Acyclic cluster algebras with dense $g$-vector fans

The $g$-vector fans play an important role in studying cluster algebras and $\tau$-tilting theory. We survey cluster algebras with dense $g$-vector fans. In particular, we show that a connected acyclic cluster algebra has a dense $g$-vector fan if and only if it is either finite type or affine type. As an application, we classify finite dimensional hereditary algebras with dense $g$-vector fans.