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Transactions of the American Mathematical Society

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The energy of equivariant maps and a fixed-point property for Busemann nonpositive curvature spaces


Author: Mamoru Tanaka
Journal: Trans. Amer. Math. Soc. 363 (2011), 1743-1763
MSC (2010): Primary 58E20, 58E40; Secondary 51F99
DOI: https://doi.org/10.1090/S0002-9947-2010-05238-9
Published electronically: November 5, 2010
MathSciNet review: 2746663
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Abstract: For an isometric action of a finitely generated group on the ultralimit of a sequence of global Busemann nonpositive curvature spaces, we state a sufficient condition for the existence of a fixed point of the action in terms of the energy of equivariant maps from the group into the space. Furthermore, we show that this energy condition holds for every isometric action of a finitely generated group on any global Busemann nonpositive curvature space in a family which is stable under ultralimit, whenever each of these actions has a fixed point.

We also discuss the existence of a fixed point of affine isometric actions of a finitely generated group on a uniformly convex, uniformly smooth Banach space in terms of the energy of equivariant maps.


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Additional Information

Mamoru Tanaka
Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
Email: sa5m10@math.tohoku.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2010-05238-9
Keywords: Global Busemann NPC spaces, fixed point, ultralimits, uniformly convex, uniformly smooth Banach space
Received by editor(s): May 16, 2008
Published electronically: November 5, 2010
Additional Notes: The author was supported by Grant-in-Aid for JSPS Fellows (21$·$1062).
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.