Abderezak Ould Houcine, Françoise Point
July 1, 2013
Abstract. The famous Tits alternative states that a linear group either contains a non-abelian free group or is soluble-by-(locally finite). In this paper we study similar alternatives in pseudofinite groups. We show, for instance, that an ℵ 0 -saturated pseudofinite group either contains a subsemigroup of rank 2 or is nilpotent-by-(uniformly locally finite). We call a class of finite groups G weakly of bounded rank if the radical has a bounded Prüfer rank and the index of the socle of is bounded. We show that an ℵ 0 -saturated pseudo-(finite weakly of bounded rank) group either contains a non-abelian free group or is nilpotent-by-abelian-by-(uniformly locally finite). We also obtain some relations between these kind of alternatives and amenability.