We study $g$-vector cones associated with clusters of cluster algebras defined from a marked surface $(S,M)$ of rank $n$. We determine the closure of the union of $g$-vector cones associated with all clusters. It is equal to $\mathbb{R}^n$ except for a closed surface with exactly one puncture, in which case it is equal to the half space of a certain explicit hyperplane in $\mathbb{R}^n$. Our main ingredients are laminations on $(S,M)$, their shear coordinates and their asymptotic behavior under Dehn twists. As an application, if $(S,M)$ is not a closed surface with exactly one puncture, the exchange graph of cluster tilting objects in the corresponding cluster category is connected. If $(S,M)$ is a closed surface with exactly one puncture, it has precisely two connected components.

中文翻译：

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来自三角化表面的 g 向量锥体的密度

我们研究了与从秩为 $n$ 的标记表面 $(S,M)$ 定义的簇代数簇相关的 $g$-vector 锥。我们确定与所有集群相关联的 $g$-vector 锥体的并集的闭包。它等于 $\mathbb{R}^n$，除了一个封闭曲面只有一个穿孔，在这种情况下，它等于 $\mathbb{R}^n$ 中某个显式超平面的一半空间。我们的主要成分是 $(S,M)$ 上的叠层、它们的剪切坐标以及它们在 Dehn 扭曲下的渐近行为。作为一个应用，如果$(S,M)$不是一个恰好有一个穿孔的闭合曲面，则连接相应簇类别中的簇倾斜对象的交换图。如果 $(S,M)$ 是一个恰好有一个穿孔的封闭曲面，则它恰好有两个连通分量。