Only their cardinality can prevent groups from being quotients of weakly branch groups

Abstract

It is a well-known fact that every proper quotient of a branch group is a virtually abelian group. We record that every group of cardinality at most $2^{\aleph_0}$ is a quotient of a weakly branch group; in fact, there are weakly branch groups that are split extensions with free complements. Furthermore, if a group is generated by $d$ elements, there is a weakly branch group generated by $(d+2)$ elements mapping onto it. As applications of this fact, we resolve a question of Thillaisundaram on the existence of weakly branch but non-branch groups with maximal subgroups of infinite index and, partially, a question of Cox and Thillaisundaram on the existence of weakly branch groups that are not invariably generated.