On groups that are residually of finite rank (Q1275681)

A group has rank \(r\), \(r\) a positive integer, if every finitely generated subgroup of \(G\) can be generated by \(r\) elements, and if no smaller integer has this property. The authors study groups that are residually of rank \(r\). For example, Theorem 1 states that a periodic group that is residually (locally finite and of rank \(r\)), is locally finite. Theorem 2 is a (partial) structure result for groups that are residually (locally soluble and of rank \(r\)). Theorem 4 implies that a locally soluble group is hyperabelian whenever it is residually of rank \(r\). The authors illustrate their results with examples. For example, the standard wreath product of two infinite cyclic groups has infinite rank, but is residually of rank 2 (weakish form of Theorem 8).