Bound-word subgroup

Definition with symbols
A quantified word-letter pair is a word with a distinguished letter, with a sequence of (possibly nested) existential and universal quantifiers on all the other letters. An element $$g$$ in a group $$G$$ satisfies this pair if setting the distinguished letter equal to $$g$$, the nested expression is true in the group (where existential and universal quantifiers are interpreted over $$G$$). The subgroup corresponding to a quantified word-letter pair is defined as the subgroup generated by all elements of the group satisfying that pair.

A bound-word subgroup is a possibly arbitrary join of finite intersections of subgroups corresponding to quantified word-letter pairs.