JournalsggdVol. 7, No. 4pp. 883–910

Free groups of interval exchange transformations are rare

  • François Dahmani

    Université de Grenoble I, Saint-Martin-D'hères, France
  • Koji Fujiwara

    Kyoto University, Japan
  • Vincent Guirardel

    Université de Rennes 1, Rennes, France
Free groups of interval exchange transformations are rare cover
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Abstract

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.

Cite this article

François Dahmani, Koji Fujiwara, Vincent Guirardel, Free groups of interval exchange transformations are rare. Groups Geom. Dyn. 7 (2013), no. 4, pp. 883–910

DOI 10.4171/GGD/209