Symplectic groupoid and cluster structure on the Teichmueller space of closed genus two surfaces

Symplectic groupoid of unipotent nxn upper-triangular matrices is formed by pairs (B, A) where B is a nondegenerate nxn matrix, A is a unipotent upper-triangular nxn matrix, and BAB^t is unipotent upper triangular. The symplectic groupoid is equipped with the natural symplectic form defined by Weinstein, which induces a Poisson bracket on the space of upper triangular unipotent matrices studied by Bondal, Dubrovin-Ugaglia, and others. We compute the cluster structure compatible with the Poisson structure and discuss its connection with the Teichmueller space of genus g curves with one or two holes equipped with a Goldman Poisson bracket. As an unexpected byproduct, we obtain a cluster structure on the Teichmueller space of closed genus two curves unknown earlier. This is a joint project with L. Chekhov.