CAT（0）空间上群上模的极限集--从欧氏到双曲
Limit sets for modules over groups on CAT(0) spaces -- from the Euclidean to the hyperbolic
摘要：
Bieri-Neumann-Strebel-Renz的0维几何不变量$Σ^{0}(G;A)$可以被解释为一个准球面极限集，这一观察打开了一条从莫比乌斯变换的离散群$Γ$的庞加莱极限集$Λ(Γ)$（其中包含$Γ$的准球面极限集）到热带几何根（当G是阿贝尔时与$Σ^{0}(G;A)$密切相关）的直接线索。我们通过引入G-模A的次球极限集$Σ(M;A)$来探索这一线索，当G通过等距作用在一个真CAT（0）度量空间M上时。这是M在无穷远处的边界的子集。在此过程中，我们遇到了这样的例子：$Σ(M;A)$是所有圆锥极限点的集合，球形建筑的补集，热带簇的径向投影的补集，或者（通过Bieri-Neumann-Strebel不变量）它与Thurston范数密切相关。
英文摘要：
The observation that the 0-dimensional Geometric Invariant $Σ^{0}(G;A)$ of Bieri-Neumann-Strebel-Renz can be interpreted as a horospherical limit set opens a direct trail from Poincaré's limit set $Λ(Γ)$ of a discrete group $Γ$ of Möbius transformations (which contains the horospherical limit set of $Γ$) to the roots of tropical geometry (closely related to $Σ^{0}(G;A)$ when G is abelian). We explore this trail by introducing the horospherical limit set, $Σ(M;A)$, of a G-module A when G acts by isometries on a proper CAT(0) metric space M. This is a subset of the boundary at infinity of M. On the way we meet instances where $Σ(M;A)$ is the set of all conical limit points, the complement of a spherical building, the complement of the radial projection of a tropical variety, or (via the Bieri-Neumann-Strebel invariant) where it is closely related to the Thurston norm.
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