Dual Teichmuller and lamination spaces

By V. Fock, A. Goncharov. 2005

164 citations recorded by the source corpus.

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 boundary. Main features, such as mapping class group action, Poisson and symplectic structures and others, are described in these terms. The lamination spaces are interpreted as the tropical limits of the Teichmuller ones. Canonical pairings between lamination and Teichmuller spaces are constructed. The paper could serve as an introduction to higher Teichmuller theory developed by the authors in math.AG/0311149, math.AG/0311245.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7eee3841a53aca8a4839d95b650a9d904d5f09c8

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In Geometry and Topology

Citations in the corpus

On cluster algebras arising from unpunctured surfaces II (2008, methodology): “Inspired by the works of Fock and Goncharov [19,18,20] and Gekhtman, Shapiro and Vainshtein [27,28] which discovered cluster structures in the context of Teichmüller theory, Fomin, Shapiro and Thurston [21] and [22] initiated a systematic study of the cluster algebras arising from oriented Riemann s”
Quivers with potentials associated to triangulated surfaces (2008, background): “After the foundational papers [10], [11] and [16] on the one hand, and [7] on the other, both surface cluster algebras and quivers with potentials have attracted the attention of several authors (e.”
Framed BPS states (2010, unknown): “We note that the defi- nition we have given is very closely related to the universal Laurent polynomials introduced by Fock and Goncharov in their study of cluster ensembles [17].”
Geometry of canonical bases and mirror symmetry (2013, background): “my of an SL2-local system Lover a loop αon S. The value of the function Ml on Lis given by Ml(L) := Y i Tr(Monni αi(L)). 2Laminations on decorated surfaces were investigated in [FG1], Section 12, and [FG3]. However the two types of laminations considered there, the A- and X-laminations, are diﬀerent”
Wall-crossing in coupled 2d-4d systems (2011, methodology): “There is a well-known algorithm [66, 65, 67, 68], reviewed in our context in [2, 3], for decomposing the trace of the parallel transport around a loop ℘ as a linear combination of the Yγ .”
Matrix formulae and skein relations for cluster algebras from surfaces (2011, methodology): “Comparing our formulas to those of Fock and Goncharov [FG3], we observe the following.”
Introduction to Cluster Algebras. Chapters 1-3 (2016, unknown): “, [17] and references therein); • recursively defined sequences of points on elliptic curves, and associated Somos-like sequences, cf.”
Branes and Quantization (2008, methodology): “by another method based on real polarizations [ 41 ].”
Total positivity and cluster algebras (2010, background): “The nascent “higher Teichmüller theory” [7, 17] is one prominent expression of this paradigm.”
Bases for cluster algebras from surfaces (2011, unknown): “Thurston’s theory of measured laminations [Th1] gives a concrete way to think about coefficients, as described in [FT, Sections 11-12] (see also [FG3]).”