Andrews-Curtis equivalent subsets

Symbol-free definition
Two subsets of a finitely generated free group are said to be Andrews-Curtis equivalent if we can go from one subset to the other by a sequence of Nielsen transformations composed with inner automorphisms in the free group (in other words, they are in the same orbit of the natural action of the Andrews-Curtis group on subsets).

Andrews-Curtis equivalent subsets turn up in the context of the (As yet unsettled) Andrews-Curtis conjecture.