University of California at Santa Barbara 
Math 260P: Representations of Surface Groups
Spring 2014
Course Details
Location: South Hall 4607
Time: Tue & Thu 2-3:15p
The students in the class will be blogging the notes from each lecture. The notes can be found on the 260P blog.
Welcome to the course website for Math 260P. This course will focus on the interplay of topology/geometry of surfaces and representation theory of the fundamental groups of surfaces into a variety of target groups. An example of this interplay is found in Stalling's simple topological proof of a, rather difficult, theorem of Marshall Hall Jr. about the nature of finitely generated subgroups of a free group. Henry Wilton has a very nice blog post summarizing these ideas that I highly recommend reading.
Here is a rough list of topics to be covered in the course:
- Residual finiteness and subgroup separability: A group is residually finite if its non-identity elements can be distinguished from the identity in some finite quotient. Subgroup separability is a related notion where we now try to find finite index subgroups that separate elements from entire subgroups. When the groups in question are fundamental groups of (sufficiently nice) topological spaces, residual finiteness and subgroup separability affect what finite covers of the space "look-like." In the case where the topological space is a surface we will show that the fundamental group has very strong subgroup separability properties.
- Hyperbolic structures and Teichmuller theory: When $S$ is a surface with negative Euler characteristic then $S$ admits a complete hyperbolic structure. From such a structure one gets a (conjugacy class of) representation $\rho:\pi_1(S)\to \text{PSL}_2(\mathbb{R})$. Typically $S$ will admit uncountably many different hyperbolic structures each giving rise to a different conjugacy class of representation into $\text{PSL}_2(\mathbb{R})$. The moduli space of these structures (Teichmuller space) is parameterized by a component the character variety (roughly speaking this is the space of conjugacy classes of representations from $\pi_1(S)$ into $\text{SL}_2(\mathbb{R})$). We will discuss this correspondence and see how properties of the representation give us information about the hyperbolic geometry on the surface and vice-versa.
- Convex projective structures: Hyperbolic structures are a specific instance of convex projective structures. Roughly speaking a convex projective structure on a surface is a realization of that surface as the quotient of a "nice" domain in $\mathbb{RP}^2$ by a discrete subgroup of $\text{PGL}_3(\mathbb{R})$. In the case of hyperbolic structures this nice domain is an ellipse in $\mathbb{RP}^2$. Again these structures are parameterized by a certain component of a character variety and properties of the representations manifest themselves as geometric properties of the structure.
- Higher Teichmuller theory: We have seen a few instances of conjugacy classes of "nice representations" corresponding to geometric structures on surfaces. One aspect of higher Teichmuller theory is to understand what sort of "geometric structures" are these representations trying to parameterize for various choices of target Lie group. We will discuss aspects of this problem with a focus on the $\text{SL}_4(\mathbb{R})$ case.
Prerequisites for this course should be pretty minimal, but include basic algebraic topology (i.e. fundamental groups, covering spaces, etc.), linear algebra, and a bit of abstract algebra (i.e. homomorphisms and modules). To summarize, graduate students who are interested in geometry and/or topology as well as advanced undergraduates are encouraged to enroll. If you have any questions about the course feel free to email me.
The will be an active participation component for those involved in the course. The students in the class may also be asked to present material on certain topics in lecture. This will be arranged in advance and I can assist you with your preparation of material to be presented. We will discuss the logistics of this on the first day.
Office Hours
South Hall
6702
Tue & Wed 9-10:30a