Department of Mathematics,
University of California San Diego
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Math 211A: Seminar in Algebra
Dr. Hugo Jenkins
UC San Diego
Relative \({\rm SL}_d\)-representation varieties of a surface
Abstract:
Let \(\Sigma\) be a genus \(g\) surface with \(n\) punctures. We will define a variety that parametrizes \({\rm SL}_d\)-representations of \(\Sigma\) in which the loops around the punctures have fixed characteristic polynomial. We will discuss two properties, geometric irreducibility and smoothness. The proof of the former uses a method due to Liebeck-Shalev involving characters of the finite group \({\rm SL}_d(\mathbb{F}_q)\) and the Lang-Weil theorem from algebraic geometry. The proof of the second applies linear algebra to the differentials of the commutator and characteristic polynomial maps. Time permitting, we will define the action of the pure mapping class group of \(\Sigma\) on our variety and indicate how our two results are used in studying the orbits.
Host: Alireza Golsefidy
November 10, 2025
3:00 PM
APM 7321
Research Areas
Algebra Algebraic Geometry Representation Theory****************************
