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**Free groups of interval exchange transformations are rare.**
*(English)*
Zbl 1371.37075

Summary: We study the group IET of all interval exchange transformations. Our first main result is that the group generated by a generic pairs of elements of IET is not free (assuming a suitable irreducibility condition on the underlying permutation). Then we prove that any connected Lie group isomorphic to a subgroup of IET is abelian. Additionally, we show that IET contains no infinite Kazhdan group. We also prove residual finiteness of finitely presented subgroups of IET and give an example of a two-generated subgroup of IET of exponential growth that contains an isomorphic copy of every finite group and which is therefore not linear.

### MSC:

37E05 | Dynamical systems involving maps of the interval |

20E07 | Subgroup theorems; subgroup growth |

37A15 | General groups of measure-preserving transformations and dynamical systems |

20F38 | Other groups related to topology or analysis |

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\textit{F. Dahmani} et al., Groups Geom. Dyn. 7, No. 4, 883--910 (2013; Zbl 1371.37075)

### References:

[1] | J. W. Cannon, W. J. Floyd, and W. R. Parry, Introductory notes on Richard Thompson’s groups. Enseign. Math. (2) 42 (1996), 215-256. · Zbl 0880.20027 |

[2] | Y. Cornulier, A sofic group away from amenable groups. Math. Ann. 350 (2011), 269-275. · Zbl 1247.20039 |

[3] | C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras . AMS Chelsea Publishing, Providence, RI, 2006. · Zbl 1093.20003 |

[4] | M. Gromov, Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. (JEMS) 1 (1999), 109-197. · Zbl 0998.14001 |

[5] | A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical sys- tems . Encyclopedia Math. Appl. 54, Cambridge University Press, Cambridge 1995. · Zbl 0878.58020 |

[6] | M. Kuranishi, On everywhere dense imbedding of free groups in Lie groups. Nagoya Math. J. 2 (1951), 63-71. · Zbl 0045.31003 |

[7] | R. C. Lyndon and P. E. Schupp, Combinatorial group theory . Ergeb. Math. Grenzgeb. 89, Springer, Berlin 1977. · Zbl 0368.20023 |

[8] | D. Marker, Model theory . Graduate Texts in Math. 217, Springer, New York 2002. · Zbl 1003.03034 |

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