Do representations of Fuchsian groups have unitary deformations? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T14:40:27Z http://mathoverflow.net/feeds/question/74791 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74791/do-representations-of-fuchsian-groups-have-unitary-deformations Do representations of Fuchsian groups have unitary deformations? anton 2011-09-07T20:28:22Z 2011-09-07T20:41:01Z <p>Let $G$ be $SL_2({\mathbb C})$ and for $a,b\in G$ let $[a,b]=aba^{-1}b^{-1}$ be the commutator bracket. Let $n$ be a natural number $\ge 2$ and let $X\subset G^{2n}$ be the set of all $g\in G^{2n}$ such that $$[g_1,g_2]\cdots[g_{2n-1},g_{2n}]=1.$$ The first question is, whether $X$ is connected. If not, can one give a list of the connected components? Finally, does the subset $SU(2)^{2n}\cap X$ meet every connected component?</p> <p>If the last question has an affirmative answer, every $G$-valued representation of the fundamental group $\Gamma$ of a compact Riemann surface of genus $n$ can be deformed to a unitary one, which explains the title of my question. </p> http://mathoverflow.net/questions/74791/do-representations-of-fuchsian-groups-have-unitary-deformations/74793#74793 Answer by Richard Kent for Do representations of Fuchsian groups have unitary deformations? Richard Kent 2011-09-07T20:35:27Z 2011-09-07T20:41:01Z <p>$X$ is the $SL_2(\mathbb{C})$--representation variety of the surface group, and, by Goldman's thesis, it is irreducible, and so connected.</p> <p>See </p> <p>Goldman, Topological components of spaces of representations. Invent. Math. 93 (1988), no. 3, 557–607.</p> <p>If you take $G$ to be $PSL_2(\mathbb{C})$, then there are two components (see also Goldman), one for each Stiefel-Whitney class.</p> <p>Edit: I should say that I recall that this is perhaps not so easy to find in Goldman's paper as he doesn't state it explicitly, but at some point he proves that the smooth locus of $X$ is connected, which gives the result.</p>