Extensive amenability and an application to interval exchanges

Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group IET of interval exchange transformations that have angular components of rational rank~${\leq 2}$. In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on IET and show that there are subgroups $G <IET$ admitting no finitely supported measure with trivial boundary.Comment: 28 page

Gabor Frames for Quasicrystals, KK-theory, and Twisted Gap Labeling

We study the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal $\Lambda,$ and the $K$-theory of the twisted groupoid $C^*$-algebra $\mathcal{A}_\sigma$ arising from a quasicrystal. In particular, we construct a finitely generated projective module \mathcal{H}_\L over $\mathcal{A}_\sigma$ related to time-frequency analysis, and any multiwindow Gabor frame for $\Lambda$ can be used to construct an idempotent in $M_N(\mathcal{A}_\sigma)$ representing \mathcal{H}_\L in $K_0(\mathcal{A}_\sigma).$ We show for lattice subsets in dimension two, this element corresponds to the Bott element in $K_0(\mathcal{A}_\sigma),$ allowing us to prove a twisted version of Bellissard's gap labeling theorem