Earthquake theorem for cluster algebras of finite type

Abstract

We introduce a cluster algebraic generalization of Thurston's earthquake map for the cluster algebras of finite type, which we call the \emph{cluster earthquake map}. It is defined by gluing exponential maps, which is modeled after the earthquakes along ideal arcs. We prove an analogue of the earthquake theorem, which states that the cluster earthquake map gives a homeomorphism between the spaces of ${\mathbb{R}}^{\mathrm{trop}}$- and ${\mathbb{R}}_{>0}$-valued points of the cluster $\mathcal{X}$-variety. For those of type $A_n$ and $D_n$, the cluster earthquake map indeed recovers the earthquake maps for marked disks and once-punctured marked disks, respectively. Moreover, we investigate certain asymptotic behaviors of the cluster earthquake map, which give rise to "continuous deformations" of the Fock--Goncharov fan.

38 pages, 9 figures. We modified the topological condition for marked surfaces in Appendix A and the definition of the earthquake map in Appendix B from the journal version