##
**Solvable groups of interval exchange transformations.**
*(English.
French summary)*
Zbl 1458.37054

The authors denote by \(\mathrm{IET}\) the group of interval exchange transformations, namely the group of all
bijections of the interval \([0, 1)\) that are piecewise translations with finitely many discontinuity points.
The group \(\mathrm{IET}(\mathcal{D})\) of interval exchange
transformations on a domain \(\mathcal{D}\) consisting of a disjoint union of finitely
many oriented circles and oriented half-open intervals (closed on the left) is here used to describe
some elementary examples of subgroups of \(\mathrm{IET}.\)
In particular, the authors consider some solvable subgroups of \(\mathrm{IET}.\)
Some insights are provided in the papers [F. Dahmani et al., Groups Geom. Dyn. 7, No. 4, 883–910 (2013; Zbl 1371.37075); K. Juschenko and N. Monod, Ann. Math. (2) 178, No. 2, 775–787 (2013; Zbl 1283.37011)].

First, it is shown that if there exists a subgroup of finite index in a group \(G\) which embeds in \(\mathrm{IET},\) then \(G\) embeds in \(\mathrm{IET}\) too (see Proposition 1.2). This implies that every finitely generated virtually abelian group embeds in \(\mathrm{IET}.\) The next attempt is done for virtually polycyclic groups. For this, it is shown that every finitely generated torsion-free solvable subgroup of \(\mathrm{IET}\) is virtually abelian (see Theorem 3.1). As a consequence, it is deduced that any virtually polycyclic group embeds into \(\mathrm{IET}\) exactly if it is virtually abelian. For the torsion case, a much greater variety of subgroups is constructed. It is proved that for any finite abelian group \(A\) and any \(k \geq 1,\) the wreath product \(A \wr {\mathbb{Z}^k}\) embeds in \(\mathrm{IET}\) (see Propositions 4.1 and 4.2). Moreover, it is pointed out that the abelian condition for the group \(A\) is crucial (see Theorem 4.4). Finally, the authors consider possible finitely generated wreath products which may be embedded in \(\mathrm{IET}.\) This task, which is based on [K. Juschenko et al., Ergodic Theory Dyn. Syst. 38, No. 1, 195–219 (2018; Zbl 1387.37030)], provides a huge variety of isomorphism classes of solvable subgroups in \(\mathrm{IET}.\) Indeed, it is proved that there exist uncountably many isomorphism classes of subgroups of \(\mathrm{IET}\) that are generated by three elements, and that are solvable of derived length 3 (see Theorem 4.9).

First, it is shown that if there exists a subgroup of finite index in a group \(G\) which embeds in \(\mathrm{IET},\) then \(G\) embeds in \(\mathrm{IET}\) too (see Proposition 1.2). This implies that every finitely generated virtually abelian group embeds in \(\mathrm{IET}.\) The next attempt is done for virtually polycyclic groups. For this, it is shown that every finitely generated torsion-free solvable subgroup of \(\mathrm{IET}\) is virtually abelian (see Theorem 3.1). As a consequence, it is deduced that any virtually polycyclic group embeds into \(\mathrm{IET}\) exactly if it is virtually abelian. For the torsion case, a much greater variety of subgroups is constructed. It is proved that for any finite abelian group \(A\) and any \(k \geq 1,\) the wreath product \(A \wr {\mathbb{Z}^k}\) embeds in \(\mathrm{IET}\) (see Propositions 4.1 and 4.2). Moreover, it is pointed out that the abelian condition for the group \(A\) is crucial (see Theorem 4.4). Finally, the authors consider possible finitely generated wreath products which may be embedded in \(\mathrm{IET}.\) This task, which is based on [K. Juschenko et al., Ergodic Theory Dyn. Syst. 38, No. 1, 195–219 (2018; Zbl 1387.37030)], provides a huge variety of isomorphism classes of solvable subgroups in \(\mathrm{IET}.\) Indeed, it is proved that there exist uncountably many isomorphism classes of subgroups of \(\mathrm{IET}\) that are generated by three elements, and that are solvable of derived length 3 (see Theorem 4.9).

Reviewer: Alireza Najafizadeh (Tabriz)

### MSC:

37E05 | Dynamical systems involving maps of the interval |

37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |

20K15 | Torsion-free groups, finite rank |

20E05 | Free nonabelian groups |

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\textit{F. Dahmani} et al., Ann. Fac. Sci. Toulouse, Math. (6) 29, No. 3, 595--618 (2020; Zbl 1458.37054)

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