Alternatives for pseudofinite groups. (Q2869859)

Tits' alternative states that a linear group either contains a non-Abelian free group or is soluble-by-(locally finite). In this paper, the authors study alternatives for pseudofinite groups, i.e.\ groups satisfying the first-order theory of finite groups (or elementarily equivalent to a non-principal ultraproduct of finite groups). Using some uniform definability results, they show that an \(\aleph_0\)-saturated pseudofinite group either contains a free sub-semigroup of rank \(2\) or is nilpotent-by-(uniformly locally finite). They also obtain results about local embeddability (called \textit{approximability}), and relations between the non-existence of a free sub-(semi)-group of rank two and amenability and supramenability.