Varieties of groups and the problem on conciseness of words

A group-word w is said to be concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is called concise in a class of groups X if whenever the set of w-values is finite for a group G belonging to the class X, it always follows that w(G) is finite. P. Hall asked whether every word is concise. Due to S. Ivanov the answer to this problem is known to be negative. On the other hand, for residually finite groups the problem remains wide open. Our talk will be about recent developments with respect to that problem. Based on a joint work with P. Shumyatsky (University of Brasilia).