Interval exchanges that do not embed in free groups

TL;DR

This paper proves that disjoint rotation maps, a specific type of interval exchange transformation, cannot be part of any subgroup of the group of all IETs that is isomorphic to a non-abelian free group, shedding light on the structure of IET groups.

Contribution

It establishes that disjoint rotation maps do not embed in free groups within the IET group, advancing understanding of the algebraic structure of interval exchange transformations.

Findings

Disjoint rotation maps cannot occur in free subgroups of IETs.

The structure of IET groups excludes certain types of subgroups.

Provides insight into the algebraic properties of interval exchange transformations.

Abstract

A disjoint rotation map is an interval exchange transformation (IET) on the unit interval that acts by rotation on a finite number of invariant subintervals. It is currently unknown whether the group E of all IETs possesses any non-abelian free subgroups. It is shown that it is not possible for a disjoint rotation map to occur in a subgroup of E that is isomorphic to a non-abelian free group.