松井 宏樹

<p lang="fr">&lt;p style='text-indent:20px;'&gt;When a pair of étale groupoids &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathcal{G} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \mathcal{G}' $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; on totally disconnected spaces are related in some way, we discuss the difference of their homology groups. More specifically, we treat two basic situations. In the subgroupoid situation, &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \mathcal{G}' $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is assumed to be an open regular subgroupoid of &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \mathcal{G} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. In the factor groupoid situation, we assume that &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \mathcal{G}' $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is a quotient of &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ \mathcal{G} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and the factor map &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ \mathcal{G}\to\mathcal{G}' $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is proper and regular. For each, we show that there exists a long exact sequence of homology groups. We present examples which arise from SFT groupoids and hyperplane groupoids.&lt;/p&gt;</p>

<title>Abstract</title> Toward the complete classification of poly-${\mathbb{Z } }$ group actions on Kirchberg algebras, we prove several fundamental theorems that are used in the classification. In addition, as an application of them, we classify outer actions of poly-${\mathbb{Z } }$ groups of Hirsch length not greater than three on unital Kirchberg algebras up to $KK$-trivial cocycle conjugacy.

In this paper, we will study representations of the continuous full group Gamma(A) of a one-sided topological Markov shift (X-A, sigma(A)) for an irreducible matrix A with entries in {0, 1} as a generalization of Higman-Thompson groups V-N, 1 &lt; N is an element of N. We will show that the group Gamma(A) can be represented as a group Gamma(tab)(A) A of matrices, called A-adic tables, with entries in admissible words of the shift space X-A, and a group Gamma(PL)(A) A of right continuous piecewise linear functions, called A-adic PL functions, on [0, 1] with finite singularities.