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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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