### Abstract

For an isometric action of a finitely generated group on the ultra-limit 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.

Original language | English |
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Pages (from-to) | 1743-1763 |

Number of pages | 21 |

Journal | Transactions of the American Mathematical Society |

Volume | 363 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2011 Apr 1 |

### Keywords

- Fixed point
- Global Busemann NPC spaces
- Ultralimits
- Uniformly convex
- Uniformly smooth Banach space

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics