aussi que le nombre de tels groupes librement indécomposables croît au moins exponentiellement avec la borne sur l’aire systolique. De plus, nous prouvons une inégalité systolique uniforme pour tous les $2$-complexes de groupe fondamental non-libre qui améliore les bornes précédemment connues dans cette dimension.

We prove a finiteness result for the systolic area of groups. Namely, we show that there are only finitely many possible unfree factors of fundamental groups of $2$-complexes whose systolic area is uniformly bounded. We also show that the number of freely indecomposable such groups grows at least exponentially with the bound on the systolic area. Furthermore, we prove a uniform systolic inequality for all $2$-complexes with unfree fundamental group that improves the previously known bounds in this dimension.

[1] Aleksandrov, A. D.; Zalgaller, V. A. Intrinsic geometry of surfaces, Translations of Mathematical Monographs, 15, American Mathematical Society, Providence, R.I., 1967 | MR | Zbl

[2] Balacheff, F. Sur des problèmes de la géométrie systolique, Sémin. Théor. Spectr. Géom., Volume 22 (2004), pp. 71-82 | EuDML | Numdam | MR | Zbl

[3] Bangert, V.; Croke, C.; Ivanov, S.; Katz, M. Filling area conjecture and ovalless real hyperelliptic surfaces, Geom. Funct. Anal., Volume 15 (2005), pp. 577-597 | DOI | MR | Zbl

[4] Bavard, C. Une remarque sur la géométrie systolique de la bouteille de Klein, Arch. Math., Volume 87 (2006) no. 1, pp. 72-74 | DOI | MR | Zbl

[5] Bochnak, J.; Coste, M.; Roy, M.-F. Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, 36, Springer-Verlag, 1998 no. 3 | MR | Zbl

[6] Burago; Zalgaller, V. A. Geometric inequalities, Grundlehren der Mathematischen Wissenschaften, 285, Springer-Verlag, Berlin, 1988 | MR | Zbl

[7] Buser, P.; Sarnak, P. On the period matrix of a Riemann surface of large genus. With an appendix by J. H. Conway and N. J. A. Sloane, Invent. Math., Volume 117 (1994) no. 1, pp. 27-56 | DOI | EuDML | MR | Zbl

[8] Croke, C.; Katz, M. Universal volume bounds in Riemannian manifolds, Surveys in Differential Geometry, Volume 8, S.T. Yau, 2002, pp. 109-137 Lectures on Geometry and Topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck at Harvard University, May 3–5, 2002, edited by S.T. Yau (Somerville, MA: International Press, 2003). See arXiv:math.DG/0302248 | MR | Zbl

[9] Delzant, T. Décomposition d’un groupe en produit libre ou somme amalgamée, J. Reine Angew. Math., Volume 470 (1996), pp. 153-180 | DOI | EuDML | Zbl

[10] Federer, H. Geometric measure theory, Grundlehren der mathematischen Wissenschaften, Volume 153, Springer–Verlag, Berlin, 1969 | MR | Zbl

[11] Gromov, M. Filling Riemannian manifolds, J. Diff. Geom., Volume 18 (1983), pp. 1-147 | MR | Zbl

[12] Gromov, M. Systoles and intersystolic inequalities, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), 291–362, Sémin. Congr., 1, Soc. Math. France, Paris, 1996 | MR | Zbl

[13] Gromov, M. Metric structures for Riemannian and non-Riemannian spaces, Progr. in Mathematics, 152, Birkhäuser, Boston, 1999 | MR | Zbl

[14] Hebda, J. Some lower bounds for the area of surfaces, Invent. Math., Volume 65 (1982), pp. 485-490 | DOI | EuDML | MR | Zbl

[15] Kapovich, I.; Schupp, P. Delzant’s T-invariant, Kolmogorov complexity and one-relator groups, Comment. Math. Helv., Volume 80 (2005) no. 4, pp. 911-933 | DOI | Zbl

[16] Katz, M. Systolic geometry and topology. With an appendix by Jake P, Math. Surveys Monographs, 137, Amer. Math. Soc., 2007 | MR | Zbl

[17] Katz, M.; Rudyak, Y.; Sabourau, S. Systoles of 2-complexes, Reeb graph, and Grushko decomposition, Int. Math. Res. Not., 2006 (Art. ID 54936, 30 pp. See arXiv:math.DG/0602009) | MR | Zbl

[18] Katz, M.; Sabourau, S. Entropy of systolically extremal surfaces and asymptotic bounds, Ergodic Theory and Dynamical Systems, Volume 25 (2005), pp. 1209-1220 (See arXiv:math.DG/0410312) | DOI | MR | Zbl

[19] Katz, M.; Sabourau, S. Hyperelliptic surfaces are Loewner, Proc. Amer. Math. Soc., Volume 134 (2006) no. 4, pp. 1189-1195 | DOI | MR | Zbl

[20] Katz, M.; Sabourau, S. An optimal systolic inequality for CAT(0) metrics in genus two, Pacific J. Math., Volume 227 (2006) no. 1, pp. 95-107 (See arXiv:math.DG/0501017) | DOI | MR | Zbl

[21] Katz, M.; Schaps, M.; Vishne, U. Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups, J. Diff. Geom., Volume 76 (2007) no. 3, pp. 399-422 (Available at arXiv:math.DG/0505007) | MR | Zbl

[22] Ohshika, K. Discrete groups, Iwanami Series in Modern Mathematics, American Mathematical Society, Providence, RI, 2002 (Translated from the 1998 Japanese original by the author. Translations of Mathematical Monographs, 207.) | MR | Zbl

[23] Pu, P. M. Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math., Volume 2 (1952), pp. 55-71 | MR | Zbl

[24] Sabourau, S. Asymptotic bounds for separating systoles on surfaces, Comment. Math. Helv. (to appear) | MR | Zbl

[25] Stallings, John R. A topological proof of Grushko’s theorem on free products, Math. Z., Volume 90 (1965), pp. 1-8 | DOI | EuDML | Zbl