# Density of $g$-vector cones from triangulated surfaces

## TL;DR

This paper characterizes the union of $g$-vector cones in cluster algebras from surfaces, revealing its geometric structure and implications for the connectivity of the associated exchange graph.

## Contribution

It determines the closure of the union of $g$-vector cones for all clusters in surface-based cluster algebras, linking geometric and combinatorial properties.

## Key findings

- Union of $g$-vector cones covers $ ext{R}^n$ except for a punctured surface case.
- Connectedness of the exchange graph depends on the surface's puncture configuration.
- Explicit description of the hyperplane for the punctured surface case.

## 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.