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Representation varieties and geometric structures in low dimensions

2–6 July 2018

Organisers: John Parker, Ser Peow Tan, Brian Bowditch

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Symposium Poster

Workshop Poster

Schedule, Titles and Abstracts

The Teichmüller space of a surface parameterises hyperbolic structures on that surface. Via the holonomy representation, one sees it as a component of the $PSL(2, \Bbb{R})$-representation variety; the space of all representations of the fundamental group of the surface, $H$, into the Lie group $PSL(2,\Bbb{R})$. More generally, for any group $H$ and Lie group $G$, the $G$-representation variety of $H$ is the space of representations of $H$ into $G$. The subspaces of discrete faithful representations are of central interest in geometric topology and geometric group theory as they often parameterise geometric structures on objects on low-dimensional or combinatorial objects (e.g.~graphs, surfaces, three-manifolds, cubical complexes). On the other hand, since these varieties are algebro-geometric in nature, representation varieties can be studied using many tools: algebraic, geometric, dynamical, computational, etc.

This workshop will bring together researchers working on these different facets of representation varieties. One focus will be the many connections between classical hyperbolic geometry and other geometric structures. This includes: (1) Strengthening analogies, for example, studying Anosov representations and stability properties via the analogy with convex cocompactness for Kleinian groups, and analysing representations of surface groups into $PU(2,1)$ via analogies with representations into $SO(3,1)$; and (2) Direct applications of classical geometry, for example, further developing the theory of discrete representations into $SO(n,1) \times SO(n,1)$ that act properly on $SO(n,1)$ via equivariant Lipschitz maps between hyperbolic space, and studying affine deformations of hyperbolic structures. This last topic also connects, via work of Danciger-Gueritaud-Kassel, to arc complexes of surfaces in striking ways that we will further pursue.


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See also:
Mathematics Research Centre
Mathematical Interdisciplinary Research at Warwick (MIR@W)
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