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Lipschitz 1-connectedness for Some Solvable Lie Groups

Part of: Special aspects of infinite or finite groups Lie groups

Published online by Cambridge University Press:  09 January 2019

David Bruce Cohen
David Bruce Cohen*
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA Email: davidbrucecohen@gmail.com
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Abstract

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A space X is said to be Lipschitz 1-connected if every Lipschitz loop 𝛾 : S1 → X bounds a O (Lip(𝛾))-Lipschitz disk f : D2 → X. A Lipschitz 1-connected space admits a quadratic isoperimetric inequality, but it is unknown whether the converse is true. Cornulier and Tessera showed that certain solvable Lie groups have quadratic isoperimetric inequalities, and we extend their result to show that these groups are Lipschitz 1-connected.

Keywords

Dehn functionsolvable groupLipschitz 1-connectedness

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 22E25: Nilpotent and solvable Lie groups
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 3 , June 2019 , pp. 533 - 555
Copyright
© Canadian Mathematical Society 2018 

Footnotes

This work was supported by NSF award 1148609.

References

Abels, Herbert, Finite presentability of S-arithmetic groups. In: Compact presentability of solvable groups. Lecture Notes in Mathematics, 1261. Springer-Verlag, Berlin, 1985. https://doi.org/10.1007/BFb0079708.Google Scholar
Borel, Armand, Linear algebraic groups. Second edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991.Google Scholar
Bridson, Martin R. and Haefliger, André, Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften, 319. Springer-Verlag, Berlin, 1999. https://doi.org/10.1007/978-3-662-12494-9.Google Scholar
Cornulier, Yves and Tessera, Romain, Geometric presentations of Lie groups and their Dehn functions . Publ. Math. Inst. Hautes Études Sci. 125(2017), 79–219. https://doi.org/10.1007/s10240-016-0087-3.Google Scholar
Epstein, David, Paterson, Mike S., Cannon, James W., Holt, Derek F., Levy, Silvio V, and Thurston, William P, Word processing in groups. Jones and Bartlett, Boston, MA, 1992.Google Scholar
Gromov, Mikhail, Asymptotic invariants of infinite groups. In: Geometric group theory. Vol. 2, London Math. Soc. Lecture Notes Series, 182, Cambridge University Press, Cambridge, 1993.Google Scholar
Kobayashi, Shoshichi and Nomizu, Katsumi, Foundations of differential geometry. Volume 1. Interscience Publishers, New York, 1963.Google Scholar
Lytchak, Alexander, Wenger, Stefan, and Young, Robert, Dehn functions and Hölder extensions in asymptotic cones. arxiv:1608.00082, 2016.Google Scholar
Mitsuishi, Ayato and Yamaguchi, Takao, Locally Lipschitz contractibility of Alexandrov spaces and its applications . Pacific J. Math. 270(2014), no. 2, 393–421. https://doi.org/10.2140/pjm.2014.270.393.Google Scholar
Young, Robert, The Dehn function of SL (n, Z) . Ann. of Math. 177(2013), no. 3, 969–1027. https://doi.org/10.4007/annals.2013.177.3.4.Google Scholar