Abstract
We give the centralizers of irreducible representations from a finitely generated group \(\varGamma \) to \(PSL(p,\mathbb {C})\) where p is a prime number. This leads to a description of the singular locus (the set of conjugacy classes of representations whose centralizers strictly contain the center of the ambient group) of the irreducible part of the character variety \(\chi ^i(\varGamma ,PSL(p,\mathbb {C}))\). When \(\varGamma \) is a free group of rank \(l\ge 2\) or the fundamental group of a closed Riemann surface of genus \(g\ge 2\), 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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Acknowledgements
The author wants to thank his former Ph.D. advisor Olivier Guichard for his support during the writing of this paper. He also wants to thank an anonymous referee for his valuable advice.
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Guérin, C. Bad irreducible subgroups and singular locus for character varieties in \(PSL(p,\mathbb {C})\) . Geom Dedicata 195, 23–55 (2018). https://doi.org/10.1007/s10711-017-0275-4
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DOI: https://doi.org/10.1007/s10711-017-0275-4
Keywords
- Representation variety
- Character variety
- Irreducible representations
- Centralizer of irreducible representations
- Orbifolds
- Fuchsian groups representations