%0 Journal Article
%T Alternatives for pseudofinite groups
%A Abderezak Ould Houcine
%A Fran£żoise Point
%J Mathematics
%D 2012
%I arXiv
%X The famous Tits' alternative states that a linear group either contains a nonabelian free group or is soluble-by-(locally finite). We study in this paper similar alternatives in pseudofinite groups. We show for instance that an $\aleph_{0}$-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 $rad(G)$ has a bounded Pr\"ufer rank and the index of the sockel of $G/rad(G)$ is bounded. We show that an $\aleph_{0}$-saturated pseudo-(finite weakly of bounded rank) group either contains a nonabelian free group or is nilpotent-by-abelian-by-(uniformly locally finite). We also obtain some relations between this kind of alternatives and amenability.
%U http://arxiv.org/abs/1205.3533v1