Distortion in groups of affine interval exchange transformations

Abstract

In this paper, we study distortion in the group A of Affine Interval Exchange Transformations (AIET). We prove that any distorted element f of A, has an iterate f k that is conjugate by an element of A to a product of infinite order restricted rotations, with pairwise disjoint supports. As consequences we prove that no Baumslag-Solitar group, BS(m, n) with |m| = |n|, acts faithfully by elements of A; every finitely generated nilpotent group of A is virtually abelian and there is no distortion element in A Q , the subgroup of A consisting of rational AIETs.

Group Actions on 1-Manifolds: A List of Very Concrete Open Questions

“…In the 1-dimensional setting, Avila proved that irrational rotations are distorted (in a very strong way) in the group of C ∞ diffeomorphism [5]. However, despite some partial progress [59], the following question is open.…”

Distortion in the group of circle homeomorphisms

“…The problem of the existence of distorted elements in some groups of homeomorphisms has been intensively studied for many years (see [2,4,5,7,9,10]). Substantial progress has been achieved for groups of diffeomorphisms of manifolds.…”

“…So far it has not even been known whether there exist distorted elements in PAff + (S 1 ). Now from [7] it follows that distorted elements, if they exist, are conjugate to rotations.…”

Property FW, differentiable structures and smoothability of singular actions

“…Namely, we will exhibit an isometric action of PL + (S 1 ) on the Hilbert space 2 (S 1 ), with linear part defined by the action of the group on S 1 : we twist the linear action with a cocycle that measures the failure of elements to be affine. This cocycle has been widely used (implicitly or explicitly) for understanding many properties of groups of PL homeomorphisms [20,21,30,34] (just to cite a few). This is to be compared to the result by Farley and Hughes [16,25] that Thompson's group V has the Haagerup property, which is also proved by exhibiting an explicit proper action of V on a Hilbert space (Farley's proof of this result in [16] had a gap and has been fixed by Hughes [25]).…”