Conciseness of words in residually finite and profinite groups

The notion of conciseness of words in groups was introduced by Phillip Hall at the beginning of the second half of the past century.

Hall conjectured that every word is concise in the class of all groups, but this was proved to be false in 1989 by Ivanov in its general form.

However, the question whether every word is concise in the class of residually finite groups, raised by Andrei Jaikin-Zapirain, is still open, and is currently the main conjecture in the topic.In recent years, the notion of strong conciseness in profinite groups has also been introduced, and in this context, an analogous conjecture has been proposed, namely, that every word is strongly concise in the class of all profinite groups.This proposal is thus devoted to the study of conciseness and strong conciseness of words, as well as some related notions, in residually finite and profinite groups.

For the purpose of getting closer to the proofs of the aforementioned conjectures, we suggest a multidisciplinary approach by combining group theoretical, topological, and measure theoretical methods.

For instance, we propose studying conciseness and strong conciseness of words in certain classes of pro-p groups, such as p-adic analytic pro-p groups; or introducing the notion of Hausdorff conciseness by relating the Hausdorff dimension of the set of word values with the Hausdorff dimension of the verbal subgroup.This project proposal is a natural continuation of the applicant’s research career, who has already worked in several word-related problems, and will highly contribute to strengthening the candidate’s research skills, as well as to bringing novel and interesting ideas to the host organisations.