## Abstract

We consider diagram groups as defined by Guba and Sapir [Mem. Amer. Math. Soc. 130 (1997)]. A diagram group G acts on the associated cube complex K by isometries. It is known that if a cube complex L is of a finite dimension, then every isometry g of L is semi-simple: inf{d(x,gx) : x ε L} is attained. It was conjectured by Farley that in the case of a diagram group G the action of G on the associated cube complex K is by semisimple isometries also when K has an infinite dimension. In this paper we give a counter-example to Farley's conjecture by showing that R. Thompson's group F , considered as a diagram group, has some elements which act as parabolic (not semi-simple) isometries on the associated cube complex.

Original language | English |
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Pages (from-to) | 965-984 |

Number of pages | 20 |

Journal | Journal of Group Theory |

Volume | 16 |

Issue number | 6 |

DOIs | |

State | Published - Nov 2013 |

Externally published | Yes |

## ASJC Scopus subject areas

- Algebra and Number Theory