  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.

Representation Theory;Combinatorics;Rings and Algebras