Periodic Reporting for period 1 - GroupConciseness (Conciseness of words in residually finite and profinite groups)
Reporting period: 2023-02-01 to 2025-01-31
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.
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.
In WP1, we first tried to show that the word [x^p,y^p]^p is concise in the class of residually finite groups. This particular word is especially interesting, as taking powers of variables inside or outside commutators is one of the main obstacles when proving that a given word is concise. Also, for the purpose of working with such a word, we made use of the powerful Lie algebra methods introduced by Efim Zelmanov in his solution to the Restricted Burnside Problem. However, many technical problems arise, mainly coming from the fact that xp is not a commutator-closed word (meaning that [x^p,y^p] is not a pth power). However, we could prove that if w is a commutator-closed word such that [w,w]p takes finitely many values in a residually finite group G, then G virtually satisfies the word [w,w]. This is very closed of being concise, and actually, it already proves conciseness for every commutator-closed word which implies nilpotency or abelian-by-nilpotency.
Another approach to Objective 1 has been proving that the word x^p is strongly concise in the class of all profinite groups. Even if, again, many technical difficulties arose, we could reduce the problem to a very specific class of groups, namely, to free (central elementary abelian)-by-(p exponent) groups. The idea is now to show that this groups are equationally Noetherian (see Work Package 3 bellow), so that the strong conciseness of x^p will immediately follow.
In WP 3, we addressed, together with Dr. Andoni Zozaya (University of Ljubljana), the problem of whether every word is strongly concise in the class of profinite linear groups. We were able to completely solve this problem, proving that this is always the case not only for profinite linear groups, but for every equationally Noetherian group (linear groups are particular examples of equationally Noetherian groups). This notion relates group theory with algebraic geometry, and rises the following question: which profinite groups are equationally Noetherian? For instance, whereas it is known that finitely generated abelian-by-polycyclic groups are equationally Noetherian, it is not known whether topologically finitely generated profinite completions of abelian-by-polycyclic groups are equationally Noetherian. Thus, proving such a result would automatically show that every word is strongly concise in such a class.
In WP4, we considered, together with Prof. Benjamin Klopsch (University of Düsseldorf) and Prof. Anitha Thillaisundaram (University of Lund), the problem of whether, in a p-adic analytic pro-p group G, the Hausdorff spectrum of G with respect to the filtration series defined by the lower p-central series of G is finite (even if this was already proved for the rest of the standard filtration series of a finitely generated pro-p group, the result for the lower p-central series was still open). Thus, we proved such a result in a very strong form: we gave an explicit description of the lower p-central series of a general p-adic analytic pro-p group. This later result is actually much more important than our original goal, as it may be applied for addressing other open problems regarding p-adic analytic groups. We also proved that in a free pro-p group F, the Hausdorff dimension of F with respect to the lower p-central series of G coincides with the whole interval [0,1], in accordance with what happens with other filtration series. These results were obtained by making use of the theory of Lie algebras developed by E. Zelmanov in his proof of the Restricted Burnside Problem.
Finally, we also addressed some other related research problems as parts of an independent work package dealing with lattices of profinite groups and with long commutator words of maximal order.
On the one hand, together with Dr. Marco Trombetti and Prof. Francesco De Giovanni (University of Naples), we extended the classical theory of the subgroup lattice of finite and infinite groups to profinite groups. Thus, we considered the lattices of open and closed subgroups of a profinite group G and developed a completely new theory in analogy to the well-known classical results. All this has been published in the International Journal of Algebra and Computation. On the other hand, together with Federico Di Conciglio (University of Salerno), we studied groups in which the number of long commutators of maximal order is bounded. We extended results by Longobardi, Maj, Shumyatsky and Traustason by proving, among other results, that if a finitely generated group G contains m k-commutators of maximal prime power order, then the k-th term of the lower central series of G is finite of (m,k)-bounded order and there exists a k-step nilpotent subgroup of G of (m,k)-bounded index in G. This is currently work in progress, but we expect to publish a research article with these results in the Journal of Algebra in the near future.
Another approach to Objective 1 has been proving that the word x^p is strongly concise in the class of all profinite groups. Even if, again, many technical difficulties arose, we could reduce the problem to a very specific class of groups, namely, to free (central elementary abelian)-by-(p exponent) groups. The idea is now to show that this groups are equationally Noetherian (see Work Package 3 bellow), so that the strong conciseness of x^p will immediately follow.
In WP 3, we addressed, together with Dr. Andoni Zozaya (University of Ljubljana), the problem of whether every word is strongly concise in the class of profinite linear groups. We were able to completely solve this problem, proving that this is always the case not only for profinite linear groups, but for every equationally Noetherian group (linear groups are particular examples of equationally Noetherian groups). This notion relates group theory with algebraic geometry, and rises the following question: which profinite groups are equationally Noetherian? For instance, whereas it is known that finitely generated abelian-by-polycyclic groups are equationally Noetherian, it is not known whether topologically finitely generated profinite completions of abelian-by-polycyclic groups are equationally Noetherian. Thus, proving such a result would automatically show that every word is strongly concise in such a class.
In WP4, we considered, together with Prof. Benjamin Klopsch (University of Düsseldorf) and Prof. Anitha Thillaisundaram (University of Lund), the problem of whether, in a p-adic analytic pro-p group G, the Hausdorff spectrum of G with respect to the filtration series defined by the lower p-central series of G is finite (even if this was already proved for the rest of the standard filtration series of a finitely generated pro-p group, the result for the lower p-central series was still open). Thus, we proved such a result in a very strong form: we gave an explicit description of the lower p-central series of a general p-adic analytic pro-p group. This later result is actually much more important than our original goal, as it may be applied for addressing other open problems regarding p-adic analytic groups. We also proved that in a free pro-p group F, the Hausdorff dimension of F with respect to the lower p-central series of G coincides with the whole interval [0,1], in accordance with what happens with other filtration series. These results were obtained by making use of the theory of Lie algebras developed by E. Zelmanov in his proof of the Restricted Burnside Problem.
Finally, we also addressed some other related research problems as parts of an independent work package dealing with lattices of profinite groups and with long commutator words of maximal order.
On the one hand, together with Dr. Marco Trombetti and Prof. Francesco De Giovanni (University of Naples), we extended the classical theory of the subgroup lattice of finite and infinite groups to profinite groups. Thus, we considered the lattices of open and closed subgroups of a profinite group G and developed a completely new theory in analogy to the well-known classical results. All this has been published in the International Journal of Algebra and Computation. On the other hand, together with Federico Di Conciglio (University of Salerno), we studied groups in which the number of long commutators of maximal order is bounded. We extended results by Longobardi, Maj, Shumyatsky and Traustason by proving, among other results, that if a finitely generated group G contains m k-commutators of maximal prime power order, then the k-th term of the lower central series of G is finite of (m,k)-bounded order and there exists a k-step nilpotent subgroup of G of (m,k)-bounded index in G. This is currently work in progress, but we expect to publish a research article with these results in the Journal of Algebra in the near future.
In this project, we addressed mainly group theoretical problems concerning conciseness of words and Hausdorff dimension in profinite groups. The main results that we have proved are, on the one hand, that every word is strongly concise in the class of all profinite linear groups, and, on the other hand, the explicit description of the lower p-central series of p-adic analytic pro-p groups. Both problems have been solved, moreover, by using a multidisciplinary approach, as we have combined group theoretical, topological, measure theoretical, Lie algebra and algebraic geometry methods. These results are concern with basic and fundamental problems of group theory, and both of them may have a strong impact in their respective fields.
Apart from this, we also obtained several related results that extend the knowledge in areas such as finite conditions in long commutator words (by finding new nilpotent-like conditions in infinite groups) or profinite groups (by creating, in analogy with the non-profinite case, a theory concerning subgroup lattices). Moreover, most of the results obtained in this project have been or will be collected in research papers, and have been or will be published in international mathematical peer-reviewed journals.
Apart from this, we also obtained several related results that extend the knowledge in areas such as finite conditions in long commutator words (by finding new nilpotent-like conditions in infinite groups) or profinite groups (by creating, in analogy with the non-profinite case, a theory concerning subgroup lattices). Moreover, most of the results obtained in this project have been or will be collected in research papers, and have been or will be published in international mathematical peer-reviewed journals.