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Register a new userLocal representations of the quantum Teichmuller space
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Authors
Hua Bai
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We introduce a certain type of representations for the quantum Teichmuller space of a punctured surface, which we call local representations. We show that, up to finitely many choices, these purely algebraic representations are classified by classical geometric data. We also investigate the family of intertwining operators associated to such a representations. In particular, we use these intertwiners to construct a natural fiber bundle over the Teichmuller space and its quotient under the action of the mapping class group. This construction also offers a convenient framework to exhibit invariants of surface diffeomorphisms.
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Kashaev algebra associated to a surface is a noncommutative deformation of the algebra of rational functions of Kashaev coordinates. For two arbitrary complex numbers, there is a generalized Kashaev algebra. The relationship between the shear coordinates and Kashaev coordinates induces a natural relationship between the quantum Teichmuller space and the generalized Kashaev algebra.
In this chapter, we survey the algebraic aspects of quantum Teichmuller space, generalized Kashaev algebra and a natural relationship between the two algebras.
Given a surface of infinite topological type, there are several Teichmuller spaces associated with it, depending on the basepoint and on the point of view that one uses to compare different complex structures. This paper is about the comparison between the quasiconformal Teichmuller space and the length-spectrum Teichmuller space. We work under this hypothesis that the basepoint is upper-bounded and admits short interior curves. There is a natural inclusion of the quasiconformal space in the length-spectrum space. We prove that, under the above hypothesis, the image of this inclusion is nowhere dense in the length-spectrum space. As a corollary we find an explicit description of the length-spectrum Teichmuller space in terms of Fenchel-Nielsen coordinates and we prove that the length-spectrum Teichmuller space is path-connected.
We prove that the every quasi-isometry of Teichmuller space equipped with the Teichmuller metric is a bounded distance from an isometry of Teichmuller space. That is, Teichmuller space is quasi-isometrically rigid.
Let X be quasi-isometric to either the mapping class group equipped with the word metric, or to Teichmuller space equipped with either the Teichmuller metric or the Weil-Petersson metric. We introduce a unified approach to study the coarse geometry of these spaces. We show that the quasi-Lipschitz image in X of a box in R^n is locally near a standard model of a flat in X. As a consequence, we show that, for all these spaces, the geometric rank and the topological rank are equal. The methods are axiomatic and apply to a larger class of metric spaces.
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