Local rigidity of lattices
Abstract
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The goal of this lectures is to give a proof of the local rigidity theorem for lattices of semisimple Lie groups. Let \(G\) be a simple Lie group not locally isomorphic to \(\textrm{SL}_2(\mathbb{R})\). Let \(\Gamma\) be a cocompact lattice in \(G\), i.e. a discrete subgroup such that \(G/\Gamma\) is compact. The theorem states that the inclusion of \(\Gamma\) in \(G\) is an isolated point of the character variety \(\textrm{Hom}(\Gamma,G)/G\). We will present a beautiful proof due to Andre Weil that you can find in the book "Discrete subgroups of Lie groups" of Raghunathan. During the proof, we will develop tools that allow to compute \(H^1(\Gamma,V)\) for \(V\) a linear representation of \(G\). The lecture will mainly be about flat vector bundles and bundle-valued differential forms, which we will cover from scratch.
KeywordsDifferential geometry, symmetric spaces, discrete subgroups of Lie groups
Prerequisites
We will only need basic facts from from Lie theory and differential geometry.
Remarks and notes
There will be notes on my website.
