Tame Algebras Have Dense g-Vector Fans

Abstract

Abstract The $\textbf{g}$-vector fan of a finite-dimensional algebra is a fan whose rays are the $\textbf{g}$-vectors of its two-term presilting objects. We prove that the $\textbf{g}$-vector fan of a tame algebra is dense. We then apply this result to obtain a near classification of quivers for which the closure of the cluster $\textbf{g}$-vector fan is dense or is a half-space, using the additive categorification of cluster algebras by means of Jacobian algebras. As another application, we prove that for quivers with potentials arising from once-punctured closed surfaces, the stability and cluster scattering diagrams only differ by wall-crossing functions on the walls contained in a separating hyperplane. The appendix is devoted to the construction of truncated twist functors and their adjoints.