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Register a new userAn invitation to higher Teichmuller theory
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Authors
Anna Wienhard
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The goal of this article is to invite the reader to get to know and to get involved into higher Teichmuller theory by describing some of its many facets.
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We derive generalizations of McShanes identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen which generalize the notion of horocycle lengths. In particular, we obtain McShane-type identities for finite-area cusped convex real projective surfaces by generalizing the Birman--Series geodesic scarcity theorem. More generally, we establish McShane-type identities for positive surface group representations with loxodromic boundary monodromy, as well as McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are systematically expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple and cross ratios. We apply our identities to derive the simple spectral discreteness of unipotent-bordered positive representations, collar lemmas, and generalizations of the Thurston metric.
We introduce $Theta$-positivity, a new notion of positivity in real semisimple Lie groups. The notion of $Theta$-positivity generalizes at the same time Lusztigs total positivity in split real Lie groups as well as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of Lie groups, SO(p,q) for p<q, and a family of exceptional Lie groups, which admit a $Theta$-positive structure. We describe key aspects of $Theta$-positivity and make a connection with representations of surface groups and higher Teichmuller theory.
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.
We prove that the Teichmuller space of surfaces with given boundary lengths equipped with the arc metric (resp. the Teichmuller metric) is almost isometric to the Teichmuller space of punctured surfaces equipped with the Thurston metric (resp. the Teichmuller metric).
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.
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