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: inf¹d.x; gx/ W x 2 Lº 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.

: We present a simple constructive proof of the fact that every abelian discrete group is uniformly amenable. We improve the growth function obtained earlier and find the optimal growth function in a particular case. We also compute a growth function for some non-abelian uniformly amenable group.

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Abstract. The radius of starlikeness for polynomials with zeroes distributed at certain curves in the unit disc as well as the case in which zeroes are concentrated at a single point are considered and sharp bounds are obtained.Mathematics subject classification (2010): Primary 30C45; Secondary 30C80.

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