Density of $g$-vector cones from triangulated surfaces

Abstract

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.

22 pages

Cited by in corpus (9)

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• Denominator vectors and dimension vectors from triangulated surfaces
• Denseness of $g$-vector cones from weighted orbifolds
• Stability scattering diagrams and quiver coverings
• $\tau$-tilting finiteness and $\mathbf{g}$-tameness: Incidence algebras of posets and concealed algebras