Free groups of interval exchange transformations are rare

TL;DR

This paper investigates the algebraic structure of the group of interval exchange transformations (IET), showing that generic pairs do not generate free groups, and that IET subgroups have restrictive properties such as abelianity and residual finiteness.

Contribution

It provides new insights into the algebraic properties of IET, including non-freeness of generic pairs, restrictions on Lie subgroups, and examples of complex subgroups with specific growth and finiteness properties.

Findings

Generic pairs in IET do not generate free groups

Any connected Lie subgroup of IET is abelian

IET contains no infinite Kazhdan groups

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.