On diagonal actions of free group on the Cantor set

Abstract

We study diagonal actions $\phi :{\mathbb{F}}_2\curvearrowright \partial {\mathbb{F}}_2\times K$ on the Cantor set which are given by ${\phi }_a={\partial }_a\times \alpha ,{\phi }_b={\partial }_b\times \beta$. Under some restrictions on $\alpha ,\beta$ we compute $K_{\ast }(C(\partial {\mathbb{F}}_2\times K){⋊}_r{\mathbb{F}}_2)$. As an application in the case of $\alpha$ is Denjoy homeomorphism of the Cantor set and $\beta =id$ we will show that $C(\partial {\mathbb{F}}_2\times K){⋊}_r{\mathbb{F}}_2$ is Kirchberg algebra with $K_{\ast }(C(\partial {\mathbb{F}}_2\times K){⋊}_r{\mathbb{F}}_2)=({\mathbb{Z}}^{\infty },{\mathbb{Z}}^{\infty })$. Also we will check that $C^{\ast }$-crossed product by Denjoy homeomorphism on the Cantor set is $C^{\ast }$-algebra generated by weighted shift, namely $C(K)⋊\mathbb{Z}\cong C^{\ast }(T_x)$ where $x\in \{1,2{\}}^{\mathbb{Z}}$ is two-sided Fibonacci sequence.

13 pages

References in corpus (1)

• Amenable minimal Cantor systems of free groups arising from diagonal actions