Groups, Geometry, and Dynamics

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Volume 7, Issue 4, 2013, pp. 883–910
DOI: 10.4171/GGD/209

Published online: 2013-11-20

Free groups of interval exchange transformations are rare

François Dahmani[1], Koji Fujiwara[2] and Vincent Guirardel[3]

(1) Université de Grenoble I, Saint-Martin-D'hères, France
(2) Kyoto University, Japan
(3) Université de Rennes 1, Rennes, France

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.

Keywords: Groups of diffeomorphisms, interval exchange transformation, Lie groups, free groups

Dahmani François, Fujiwara Koji, Guirardel Vincent: Free groups of interval exchange transformations are rare. Groups Geom. Dyn. 7 (2013), 883-910. doi: 10.4171/GGD/209