On groups with large verbal quotients

Let $w=w(x_1,...,x_n)$ be a word, i.e. an element of the free group $F = \langle x_1,...,x_n \rangle$. The verbal subgroup $w(G)$ of a group $G$ is the subgroup generated by the set $\{ w(x_1,...,x_n) : x_1,...,x_n \in G \}$ of all $w$-values in $G$. Following J. Gonz\'alez-S\'anchez and B. Klopsch, a group $G$ is $w$-maximal if $|H:w(H)| < |G:w(G)|$ for every $H<G$. In this paper we give new results on $w$-maximal groups, and study the weaker condition in which the previous inequality is not strict. Some applications are given: for example, if a finite group has a solvable (resp. nilpotent) section of size $n$, then it has a solvable (resp. nilpotent) subgroup of size at least $n$.