Alternatives for pseudofinite groups

TL;DR

This paper explores alternative structural properties of pseudofinite groups, showing they either contain free substructures or are close to nilpotent or finite, extending classical group theory results.

Contribution

It establishes new alternative theorems for pseudofinite groups, including conditions under which they contain free subgroups or are near nilpotent, generalizing Tits' alternative.

Findings

An $eth_{0}$-saturated pseudofinite group either contains a rank 2 subsemigroup or is nilpotent-by-locally finite.

In weakly bounded rank cases, such groups either contain a free subgroup or are nilpotent-by-abelian-by-locally finite.

Connections between these alternatives and amenability are also discussed.

Abstract

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.