Math 569: Representations of surface groups
University of Illinois at Chicago
||David Dumas (email@example.com)
Tue & Thu, 3:30–4:45pm in Taft Hall 300
Mon (Apr 10, 17, 24), 1:00pm in SEO 1227
Mon & Thu 2–3pm
||F. Labourie, Lectures on Representations of Surface Groups.|
European Mathematical Society, 2013. ISBN 978-3-03719-127-9
Purchase from AMS / Not available on Amazon / Not in UIC library
We will study the space of
representations of the fundamental group of a surface (i.e. a real
2-manifold) into a Lie group G. We will investigate the local and
global geometry of this space, including questions about
connectedness, smoothness, singularities, and complex and symplectic
structures. Following the main thread of the textbook, we will
discuss a remarkable formula that gives the symplectic volume of the
space of representations when G is compact.
After that, we will branch out
and consider some special classes of surface group representations
with nice geometric properties, such as representations in
SL(2,R) coming from hyperbolic structures, representations in
SL(2,C) associated to complex projective structures and to
bending deformations of surfaces in hyperbolic 3-space, and
representations in SL(3,R) associated to convex real projective
structures. The idea of studying these examples is to see how some
general phenomena play out in these concrete situations, and also to
discuss the unique features of each one.
As described in the syllabus, each student taking the course for credit must prepare a 30-minute final presentation on a topic to be approved by the instructor.
The presentations will be held in three sessions:
- Tue Apr 25, in class (Lecture 29), one presentation
- Thu Apr 27, in class (Lecture 30), two presentations
- Fri May 5, 1:00-3:00pm (final exam period), three presentations
The schedule of presenters was announced by email to the class on April 6.
Links and resources
Here we collect citations for further reading about course material and links to relevant online materials.
- General references related to course material:
- The classification of surfaces:
- Character/representation varieties:
- Fundamental group by open covers
- A. Hatcher, Algebraic Topology. Cambridge University Press, 2002.
- The nerve is defined in section 3.3, and the homotopy equivalence of a space with the nerve of a good cover is proved in section 4.G.
- J. Cannon, Geometric Group Theory. Chapter 6 in Handbook of Geometric Topology, Elsevier, 2001.
- A direct and elementary proof of the nerve description of π1(X) for an open cover with Ui and Ui ∪ Uj simply connected is given in section 2, pp260-271.
- Good covers of manifolds are usually constructed by taking an open cover by geodesically convex balls; existence of these is proved e.g. in:
- Do Carmo, Riemannian Geometry, Birkhäuser, 1992. Section 3.4.
- Spivak, A Comprehensive Introduction to Differential Geometry, Volume 1, Publish or Perish, 1999. Chapter 9, problem 32.
- Vector bundles
- Connections and curvature
- Construction of the moduli space
- Moduli spaces of polygons in Euclidean space
- M. Kapovich and J. Millson, On the moduli space of polygons in the Euclidean plane. J. Diff. Geom., 1995.
- Definitions of the moduli spaces, length-angle duality, hyperbolic cone structure, relation to stable measures on S1.
- M. Kapovich and J. Millson, The symplectic geometry of polygons in Euclidean space. J. Diff. Geom., 1996.
- Fixed-length moduli space of polygons in R3; link to moduli space of marked points on CP1. Construction of the moduli space as a symplectic reduction. Bending flows are Hamiltonian.
- S. Kojima, H. Nishi, and Y. Yamashita, Configuration spaces of points on the circle and hyperbolic Dehn fillings. Topology, 1999. (Also on the preprint arxiv.)
- Detailed discussion of topology and hyperbolic structure for the equilateral case. Deformations of the edge length vector give a locally injective map to the character variety for n=5 and n=6.
- S. Kojima, H. Nishi, and Y. Yamashita, Configuration Spaces of Points on the Circle and Hyperbolic Dehn Fillings, II. Geometriae Dedicata, 2002.
- Global injectivity of the map to the character variety arising from deformations of edge lengths for n=5 and n=6.
- T. Hinokuma and H. Shiga, Topology of the Configuration Space of Polygons as a Codimension One Submanifold of a Torus. Publ. Res. Inst. Math. Sci., 1998.
- Morse-theoretic methods are used to study the topology of the moduli space of equilateral polygons (and more generally, polygons with (n-1) equal side lengths). In particular the homology and the fundamental group are computed.
- F. Apéry and M. Yoshida, Pentagonal structure of the configuration space of five points in the real projective line. Kyushu J. Math., 1998.
- A beautiful interpretation of the genus-four surface that parameterizes equilateral pentagons in R2 as the great dodecahedron in R3.