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Singular locus of the character variety in P SL(p, C)
Clément Guérin
IRMA Institut de Recherche Mathématiques avancée, Université de Strasbourg, 7 rue René
Descartes, F-67084 Strasbourg , France, e-mail : guerin@math.unistra.fr
Abstract
Following the terminology of Sikora, if Γ is a finitely generated group and G is
a complex semi-simple Lie group, we say that a representation ρ : Γ → G is bad
when ρ is irreducible with a non-trivial centralizer (i.e. not reduced to the center
of G). The singular locus of the character variety χ(Γ, G) is defined as the subset
of conjugacy classes of bad representations. We will describe here the singular locus
when G := P SL(p, C) and p is a prime number. We shall see that when Γ is a free
group of rank ≥ 2 or the fundamental group of a Riemann surface of genus ≥ 2 then
the singular locus is connected. If time permits, we will also explain what happens
when G := P SL(n, C) and n is any integer.
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