A parabolic action on a proper, CAT(0) cube complex

Yael Algom-Kfir, Bronislaw Wajnryb, Pawel Witowicz

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)965-984
Number of pages20
JournalJournal of Group Theory
Issue number6
StatePublished - Nov 2013
Externally publishedYes

ASJC Scopus subject areas

  • Algebra and Number Theory


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