Geometric leaf of symplectic groupoid

Abstract

We consider the symplectic groupoid of pairs $(B,A)$ with $A$ real unipotent upper-triangular matrix and $B\in G{L}_n$ being such that $\tilde{A}=BA{B}^T$ is also a unipotent upper-triangular matrix. Fock and Chekhov defined a Poisson map of Teichmüller space ${\mathcal T_{g,s}$ of genus $g$ surfaces with $s$ holes into the space of unipotent upper-triangular $n\times n$ matrices whose image forms the \emph{geometric locus}. The elements of geometric locus satisfy \emph{rank condition}. We describe the Hamiltonian reduction of the Poisson cluster variety of symplectic groupoid by the rank condition for $n=5$ and $6$. In both cases, we analyze the induced cluster structures on the results of Hamiltonian reduction and recover celebrated cluster structure on ${\mathcal{T}}_{2,1}$ for $n=5$ and ${\mathcal{T}}_{2,2}$ for $n=6$.

many misprints and references added, following reviewers comments several explanations added