Irreducible bases and subgroups of a wreath product in applying to diffeomorphism groups acting on the Möbius band

Abstract

We generalize the results presented in the book of Meldrum (Wreath products of groups and semigroups, vol 74, CRC Press, Boca Raton, 1995) about commutator subgroup of wreath products since, as well as considering regular wreath products, we consider those which are not regular (in the sense that the active group \(\mathcal { A}\) does not have to act faithfully). The commutator of such a group, its minimal generating set and the centre of such products has been investigated here. The quotient group of the restricted and unrestricted wreath product by its commutator is found. The generic sets of commutator of wreath product were investigated. The structure of wreath product with non-faithful group action is investigated. Given a permutational wreath product sequence of cyclic groups, we investigate its minimal generating set, the minimal generating set for its commutator and some properties of its commutator subgroup. We strengthen the results from the author (Skuratovskii in Algebra, topology and analysis (summer school), pp 121–123, 2016; International scientific conference. Algebraic and geometric methods of analysis, 2018; The commutator subgroup of Sylow 2-subgroups of alternating group, commutator width of wreath product, arXiv:1903.08765; Minimal generating sets of cyclic groups wreath product (in russian), vol 118, 2018) and construct the minimal generating set for the wreath product of both finite and infinite cyclic groups, in addition to the direct product of such groups. The fundamental group of orbits of a Morse function \(f:M\rightarrow \mathbb {R}\) defined upon a Möbius band M with respect to the right action of the group of diffeomorphisms \(\mathcal {D}(M)\) has been investigated. In particular, we describe the precise algebraic structure of the group \(\pi _1 O(f)\). A minimal set of generators for the group of orbits of the functions \({{\pi }_{1}}({{O}_{f}},f)\) arising under the action of the diffeomorphisms group stabilising the function f and stabilising \(\partial M\) have been found. The Morse function f has critical sets with one saddle point. We consider a new class of wreath-cyclic geometrical groups. The minimal generating set for this group and for the commutator of the group are found.

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Acknowledgements

We are grateful to Antonenko Alexandr for a graphical support and Sergey Maksymenko for Morse function description, also we thanks to Samoilovych I.

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Correspondence to Ruslan V. Skuratovskii.

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Skuratovskii, R.V., Williams, A. Irreducible bases and subgroups of a wreath product in applying to diffeomorphism groups acting on the Möbius band. Rend. Circ. Mat. Palermo, II. Ser (2020). https://doi.org/10.1007/s12215-020-00514-5

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Keywords

  • Wreath product
  • Minimal generating set of commutator subgroup
  • Center of non regular wreath product
  • Quotient by commutator subgroup of wreath product
  • Semidirect product
  • Fundamental group of orbits of one Morse function

Mathematics Subject Classification

  • 20B27
  • 20E08
  • 20B22
  • 20B35
  • 20F65
  • 20B07