How nice are representation varieties of Fuchsian groups? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T03:30:17Z http://mathoverflow.net/feeds/question/66301 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66301/how-nice-are-representation-varieties-of-fuchsian-groups How nice are representation varieties of Fuchsian groups? Greg Muller 2011-05-28T17:40:21Z 2012-03-18T09:58:56Z <h2>Background</h2> <p>Let $S_{g,n}$ be an oriented surface of genus $g$, with $n$ punctures. We explicitly prohibit the <em>non-hyperbolic</em> cases:</p> <ul> <li>$g=0$, $n=0,1,2$.</li> <li>$g=1$, $n=0$.</li> </ul> <p>Let $\Gamma$ be the fundamental group of $S_{g,n}$; a group arising this way is called a <strong>Fuchsian group</strong> (as opposed to some authors, we don't require that $\Gamma$ comes with an embedding into $PSL_2(\mathbb{R})$).</p> <p>Let $G$ be a complete reductive algebraic group over a field $k$. The <strong>representation algebra</strong> $Rep(\Gamma,G)$ is defined such that $k$-algebra maps $$Rep(\Gamma,G)\rightarrow R$$ are naturally equivalent to group maps $$\Gamma\rightarrow G(R)$$ The <strong>representation variety</strong> is then $X_{\Gamma,G}:=Spec(Rep(\Gamma,G))$, despite the fact that this scheme can be non-reduced and hence not really a 'variety'.</p> <h2>Question</h2> <p>For arbitrary groups $\Gamma$, the scheme $X_{\Gamma,G}$ can be quite bad. It is non-reduced $G=PSL_2(\mathbb{C})$ and for $\Gamma$ some Artinian groups (Kapovich-Millson, 1999) or for $\Gamma$ the fundamental group of some 3-manifolds (Kapovich, 2001).</p> <p>For $\Gamma=\mathbb{Z}^2=\pi_{1,0}$ (one of the excluded cases), the representation variety $X_{\mathbb{Z}^2,G}$ is the <em>commuting scheme</em> of $G$. The reducedness of the commuting scheme is still an open question.</p> <p>Despite this, it seems like it might be possible there are general theorems about nice properties of $X_{\Gamma,G}$, when $\Gamma$ is Fuchsian. For example, if $g=0$, then $\pi_{0,n}=F_{n-1}$, the free group on $n-1$ generators. Then $X_{F_{n-1},G}=G^{n-1}$, which is a smooth variety.</p> <p>For $\Gamma$ Fuchsian, is it known whether $X_{\Gamma,G}$ is </p> <ul> <li>reduced?</li> <li>normal?</li> <li>smooth?</li> </ul> <p>I have a vested interest in their normality, but that seems like the question least likely to be addressed directly.</p> http://mathoverflow.net/questions/66301/how-nice-are-representation-varieties-of-fuchsian-groups/91494#91494 Answer by Misha for How nice are representation varieties of Fuchsian groups? Misha 2012-03-17T21:23:21Z 2012-03-18T09:58:56Z <p>Answers to all questions are negative, however, $Hom(\Gamma, G)$ is smooth away from representations $\rho$ such that the centralizer of $\rho(\Gamma)$ in $G$ is finite. (This is due to Andre Weil, "Remarks on cohomology of groups", Annals of Math., 1964, but since then it was reproven by many others in a variety of ways. For instance, Goldman has a very clear proof using Poincare duality and vanishing of $H^2$. However, if you allow Fuchsian groups with torsion instead of just surface groups, Weil's result is the best. Incidentally, there is a way to have an interesting theory of representations of fundamental groups of surfaces with boundary: You just have to consider <em>relative</em> representation varieties where you fix the conjugacy classes of images of boundary loops. Weil deals with these as well.)</p> <p>i. There are (real) non-reduced examples: $G=PU(2,1)$ and $\rho$ is a discrete and faithful representation that lands in $PU(1,1)$, see e.g. paper of Goldman and Millson <a href="http://www2.math.umd.edu/~wmg//LocalRigidity.pdf" rel="nofollow">http://www2.math.umd.edu/~wmg//LocalRigidity.pdf</a></p> <p>The key reason is that the dimension of Zariski tangent space to $Hom(\Gamma, G)$ at all these representations is larger than the actual dimension of the representation variety. On the other hand, over algebraically closed fields it will be reduced. </p> <p>ii. Even if you look at the reduced scheme, both smoothness and normality will fail at the trivial representation to a nonabelian group, like $SL(2)$. The analytical germ of $Hom(\Gamma, G)$ at the trivial representation is given by: Take the vector space $Z^1(\Gamma, {\mathfrak g})$ (with trivial action of $\Gamma$ on the Lie algebra) and impose the quadratic equations given by vanishing of the cup-products $[\omega \cup \omega], \omega\in Z^1$. See<br> Goldman's paper "Representations of fundamental groups of surfaces", reference [5] in the paper by Goldman and Millson linked above. There is actually a much stronger result by Goldman-Millson and Simpson which describes local singularities of representation varieties of Kahler groups in terms of cup product on $H^1$. </p> <p>However, it is an open problem if $Hom(\Gamma, G)$ is smooth at representations with finite centralizers, where $\Gamma$ is a Kahler group. </p>