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Systolic invariants of groups and 2-complexes via Grushko decomposition. (English) Zbl 1142.53035

Summary: 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.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
55U10 Simplicial sets and complexes in algebraic topology
57Q99 PL-topology
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References:

[1] Aleksandrov, A. D.; Zalgaller, V. A., Intrinsic geometry of surfaces, 15 (1967) · Zbl 0146.44103
[2] Balacheff, F., Sur des problèmes de la géométrie systolique, Sémin. Théor. Spectr. Géom., 22, 71-82 (2004) · Zbl 1083.53043
[3] Bangert, V.; Croke, C.; Ivanov, S.; Katz, M., Filling area conjecture and ovalless real hyperelliptic surfaces, Geom. Funct. Anal., 15, 577-597 (2005) · Zbl 1082.53033 · doi:10.1007/s00039-005-0517-8
[4] Bavard, C., Une remarque sur la géométrie systolique de la bouteille de Klein, Arch. Math., 87, 1, 72-74 (2006) · Zbl 1109.53037 · doi:10.1007/s00013-006-1665-2
[5] Bochnak, J.; Coste, M.; Roy, M.-F., Real algebraic geometry, 36, 3 (1998) · Zbl 0912.14023
[6] Burago; Zalgaller, V. A., Geometric inequalities, 285 (1988) · Zbl 0633.53002
[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., 117, 1, 27-56 (1994) · Zbl 0814.14033 · doi:10.1007/BF01232233
[8] Croke, C.; Katz, M., Surveys in Differential Geometry, 8, 109-137 (2002) · Zbl 1051.53026
[9] Delzant, T., Décomposition d’un groupe en produit libre ou somme amalgamée, J. Reine Angew. Math., 470, 153-180 (1996) · Zbl 0836.20038 · doi:10.1515/crll.1996.470.153
[10] Federer, H., Grundlehren der mathematischen Wissenschaften, 153 (1969) · Zbl 0176.00801
[11] Gromov, M., Filling Riemannian manifolds, J. Diff. Geom., 18, 1-147 (1983) · Zbl 0515.53037
[12] Gromov, M., Systoles and intersystolic inequalities (1996) · Zbl 0877.53002
[13] Gromov, M., Metric structures for Riemannian and non-Riemannian spaces, 152 (1999) · Zbl 0953.53002
[14] Hebda, J., Some lower bounds for the area of surfaces, Invent. Math., 65, 485-490 (1982) · Zbl 0482.53028 · doi:10.1007/BF01396632
[15] Kapovich, I.; Schupp, P., Delzant’s T-invariant, Kolmogorov complexity and one-relator groups, Comment. Math. Helv., 80, 4, 911-933 (2005) · Zbl 1151.20026 · doi:10.4171/CMH/39
[16] Katz, M., Systolic geometry and topology. With an appendix by Jake P, 137 (2007) · Zbl 1149.53003
[17] Katz, M.; Rudyak, Y.; Sabourau, S., Systoles of 2-complexes, Reeb graph, and Grushko decomposition, Int. Math. Res. Not. (2006) · Zbl 1116.57001
[18] Katz, M.; Sabourau, S., Entropy of systolically extremal surfaces and asymptotic bounds, Ergodic Theory and Dynamical Systems, 25, 1209-1220 (2005) · Zbl 1097.53027 · doi:10.1017/S0143385704001014
[19] Katz, M.; Sabourau, S., Hyperelliptic surfaces are Loewner, Proc. Amer. Math. Soc., 134, 4, 1189-1195 (2006) · Zbl 1090.53045 · doi:10.1090/S0002-9939-05-08057-3
[20] Katz, M.; Sabourau, S., An optimal systolic inequality for CAT(0) metrics in genus two, Pacific J. Math., 227, 1, 95-107 (2006) · Zbl 1156.53019 · doi:10.2140/pjm.2006.227.95
[21] Katz, M.; Schaps, M.; Vishne, U., Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups, J. Diff. Geom., 76, 3, 399-422 (2007) · Zbl 1149.53025
[22] Ohshika, K., Discrete groups (2002) · Zbl 1006.20031
[23] Pu, P. M., Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math., 2, 55-71 (1952) · Zbl 0046.39902
[24] Sabourau, S., Asymptotic bounds for separating systoles on surfaces, Comment. Math. Helv. · Zbl 1142.53034
[25] Stallings, John R., A topological proof of Grushko’s theorem on free products, Math. Z., 90, 1-8 (1965) · Zbl 0135.04603 · doi:10.1007/BF01112046
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