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Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups

Published online by Cambridge University Press:  20 November 2018

Yu Kitabeppu  and
Sajjad Lakzian
Yu Kitabeppu
Affiliation:
Kyoto University e-mail: y.kitabeppu@gmail.com
Sajjad Lakzian
Affiliation:
Hausdorff Institute for Mathematics, Universität Bonn e-mail: lakzians@gmail.com
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Abstract

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In this paper, we generalize the finite generation result of Sormani to non-branching $RCD\left( 0,\,N \right)$ geodesic spaces (and in particular, Alexandrov spaces) with full supportmeasures. This is a special case of the Milnor’s Conjecture for complete non-compact $RCD\left( 0,\,N \right)$ spaces. One of the key tools we use is the Abresch–Gromoll type excess estimates for non-smooth spaces obtained by Gigli–Mosconi.

Keywords

53C2330L99Milnor conjecturenon negative Ricci curvaturecurvature dimension conditionfinitely generatedfundamental groupinfinitesimally Hilbertian
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 4 , 01 December 2015 , pp. 787 - 798
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Abresch, U. and Gromoll, D., On complete manifolds with nonnegative Ricci curvature. J. Amer. Math. Soc. 3(1990), no. 2, 355–374. http://dx.doi.Org/10.1090/S0894-0347-1990-1030656-6 Google Scholar
[2] Ambrosio, L., Gigli, N., and Savaré, G. Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163(2014), no. 7,1405-1490. http://dx.doi.Org/10.1215/00127094-2681605 Google Scholar
[3] Anderson, M. T., On the topology of complete manifolds of nonnegative Ricci curvature. Topology 29(1990), no. 1, 41–55. http://dx.doi.Org/10.1016/0040-9383(90)90024-E Google Scholar
[4] Bâcher, K. and Sturm, K. T., Localization and tensorization properties of the curvature-dimension condition for metric measure spaces. J. Funct. Anal. 259(2010), no. 1, 28–56. http://dx.doi.Org/10.1016/j.jfa.2010.03.024 Google Scholar
[5] Burago, D., Burago, Y., and Ivanov, S., A course in metric geometry. Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001.Google Scholar
[6] Cheeger, J. and Colding, T. H., Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. of Math. (2) 144(1996), no. 1,189-237. http://dx.doi.Org/10.2307/2118589 Google Scholar
[6] Cheeger, J. and Colding, T. H., Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. of Math. (2) 144(1996), no. 1,189-237. http://dx.doi.Org/10.2307/2118589 Google Scholar
[7] Cheeger, J. and Colding, T. H., On the structure of spaces with Ricci curvature bounded below. I. J. Differential Geom. 46(1997), no. 3, 406–480.Google Scholar
[8] Cheeger, J. and Colding, T. H., On the structure of spaces with Ricci curvature bounded below. II. J. Differential Geom. 54(2000), no. 1, 13–35.Google Scholar
[9] Cheeger, J. and Gromoll, D., On the structure of complete manifolds of nonnegative curvature. Ann. of Math. 96(1972), 413–443. http://dx.doi.Org/10.2307/1970819 Google Scholar
[10] Gigli, N., On the differential structure of metric measure spaces and applications. arxiv:1205.6622Google Scholar
[11] Gigli, N., The splitting theorem in non-smooth context. arxiv:1302.5555Google Scholar
[12] Gigli, N. and Mosconi, S., The Abresch-Gromoll inequality in a non-smooth setting. Discrete Contin. Dyn. Syst. 34(2014), no. 4,1481-1509.Google Scholar
[13] Kuwae, K., Machigashira, Y., and Shioya, T., Sobolev spaces, Laplacian, and heat kernel on Alexandrovspaces. Math. Z. 238(2001), no. 2, 269–316. http://dx.doi.Org/10.1007/s002090100252 Google Scholar
[14] Li, P., Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature. Ann. of Math. (2) 124(1986), no. 1,1-21. http://dx.doi.Org/10.2307/1971385 Google Scholar
[15] Liu, G., 3-manifolds with nonnegative Ricci curvature. Invent. Math. 193(2013), no. 2, 367–375. http://dx.doi.Org/10.1007/s00222-012-0428-x Google Scholar
[16] Lott, J. and Villani, C., Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169(2009), no. 3, 903–991. http://dx.doi.Org/10.4007/annals.2009.169.903 Google Scholar
[17] Milnor, J., A note on curvature and fundamental group. J. Differential Geometry 2(1968), 1–7.Google Scholar
[18] Mitsuishi, A. and Yamaguchi, T., Locally Lipschitz contractibility of Alexandrov spaces and its applications. Pacific J. Math. 270(2014), no. 2, 393–421.http://dx.doi.Org/10.2140/pjm.2014.270.393 Google Scholar
[19] Perelman, G., Spaces with curvature bounded below. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), Birkhâuser, Basel, 1995, pp. 517–525.Google Scholar
[20] Petrunin, A., Alexandrov meets Lott-Villani-Sturm. Munster J. Math. 4(2011), 53–64.Google Scholar
[21] Rajala, T. and Sturm, K. T., Non-branching geodesies and optimal maps in strong CD(K, oo)-spaces. Calc. Var. Partial Differential Equations 50(2014), no. 3-4, 831–846. http://dx.doi.Org/10.1007/s00526-013-0657-x Google Scholar
[22] Schoen, R. and Yau, S. T., Complete three-dimensional manifolds with positive Ricci curvature and scalar curvature. In: Seminar on Differential Geometry, Ann. of Math. Stud., 102, Princeton University Press, Princeton, NJ, 1982, pp. 209–228.Google Scholar
[23] Shankar, K. and Sormani, C., Conjugate points in length spaces. Adv. Math. 220(2009), no. 3, 791–830. http://dx.doi.Org/10.1016/j.aim.2008.10.002 Google Scholar
[24] Sormani, C., Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups. J. Differential Geom. 54(2000), no. 3, 547–559.Google Scholar
[25] Sturm, K. T., On the geometry of metric measure spaces. I. Acta Math. 196(2006), no. 1, 65–131. http://dx.doi.Org/10.1007/s11511-006-0002-8 Google Scholar
[26] Sturm, K. T., On the geometry of metric measure spaces. II. Acta Math. 196(2006), no. 1,133-177. http://dx.doi.Org/10.1007/s11511-006-0003-7 Google Scholar
[27] Wei, G. and Wylie, W., Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom. 83(2009), no. 2, 377–405.Google Scholar