Let $M^3$ be a three-manifold and consider the representation variety and the character variety of $M$: $$Y=\operatorname{Hom}(\pi_1(M^3),\operatorname{SL}(2,\mathbb C))$$ $$X=\operatorname{Hom}(\pi_1(M^3),\operatorname{SL}(2,\mathbb C))//\operatorname{SL}(2,\mathbb C)$$ I would like to think of these as schemes. They are usually singular; in fact the trivial representation is almost always a singular point.

Are there any known examples where $Y$ (or $X$) are nonreduced as schemes?

I really want to know the answer for $\pi_1(M^3)$; examples with just any finitely presented group would be less interesting.

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    $\begingroup$ Coincidentally I was recently reading a paper of A. Sikora, where he mentions that $sl_2$ character varieties aren't always reduced as schemes. See section 12 in arxiv.org/abs/0902.2589 However, for the 3-manifold group case, he cites work of M.Kapovich that doesn't seem to be in print. $\endgroup$ – Gjergji Zaimi Feb 28 '13 at 7:39
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    $\begingroup$ But now you know who to ask :) $\endgroup$ – Gjergji Zaimi Feb 28 '13 at 7:41
  • $\begingroup$ Since the tag 3-manifolds exists I changed your (new) three-manifolds to it. In fact I slightly prefer the word-version you used, but then to split the tag seems certainly not good. $\endgroup$ – user9072 Feb 28 '13 at 11:10

There is actually an old (ca 1986) example of nonreduced $SL(2, {\mathbb C})$-representation scheme of a 3-manifold group. Take an oriented Seifert manifold $M$ which fibers over the $S^2(3,3,3)$ orbifold (sphere with three cone points of order 3). The fundamental group of the base-orbifold is von Dyck group with presentation $$ \Gamma=\langle a, b, c | a^3, b^3, c^3, abc\rangle. $$ It is an old observation of Lubotzky and Magid (in their book "Representation varieties of finitely-generated groups") that the $SL(2, {\mathbb C})$-representation scheme of $\Gamma$ is nonreduced at a representation $\rho_0$ whose image is isomorphic to ${\mathbb Z}_3$ ($\rho_0$ sends all generators to an order $3$ element). Namely, $H^1(\Gamma, Ad \rho_0)$ is $1$-dimensional, while the representation $\rho_0$ s locally rigid. There is a nice geometric explanation of this phenomenon: Take a spherical equilateral triangle in $S^3=SU(2)$ contained in a great circle. Then this triangle is locally rigid but admits a nontrivial 1st order deformation in $S^3$. Now, $\rho_0$ lifts to an $SL(2, {\mathbb C})$-representation $\tilde\rho_0$ of the central extension $\pi=\pi_1(M)$ of the group $\Gamma$ (killing the center of $\pi$). This is your example. It is locally rigid but has 1-st order nontrivial infinitesimal deformations. A drawback of this example is that the image has large centralizer.

Below is a more difficult "universality" result:

Theorem 1. Let $X$ be an affine scheme over ${\mathbb Q}$ and $x\in X$ be a rational point. Then there exists an open subscheme $X'\subset X$ containing $x$, a natural number $N$, a closed 3-dimensional manifold $M$ with fundamental group $\pi$, a unitary representation $\rho: \pi\to SU(2)\subset SL(2, {\mathbb C})$ (whose image is dense in $SU(2)$) and an open subscheme $$ R'\subset Hom(\pi, SL(2, {\mathbb C})) $$ containing $\rho$, so that $R'$ admits a regular etale covering over $X'\times SL(2, {\mathbb C})^N$ (with abelian group of deck transformations) and the covering sends $\rho$ to $x$. In particular, the centralizer of $\rho(\pi)$ is $\{\pm 1\}$.

For instance, the analytic germ of the character scheme $$ Hom(\pi, SL(2, {\mathbb C}))//SL(2, {\mathbb C})$$ at $[\rho]$ could be isomorphic to the germ at $0$ of the scheme $$ \{x^{100}=0\} \times {\mathbb C}^{k} $$ (for some $k$). In particular, the character scheme of 3-manifold groups could be nonreduced at points of Zariski density.

Proof of Theorem 1 could be now found here. The proof is a combination of my old work with Millson (see here) with the recent theorem of Panov and Petrunin, which deserves to be better known:

Theorem 2. For every finitely presented group $\Gamma$ there exists a closed 3-dimensional orbifold $O$ so that the fundamental group of the underlying space of $O$ is isomorphic to $\Gamma$.

It is a difficult open problem if Theorem 1 holds for 3-manifolds which are homology spheres.

  • $\begingroup$ It seems like the manifold $M$ is fairly complicated. Do you know if such behavior can happen for a knot complement in $S^3$? In particular, do you know if the $SL_2(\mathbb C)$-representation scheme of a knot complement can be non-reduced? $\endgroup$ – Peter Samuelson May 30 '13 at 4:51
  • $\begingroup$ @Peter: I am fairly sure, the same could be done for knots (or, at least, links) as well, but, at the moment, it is an open problem. $\endgroup$ – Misha May 30 '13 at 10:57

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