Bad irreducible subgroups and singular locus for character varieties in $$PSL(p,\mathbb {C})$$ P S L ( p , C )

Abstract

We give the centralizers of irreducible representations from a finitely generated group Γ to P SL(p, C) where p is a prime number. This leads to a description of the singular locus (the set of conjugacy classes of representations whose centralizer strictly contains the center of the ambient group) of the irreducible part of the character variety χ i (Γ, P SL(p, C)). When Γ is a free group of rank l ≥ 2 or the fundamental group of a closed Riemann surface of genus g ≥ 2, we give a complete description of this locus and prove that this locus is exactly the set of algebraic singularities of the irreducible part of the character variety.

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