Questions tagged [character-varieties]

moduli of representations (Betti moduli), moduli of Higgs bundles (Dolbeault moduli), moduli of connections (de Rham moduli space), moduli of principal bundles (Čech moduli space), moduli of polygons, moduli of geometric structures, spin networks, skein theory, A-polynomial, higher Teichmüller theory

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15 votes
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Can the SL_2 character variety of a three-manifold be nonreduced?

Let $M^3$ be a three-manifold and consider the representation variety and the character variety of $M$: $$Y=\operatorname{Hom}(\pi_1(M^3),\operatorname{SL}(2,\mathbb C))$$ $$X=\operatorname{Hom}(\pi_1(...
John Pardon's user avatar
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31 votes
4 answers
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Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?

Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are ...
Daniel Moskovich's user avatar
12 votes
3 answers
3k views

Zariski tangent spaces to representation varieties

In Bill Goldman's paper "The Symplectic Nature of the Fundamental Groups of Surfaces" (Advances, 54, 200-225, '84) it is stated that the "Zariski tangent space" to a representation space Hom$(\pi, G)/...
Dan Ramras's user avatar
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10 votes
1 answer
841 views

What is (explicitly) known about the SL(n,C) character varieties of 3-manifolds?

The $SL(2,{\bf C})$ character variety of a 3-manifold with 1-cusp $M$ (like a knot complement in the 3-sphere) essentially coincide with the variety defined by the A-polynomial. Those polynomials are ...
Gaetan Borot's user avatar