Abstract
We announce results on a compactification of general character varieties that has good topological properties and give various interpretations of its ideal points. We relate this to the Weyl chamber length compactification and apply our results to the theory of maximal and Hitchin representations.
Highlights
Let G ≤ SLn be a connected semisimple algebraic group defined over Q and let G := G(R) (or any subgroup of finite index in G(R))
The character variety Ξ(Γ,G) of a finitely generated group Γ into G is the quotient by G-conjugation of the topological space Homred(Γ,G) of reductive representations, where the latter is equipped with the topology of pointwise convergence
Our aim here is to give an overview of our results in [13,14,15]: we study a compactification of Ξ(Γ,G) with good topological properties giving various interpretations of its ideal points (Section 2) and we apply our study to the theory of maximal (Section 3) and Hitchin representations (Section 4)
Summary
Let G ≤ SLn be a connected semisimple algebraic group defined over Q and let G := G(R) (or any subgroup of finite index in G(R)). Points in the set of discontinuity for the M C G +(Σ)-action on the real spectrum boundary of a PSL(n, R)-Hitchin component or of a component of maximal representations give rise to harmonic conformal maps into appropriate buildings. This introduction is by all means not meant to be exhaustive, but only to titillate the curiosity of the readers and entice them to read this announcement
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