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We consider diagram groups as defined by V. Guba and M. Sapir. A diagramgroup G acts on the associated cube complex K by isometries. It is known thatif a cube complex L is of a finite dimension then every isometry g of L issemi-simple, i.e. its translation length is realized. It was conjectured by D.S. Farley that in the case of a diagram group G the action of G on theassociated cube complex K is by semisimple isometries even when K has aninfinite dimension. In this paper we give a counterexample to Farley Conjectureand we show that R. Thompson's group F, considered as a diagram group, has someelements which act as parabolic (not semi-simple) isometries on the associatedcube complex.

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