Related papers: Ratio coordinates for higher Teichm\"uller spaces
In this paper, we compute the center of the balanced Fock-Goncharov algebra and determine its rank over the center when the quantum parameter is a root of unity. These results have potential applications to the study of the center and rank…
In this chapter, we survey the algebraic aspects of quantum Teichm\"uller space, generalized Kashaev algebra and a natural relationship between the two algebras.
In this article, we derive estimates of Teichm\"uller modular forms, and associated invariants. Let $\mathcal{M}_{g}$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\geq 2$, and let $\overline{M}_{g}$ be the…
The paper presents some recent results on the BMO Teichm\"uller space, its subspaces and quotient spaces. We first consider the chord-arc curve subspace and prove that every element of the BMO Teichm\"uller space is represented by its…
An almost Fuchsian 3-manifold is a quasi-Fuchsian manifold which contains an incompressible closed minimal surface with principal curvatures in the range of $(-1,1)$. Such a 3-manifold $M$ admits a foliation of parallel surfaces, whose…
We consider the Grothendieck--Teichm\"uller group under a new aspect. Using real algebraic geometry and web theory we show that it preserves dihedral symmetry relations, present in the fundamental groupoids of configuration spaces of marked…
We establish an isomorphism between the Grothendieck-Teichm\"uller Lie algebra $\mathfrak{grt}_1$ in depth two modulo higher depth and the cohomology of the two-loop part of the graph complex of internally connected graphs…
In his 1944 paper Ver\"anderliche Riemannsche Fl\"achen , Teichm\"uller defined a structure of complex manifold on the set of isomorphism classes of marked closed Riemann surfaces of genus g. The complex manifold he obtained is the space…
We describe a ring whose category of Cohen-Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. More precisely, there is a…
Let P(S) be the space of convex projective structures on a surface S with negative Euler characteristic. Goldman and Bonahon-Dreyer constructed two different sets of global coordinates for P(S), both associated to a pair of pants…
We study threefolds $Y$ fibred by $A_m$-surfaces over a curve $S$ of positive genus. An ideal triangulation of $S$ defines, for each rank $m$, a quiver $Q(\Delta_m)$, hence a $CY_3$-category $(C,W)$ for any potential $W$ on $Q(\Delta_m)$.…
We derive the quantum Teichm\"uller space, previously constructed by Kashaev and by Fock and Chekhov, from tensor products of a single canonical representation of the modular double of the quantum plane. We show that the quantum dilogarithm…
Given a natural number m and a Lie algebra g, the m th generalized Takiff Lie algebra of g is the Lie algebra gm\,:= g $\otimes$ C[T ]/T m+1 . For n $\ge$ m, we define the (m, n)-modality of an adjoint orbit $\Omega$m in gm to be the…
Given any Koszul algebra of finite global dimension one can define a new algebra, which we call a higher zigzag algebra, as a twisted trivial extension of the Koszul dual of our original algebra. If our original algebra is the path algebra…
Guided by physical needs, we deal with the rotationally isotropic Poincar\'e ball, when considering the complement of Borromean rings embedded in it. We consistently describe the geometry of the complement and realize the fundamental group…
We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the "hyperbolic directions" in that space. This boundary is a quasi-isometry invariant and thus produces…
We introduce a new class of $\mathfrak{sl}_2$-triples in a complex simple Lie algebra $\mathfrak{g}$, which we call magical. Such an $\mathfrak{sl}_2$-triple canonically defines a real form and various decompositions of $\mathfrak{g}$.…
Let S be a surface of genus g with n points removed, G a connected Lie group, and X(G) the moduli space of representations of the fundamental group of S into G. We compute the fundamental group of X(G) when n>0 and G is a real or complex…
For a representation of a connected compact Lie group G in a finite dimensional real vector space U and a subspace V of U, invariant under a maximal torus of G, we obtain a sufficient condition for V to meet all G-orbits in U, which is also…
We prove a quantitative estimate with a power saving error term for the number of points in a mapping class group orbit of Teichm\"uller space that lie within a Teichm\"uller metric ball of given center and large radius. Estimates of the…