The cluster modular group of the dimer model
Associated to a convex integral polygon N is a cluster integrable system XN constructed from the dimer model. We compute the group GN of symmetries of XN , called the (2-2) cluster modular group, showing that it is a certain abelian group conjectured by Fock and Marshakov. Combinatorially, non-torsion elements of GN are ways of shuffling the underlying bipartite graph, generalizing domino-shuffling. Algebro-geometrically, GN is a subgroup of the Picard group of a certain algebraic surface associated to N .