Related papers: Laminations from the symplectic double

Cluster varieties are geometric objects that have recently found applications in several areas of mathematics and mathematical physics. This thesis studies the geometry of a large class of cluster varieties associated to compact oriented…

We survey explicit coordinate descriptions for two (A and X) versions of Teichmuller and lamination spaces for open 2D surfaces, and extend them to the more general set-up of surfaces with distinguished collections of points on the…

One can realize higher laminations as positive configurations of points in the affine building. The duality pairings of Fock and Goncharov give pairings between higher laminations for two Langlands dual groups $G$ and $G^{\vee}$. These…

We introduce rational bounded $\mathfrak{sp}_4$-laminations on a marked surface $\boldsymbol{\Sigma}$ as a proposed topological model for the rational tropical points $\mathcal{A}_{Sp_4,\boldsymbol{\Sigma}}(\mathbb{Q}^{\mathsf{T}})$ of the…

We describe the spaces of the positive and tropical points of the moduli space $\mathcal{P}_{PGL_2,\Sigma}$ introduced by Goncharov--Shen [GS19] as certain Teichm\"uller and lamination spaces, respectively, with additional data of pinnings.…

In [Mor], we have introduced a notion of flat laminations on surfaces endowed with a flat structure, similar to geodesic laminations on hyperbolic surfaces. Here is a sequel to this article that aims at defining transversal measures on flat…

Let G be a split semi-simple algebraic group over Q. Let S be a decorated surface, that is a topological oriented surface with a finite set of marked points on the boundary, considered modulo isotopy. We introduce a moduli space D(G,S) and…

Positive configurations of points in the affine building were introduced in \cite{Le} as the basic object needed to define higher laminations. We start by giving a self-contained, elementary definition of positive configurations of points…

Generalizing the work of Fock--Goncharov on rational unbounded laminations, we give a geometric model of the tropical points of the cluster variety $\mathcal{X}_{\mathfrak{sl}_3,\Sigma}$, which we call unbounded…

Let $\mathbb F$ be a field of characteristic not $2$, and let $(A,B)$ be a pair of $n\times n$ matrices over $\mathbb F$, in which $A$ is symmetric and $B$ is skew-symmetric. A canonical form of $(A,B)$ with respect to congruence…

Let M be a compact oriented even-dimensional manifold. This note constructs a compact symplectic manifold S of the same dimension and a map f from S to M of strictly positive degree. The construction relies on two deep results: the first is…

We introduce combinatorial objects which are parameterized by the positive part of the tropical Grassmannian $Gr(k,n)$. Our method is to relate the Grassmannian to configuration spaces of flags. By work of the first author, and of Goncharov…

We introduce coordinates on the spaces of framed and decorated representations of the fundamental group of a surface with nonempty boundary into the symplectic group $Sp(2n,\mathbf R)$. These coordinates provide a noncommutative…

We propose a construction of a double quasi-Poisson bracket on the group algebra associated to the twisted fundamental group of a marked oriented surface $(S,P)$ with boundary, where $P$ is a finite set of marked points on the boundary of…

A decorated surface S is a surface with a finite set of special points on the boundary, considered modulo isotopy. Let G be a split reductive group. A pair (G, S) gives rise to a moduli space A(G, S), closely related to the space of G-local…

To investigate the degree $d$ connectedness locus, Thur\-ston studied \emph{$\sigma_d$-invariant laminations}, where $\sigma_d$ is the $d$-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials…

We prove a conjecture of Fock and Goncharov which provides a birational equivalence of a cluster variety called the cluster symplectic double and a certain moduli space of local systems associated to a surface.

Given any rational map $f$, there is a lamination by Riemann surfaces associated to $f$. Such laminations were constructed in general by Lyubich and Minsky. In this paper, we classify laminations associated to quadratic polynomials with…

In this paper we partially settle Fock-Goncharov's duality conjecture for cluster varieties associated to their moduli spaces of ${\rm G}$-local systems on a punctured surface $\frak{S}$ with boundary data, when ${\rm G}$ is a group of type…

This work uncovers the tropical analogue for measured laminations of the convex hull construction of decorated Teichmueller theory, namely, it is a study in coordinates of geometric degeneration to a point of Thurston's boundary for…