Geometric leaf of symplectic groupoid

TL;DR

This paper explores the geometric structure of a symplectic groupoid related to unipotent upper-triangular matrices, analyzing Hamiltonian reductions and cluster structures for specific cases connected to Teichmüller spaces.

Contribution

It describes the Hamiltonian reduction of the symplectic groupoid for n=5 and 6, revealing cluster structures on related Teichmüller spaces.

Findings

Hamiltonian reduction for n=5 and 6 cases.

Induced cluster structures on reduced spaces.

Recovery of known cluster structures on Teichmüller spaces.

Abstract

We consider the symplectic groupoid of pairs $(B, A)$ with $A$ real unipotent upper-triangular matrix and $B\in GL_n$ being such that $\tilde A=BAB^T$ is also a unipotent upper-triangular matrix. Fock and Chekhov defined a Poisson map of Teichm\"uller 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 $n=6$.