Higher Teichmueller Theory
Final Report Abstract
Moduli spaces of flat bundles and representation varieties play a prominent role in various areas of mathematics. Historically such spaces first arose in the study of systems of analytic differential equations. Analytic continuation of their solutions led to monodromies and linear representations of the fundamental groups of the underlying manifolds, giving rise to the first examples of local systems. Closely related, and in fact locally homeomorphic, are deformation spaces of locally homogeneous geometric structures. Formally belonging to the realm of differential geometry and topology, the study of local systems and of deformation spaces of geometric structures heavily draws from Lie theory. Teichmüller space, a central object in several areas of mathematics, is the prototype of a space of local systems which is globally homeomorphic to a deformation space of locally homogeneous geometric structures. It is a smooth cover of the moduli space of Riemann surfaces of fixed topological type and parametrizes (marked) conformal structures on a surface Σ. Via the uniformization theorem of Riemann surfaces, it identifies with the space of (marked) hyperbolic structures, and in turn also with a connected component of the space of representations of the fundamental group of Σ into PSL(2, R) consisting of (equivalence classes of) discrete embeddings. Higher Teichmüller theory emerged in the last twenty years and has become an increasingly active subject, which combines techniques from discrete subgroups of Lie groups, cluster algebras, tropical geometry, bounded cohomology, gauge theory, Higgs bundles, Hitchin systems, W-algebras, positivity and causality in Lie groups, projective and Lorentzian geometry and opers. It also sparked interest from theoretical physics: through relations to Wn -algebras suggested by Witten, connections to supersymmetric gauge theories which became apparent in recent work of Gaiotto, Moore and Neitzke and applications to the geometric Langlands program. Higher Teichmüller theory generalizes the classical concepts associated to the Lie group PSL(2) to Lie groups of higher rank, such as PSL(n). It deals with connected components of spaces of representations of fundamental groups of surfaces into Lie groups G, which consist entirely of discrete u embeddings. So far two families of such higher Teichmüller spaces are known: Hitchin components, which are defined when G is a split real simple adjoint Lie group (e.g. PSL(n, R) or PSp(2n, R)), and maximal representations, which are defined when G is a simple Lie group of Hermitian type (e.g. SU(n, m) or Sp(2n, R)). In this project we proved several new and important results about geometric structures, which are associated and related to higher Teichmüller spaces and their deformation spaces. We highlight one result from each of the four focus questions of the project. (1) Geometric structures: We showed that many cusped hyperbolic three-manifold can be deformed to convex projective structures with totally geodesic torus boundary, and that such structures can be convexly glued together whenever the geometry at the boundary matches up. From this we deduce that many doubles of cusped hyperbolic three-manifolds admit convex projective structures. (2) Coordinates: We defined generalizations of shear-coordinates for maximal representations into symplectic groups, whose coordinate changes exhibit cluster-like features. In this ongoing research project we plan to further analyse these structures. (3) Length functions: We established a collar lemma for Hitchin representations. This was a quite surprising result, which already inspired research by other scientists. (4) Causal representations We further developed the theory of weakly maximal representation characterize in terms of causal structures.
Publications
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Anosov representations and proper actions. Geometry & Topology
Anna Wienhard, with F. Guéritaud, O. Guichard, and F. Kassel
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Collar lemma for Hitchin representations. Geometry & Topology
Gye-Seon Lee, with T. Zhang
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On order preserving representations. Journal of the London Mathematical Society
Anna Wienhard, with G. Ben Simon, M. Burger, T. Hartnick, and A. Iozzi
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Convex projective structures on non-hyperbolic threemanifolds
Gye-Seon Lee, S. Ballas and J. Danciger
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Projective deformations of weakly orderable hyperbolic Coxeter orbifolds, Geometry & Topology 19 (2015), pp. 1777–1828
Gye-Seon Lee, with S. Choi
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On weakly maximal representations of surface groups. Journal of Differential Geometry Volume 105, Number 3 (2017), 375-404
Anna Wienhard, with G. Ben Simon, M. Burger, T. Hartnick, and A. Iozzi
