most recent 30 from http://mathoverflow.net 2013-05-18T21:08:51Z http://mathoverflow.net/feeds/user/8196 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109967/algorithm-to-test-for-discrete-or-quasi-fuchsian-subgroups-of-psl2-c David Dumas 2012-10-18T02:29:28Z 2012-10-18T16:37:26Z

<p>Let $\Gamma = \pi_1(S)$ denote the fundamental group of a compact surface $S$ of genus $g>1$.</p> <p>Given a representation $\rho : \Gamma \to \mathrm{PSL}(2,\mathbb{C})$, specified by matrix representatives for the images of a fixed generating set, is there an algorithm to answer either of the following questions?</p> <p>1) Is the image $\rho(\Gamma)$ a quasi-Fuchsian group?</p> <p>2) Is the image $\rho(\Gamma)$ discrete?</p> <p>There are a number of related situations where I am aware of algorithms of this type, but all are limited to special classes of two-generator subgroups of $\mathrm{PSL}(2,\mathbb{C})$. For example:</p> <ul> <li><p>For two-generator subgroups of $\mathrm{PSL}(2,\mathbb{R})$ there is the Gilman-Maskit algorithm which tests for discreteness. There are some related sufficient conditions for discreteness in the two-generator case in $\mathrm{PSL}(2,\mathbb{C})$.</p></li> <li><p>For punctured torus groups (i.e. representations of $\mathbb{F}_2$ where $abab^{-1}$ maps to a parabolic element) in $\mathrm{PSL}(2,\mathbb{C})$ there is a method of Komori, Sugawa, Wada, and Yamashita based on simultaneously testing Jorgensen's inequality (attempting to find a certificate that the group is not discrete) while also trying to find a Ford fundamental domain (of a type that would give a certificate that the group is quasi-Fuchsian).</p></li> <li><p>Also in the punctured torus case, Bowditch has a conjectural characterization of quasi-Fuchsian groups in terms of a certain subset of the infinite trivalent tree of "generating triples", which is easy to test algorithmically. As in the previous case this can be combined with Jorgensen's inequality to get a heuristic test for discreteness.</p></li> </ul> <p>Based on these cases I would especially like to know about methods for discreteness or quasi-Fuchsian testing that apply in the compact surface case without assuming that the representation maps into $\mathrm{PSL}(2,\mathbb{R})$.</p>

http://mathoverflow.net/questions/57308/families-of-fuchsian-models/57383#57383 David Dumas 2011-03-04T18:07:28Z 2011-03-04T18:07:28Z

<p>I think the first question is not well-formed. The Bers construction gives a holomorphically varying family of <strong>quasi</strong>-Fuchsian groups $G(t)$ (in $PSL_2(\mathbb{C})$) each of which has a simply-connected invariant domain $U(t)$ whose quotient is biholomorphic to the Riemann surface represented by $t$. Both the groups and the domains are varying, as is necessary to obtain a holomorphic family.</p> <p>For the second question, do you care about the actual complex structure (i.e. charts mapping open subsets of $\mathcal{T}_g$ into $\mathbb{C}^n$) or just the almost complex structure (multiplication by $i$ on a tangent space)?</p> <p>The almost complex structure is easy to see in most models for Teichmuller space, because the tangent space at a point is typically represented by a complex vector space. For example, $T_X \mathcal{T}_g$ can be seen as the linear dual of $Q(X)$, the vector space of holomorphic quadratic differentials on the Riemann surface $X$. Multiplication of differentials by the constant function $i$ induces the almost complex structure.</p>

http://mathoverflow.net/questions/34640/local-vs-infinitesimal-rigidity/34685#34685 David Dumas 2010-08-05T20:43:21Z 2010-08-05T20:43:21Z

<p>Local rigidity means that the structure in question is an isolated point in its deformation space (which is typically an algebraic set). Infinitesimal rigidity means that there are no first-order deformations of the structure in question. A first-order deformation is a nonzero element of a certain cohomology group.</p> <p>Because you can take the derivative of a path of structures and get a first-order deformation, infinitesimal rigidity implies local rigidity.</p> <p>Because a first-order deformation may or may not correspond to an actual path (due to higher-order obstructions), local rigidity does NOT necessarily imply infinitesimal rigidity.</p>

http://mathoverflow.net/questions/109967/algorithm-to-test-for-discrete-or-quasi-fuchsian-subgroups-of-psl2-c/109974#109974 David Dumas 2012-10-18T13:53:52Z 2012-10-18T13:53:52Z

Real computation is what I had in mind. Certainly I would want to allow computing words in the generators of $\rho(\Gamma)$ and comparisons between traces or matrix entries as &quot;basic operations&quot;. Thanks for your answers, which convince me there is no hope for an algorithm in general. As in the punctured torus case, when actually implementing such a test I will need to settle for heuristics that leave a thin set in the character variety &quot;undecided&quot;.

http://mathoverflow.net/questions/109967/algorithm-to-test-for-discrete-or-quasi-fuchsian-subgroups-of-psl2-c David Dumas 2012-10-18T13:26:52Z 2012-10-18T13:26:52Z

I edited the last sentence to clarify that I <i>am</i> interested in the closed surface case.