GroupConciseness · Conciseness of words in residually finite and profinite groups

Conciseness of words in residually finite and profinite groups

Within the area of group theory, a word (or a group-word) w in k variables is, by definition, an element of the free group of rank k. Thus, for every group G, the word w defines a map w_G from the direct product of k copies of G to G itself defined by substituting word variables by elements of G. The image of this map is called the set of w-values of G, while the subgroup generated by the set of w-values is called the verbal subgroup of G corresponding to w. Word problems in groups consist, mainly, on the study of the relation between the set of w-values and the verbal subgroup of a group G. Word problems in groups have been one of the main topics in group theory since the definition of the commutator word in the eve of the 20th century. It was not until the beginning of the second half of the past century, though, that P. Hall introduced, together with some famous conjectures, the notion of conciseness of words: a word is said to be concise in a class of groups C if for every G in C, the finiteness of the set of w-values implies that of the verbal subgroup. Thus, the first of those conjectures stated that every word is concise in the class of all groups. This problem received much attention during the first decades after its formulation, and much progress was made towards its resolution. However, in 1989, Ivanov found a word which was not concise in a particular group, showing that Hall’s conjecture is false in its general form. Nothing was published for several years regarding conciseness of words after Ivanov’s counterexample. Nevertheless, word problems, in general, have undergone a revival in the last two decades due to the achievement of some important results by, among others, D. Segal, N. Nikolov or A. Jaikin-Zapirain. The theory of words in groups has acquired, thus, a great maturity among researchers, and is nowadays a really active and productive research area. In this line, in 2008, A. Jaikin-Zapirain proposed a new version of Hall’s conjecture. As Ivanov's counterexample was not residually finite, and based on clear evidences on the literature, Jaikin-Zapirain conjectured that all words are concise in the class of residually finite groups (a group is said to be residually finite if the intersection of all its subgroup of finite index is trivial). Much more success has been obtained for this conjecture than for the original one, and, in recent years, a great deal of important words have been proved to be concise in the class of residually finite groups. Moreover, apart from all these results, some new related notions have been recently introduced, opening, in this way, a bunch of possibilities to explore. Therefore, seeing the current output of the topic, this proposal was devoted to the study of conciseness of words and some related notions in residually finite groups by using a multidisciplinary approach, combining different group theoretical, topological, and measure theoretical methods.

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