论文标题
对某些离散平面仿射组
Equivariant K-homology and K-theory for some discrete planar affine groups
论文作者
论文摘要
We consider the semi-direct products $G=\mathbb Z^2\rtimes GL_2(\mathbb Z), \mathbb Z^2\rtimes SL_2(\mathbb Z)$ and $\mathbb Z^2\rtimesΓ(2)$ (where $Γ(2)$ is the congruence subgroup of level 2).对于他们每个人，我们都计算了鲍姆 - 康涅狄格州的两面猜想，即分类空间的$ k $ - $ k $ - $ \ usepline $ \ usepline {e} g $，以在左侧进行适当的操作，以及降低的组$ c^*$ algebra的分析k理论。通过存在$ \ upsine {e} g $的3维模型，可以实现LHS的计算，该模型允许Bredon同源性替换Equivariant K-词素。我们要注意$ g $中扭转的存在，从而广泛研究了与有限亚组相关的墙纸组。对于第二组和第三组，$ K_0 $中的计算提供了由Baum-Connes Assembly地图匹配的明确发电机。
We consider the semi-direct products $G=\mathbb Z^2\rtimes GL_2(\mathbb Z), \mathbb Z^2\rtimes SL_2(\mathbb Z)$ and $\mathbb Z^2\rtimesΓ(2)$ (where $Γ(2)$ is the congruence subgroup of level 2). For each of them, we compute both sides of the Baum-Connes conjecture, namely the equivariant $K$-homology of the classifying space $\underline{E}G$ for proper actions on the left-hand side, and the analytical K-theory of the reduced group $C^*$-algebra on the right-hand side. The computation of the LHS is made possible by the existence of a 3-dimensional model for $\underline{E}G$, which allows to replace equivariant K-homology by Bredon homology. We pay due attention to the presence of torsion in $G$, leading to an extensive study of the wallpaper groups associated with finite subgroups. For the second and third groups, the computations in $K_0$ provide explicit generators that are matched by the Baum-Connes assembly map.
