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Register a new userFock--Goncharov coordinates for semisimple Lie groups
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S. Gilles
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Fock and Goncharov introduced cluster ensembles, providing a framework for coordinates on varieties of surface representations into Lie groups, as well as a complete construction for groups of type $A_n$. Later, Zickert, Le, and Ip described, using differing methods, how to apply this framework for other Lie group types. Zickert also showed that this framework applies to triangulated $3$-manifolds. We present a complete, general construction, based on work of Fomin and Zelevinsky. In particular, we complete the picture for the remaining cases: Lie groups of types $F_4$, $E_6$, $E_7$, and $E_8$.
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For a finite-type surface $mathfrak{S}$, we study a preferred basis for the commutative algebra $mathbb{C}[mathcal{X}_{mathrm{SL}_3(mathbb{C})}(mathfrak{S})]$ of regular functions on the $mathrm{SL}_3(mathbb{C})$-character variety, introduced by Sikora-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface $mathfrak{S}$. We show that this basis can be naturally indexed by positive integer coordinates, defined by Knutson-Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock-Goncharov, to the tropical points at infinity of the dual version of the character variety.
In a companion paper (arXiv 2011.01768) we constructed non-negative integer coordinates $Phi_mathcal{T}$ for a distinguished collection $mathcal{W}_{3, widehat{S}}$ of $mathrm{SL}_3$-webs on a finite-type punctured surface $widehat{S}$, depending on an ideal triangulation $mathcal{T}$ of $widehat{S}$. We prove that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation $mathcal{T}^prime$ is chosen then the coordinate change map relating $Phi_mathcal{T}$ and $Phi_{mathcal{T}^prime}$ is a prescribed tropical cluster transformation. Moreover, when $widehat{S}=Box$ is an ideal square, we provide a topological geometric description of the Hilbert basis (in the sense of linear programming) of the non-negative integer cone $Phi_mathcal{T}(mathcal{W}_{3, Box}) subset mathbb{Z}_{geq 0}^{12}$, and we prove that this cone canonically decomposes into 42 sectors corresponding topologically to 42 families of $mathrm{SL}_3$-webs in the square.
We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.
Divergence functions of a metric space estimate the length of a path connecting two points $A$, $B$ at distance $le n$ avoiding a large enough ball around a third point $C$. We characterize groups with non-linear divergence functions as groups having cut-points in their asymptotic cones. By Olshanskii-Osin-Sapir, that property is weaker than the property of having Morse (rank 1) quasi-geodesics. Using our characterization of Morse quasi-geodesics, we give a new proof of the theorem of Farb-Kaimanovich-Masur that states that mapping class groups cannot contain copies of irreducible lattices in semi-simple Lie groups of higher ranks. It also gives a generalization of the result of Birman-Lubotzky-McCarthy about solvable subgroups of mapping class groups not covered by the Tits alternative of Ivanov and McCarthy. We show that any group acting acylindrically on a simplicial tree or a locally compact hyperbolic graph always has many periodic Morse quasi-geodesics (i.e. Morse elements), so its divergence functions are never linear. We also show that the same result holds in many cases when the hyperbolic graph satisfies Bowditchs properties that are weaker than local compactness. This gives a new proof of Behrstocks result that every pseudo-Anosov element in a mapping class group is Morse. On the other hand, we conjecture that lattices in semi-simple Lie groups of higher rank always have linear divergence. We prove it in the case when the $mathbb{Q}$-rank is 1 and when the lattice is $SL_n(mathcal{O}_S)$ where $nge 3$, $S$ is a finite set of valuations of a number field $K$ including all infinite valuations, and $mathcal{O}_S$ is the corresponding ring of $S$-integers.
These notes grew out of our learning and applying the methods of Fock and Goncharov concerning moduli spaces of real projective structures on surfaces with ideal triangulations. We give a self-contained treatment of Fock and Goncharovs description of the moduli space of framed marked properly convex projective structures with minimal or maximal ends, and deduce results of Marquis and Goldman as consequences. We also discuss the Poisson structure on moduli space and its relationship to Goldmans Poisson structure on the character variety.
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