关于$\widetilde A_2$群的边界$C^*$ -代数的K理论
On the K-theory of boundary $C^*$-algebras of $\widetilde A_2$ groups
摘要：
设$Γ$是$\PGL_3(\mathbb K)$的一个$\widetilde A_2$子群，其中$\mathbb K$是具有$q$阶剩余域的局部域.证明了共不变模$C(\mathbb P^2_{\mathbb K},\mathbb Z)_Γ$是有限的，其中$\mathbb P^2_{\mathbb K}$是$\mathbb K$上的投影平面.若群$Γ$是Tits型群，且若$q \not\equiv 1 \pmod {3}$，则（全）交叉积$C^*$ -代数$C(Ω)\rtimesΓ$的K理论中类$[I]_{K_0}$的阶的精确值被确定，其中$Ω$是$\PGL_3(\mathbb K)$的Furstenberg边界.对于Tits型群，这证明了G. Robertson和T.斯蒂格
英文摘要：
Let $Γ$ be an $\widetilde A_2$ subgroup of $\PGL_3(\mathbb K)$, where $\mathbb K$ is a local field with residue field of order $q$. The module of coinvariants $C(\mathbb P^2_{\mathbb K},\mathbb Z)_Γ$ is shown to be finite, where $\mathbb P^2_{\mathbb K}$ is the projective plane over $\mathbb K$. If the group $Γ$ is of Tits type and if $q \not\equiv 1 \pmod {3}$ then the exact value of the order of the class $[I]_{K_0}$ in the K-theory of the (full) crossed product $C^*$-algebra $C(Ω)\rtimesΓ$ is determined, where $Ω$ is the Furstenberg boundary of $\PGL_3(\mathbb K)$. For groups of Tits type, this verifies a conjecture of G. Robertson and T. Steger.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
