Log-canonical coordinates on the symplectic groupoid of unipotent upper-triangular matrices and Teichmueller space of closed genus two surfaces

We described a system of log-canonical coordinates for the symplectic groupoid of unipotent upper-triangular matrices equipped with the natural Poisson structure. As a byproduct, we discovered a system of log-canonical coordinates and a corresponding cluster structure for the Teichmuelller space of closed genus two curves. In particular, we obtain a cluster description of the mapping class group action. This is a joint work with L. Chekhov.