Earthquake theorem for cluster algebras of finite type · arXivDesk
2206.15226Jun 30, 202238 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
Earthquake theorem for cluster algebras of finite type
We introduce a cluster algebraic generalization of Thurston's earthquake map for the cluster algebras of finite type, which we call the 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 Rtrop- and R>0-valued points of the cluster X
Nearby in the stack
-variety. For those of type
An
and
Dn
, 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.