Geometric Leaves of Symplectic Groupoids

A symplectic groupoid of triangular bilinear forms consists of pairs of matrices $(B,A)$ with $A\in \mathcal A$ - the set of real unipotent upper-triangular matrices - and $B\in {\rm SL}_n({\mathbb R})$ such that $\widetilde{A}=BAB^{\mathsf T}\in\mathcal A$. The symplectic groupoid possesses a nondegenerate symplectic structure which induces a reflection-equation Poisson algebra on $\mathcal A$. Fock and Chekhov defined a Poisson map from the Teichmüller space ${\mathcal T}_{g,s}$ of genus $g$ surfaces with $s$ (one or two) holes to the space of unipotent upper-triangular $n\times n$ matrices; the image of this map forms a geometric locus. A matrix $A$ in the geometric locus satisfies the rank condition: $\operatorname{rank}\bigl(A+A^{\mathsf T}\bigr)\le 4$ and an additional positivity condition. We describe the coisotropic reduction of the Poisson cluster variety of $\mathcal A$ by the rank condition for $n=5$ and $6$ and show that these coisotropic reductions yield the corresponding geometric loci. Moreover, we analyze in both cases the cluster structures obtained by coisotropic reductions and recover the celebrated cluster structure of ${\mathcal T}_{2,1}$ for $n=5$ and ${\mathcal T}_{2,2}$ for $n=6$. We also discuss the positivity of the geometric cluster variables for $n=3,4,5,6$.