Let be a group-word. Suppose that the set of all -values in a profinite group is contained in a union of countably many subgroups. It is natural to ask in what way the structure of the verbal subgroup depends on the properties of the covering subgroups. The present article is a survey of recent results related to that question. In particular we survey results on finite and countable coverings of word-values (mostly commutators) by procyclic, abelian, nilpotent, and soluble subgroups, as well as subgroups with finiteness conditions. The last section of the paper is devoted to relation of the described results with Hall's problem on conciseness of group-words.
DOI :
10.4171/RSMUP/137-14
Classification :
20
Mots-clés : Profinite groups, covering subgroups, commutators, verbal subgroups
Mots-clés : Profinite groups, covering subgroups, commutators, verbal subgroups
Affiliations des auteurs :
Acciarri, Cristina 1 ;
Shumyatsky, Pavel 1
@article{RSMUP_2017__137__237_0, author = {Acciarri, Cristina and Shumyatsky, Pavel}, title = {Coverings of commutators in profinite groups}, journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova}, pages = {237--257}, publisher = {European Mathematical Society Publishing House}, address = {Zuerich, Switzerland}, volume = {137}, year = {2017}, doi = {10.4171/RSMUP/137-14}, mrnumber = {3652879}, zbl = {1393.20012}, url = {http://www.numdam.org/articles/10.4171/RSMUP/137-14/} }
TY - JOUR AU - Acciarri, Cristina AU - Shumyatsky, Pavel TI - Coverings of commutators in profinite groups JO - Rendiconti del Seminario Matematico della Università di Padova PY - 2017 SP - 237 EP - 257 VL - 137 PB - European Mathematical Society Publishing House PP - Zuerich, Switzerland UR - http://www.numdam.org/articles/10.4171/RSMUP/137-14/ DO - 10.4171/RSMUP/137-14 ID - RSMUP_2017__137__237_0 ER -
%0 Journal Article %A Acciarri, Cristina %A Shumyatsky, Pavel %T Coverings of commutators in profinite groups %J Rendiconti del Seminario Matematico della Università di Padova %D 2017 %P 237-257 %V 137 %I European Mathematical Society Publishing House %C Zuerich, Switzerland %U http://www.numdam.org/articles/10.4171/RSMUP/137-14/ %R 10.4171/RSMUP/137-14 %F RSMUP_2017__137__237_0
Acciarri, Cristina; Shumyatsky, Pavel. Coverings of commutators in profinite groups. Rendiconti del Seminario Matematico della Università di Padova, Tome 137 (2017), pp. 237-257. doi : 10.4171/RSMUP/137-14. http://www.numdam.org/articles/10.4171/RSMUP/137-14/
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