User kenso - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:25:23Z http://mathoverflow.net/feeds/user/24057 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100212/discreteness-of-a-group-of-hyperbolic-isometries Discreteness of a group of hyperbolic isometries KENSO 2012-06-21T07:15:08Z 2012-06-21T21:10:23Z <p>Referring to <a href="http://mathoverflow.net/questions/100130/a-question-about-hyperbolic-double-torus" rel="nofollow">http://mathoverflow.net/questions/100130/a-question-about-hyperbolic-double-torus</a>, there is non-discrete $\Gamma= \left&lt; a,b,c,d~~|~~[a,b][c,d] \right> \subset PSL_{2}(\mathbb{R})$ where $a,b,c,d$ are hyperbolic elements. I wondered what the sufficient and necessary conditions of the representations of $a,b,c,d$ for the discreteness of $\Gamma$. This may be related to some concepts of Teichmuller space...</p> http://mathoverflow.net/questions/100130/a-question-about-hyperbolic-double-torus A question about hyperbolic double torus KENSO 2012-06-20T13:03:07Z 2012-06-20T17:18:39Z <p>Hi</p> <p>I have a question about hyperbolic 2-torus, from now on donoted by $\Sigma_{2}$</p> <p>Actually I've tried to prove that for a group $\Gamma \subset \textrm{Isom}^{+}(\mathbb{H}^{2})$ represented by $\Gamma = \left&lt; a,b,c,d~~|~~[a,b][c,d]=1 \right>$, $a$,$b$,$c$, and $d$ are hyperbolic element in $\textrm{Isom}^{+}(\mathbb{H}^{2})$(which fix 2 points only on $\partial \mathbb{H}^{2}$.) if and only if $\mathbb{H}^{2}/\Gamma =\Sigma_{2}$. "If"-part is easy so I mainly consider "only if"-part.</p> <p>It can be verified that one could make $\Sigma_{2}$ by pairing sides of a hyperbolic regular octagon which has angle $\frac{\pi}{4}$. Let the pairing isometries be $\alpha$, $\beta$, $\gamma$ and $\delta$$\in \textrm{Isom}^{+}(\mathbb{H}^{2})$. Then they generate a holonomy group, or deck transformation group of $\mathbb{H}^{2}$ with the relation $[\alpha,\beta][\gamma,\delta]=1$, where $[A,B]=ABA^{-1}B^{-1}$. Note that $\alpha$, $\beta$, $\gamma$ and $\delta$ should be hyperbolic elements.</p> <p>Now the question is following :</p> <p>For $\Gamma$, is there ALWAYS an 'corresponding' octagon(not necessarily regular) which is a fundamental domain for $\Sigma_{2}$?? I mean the side-pairing of the octagon induces $\Sigma_{2}$ and the paring maps are exactly $a$,$b$,$c$, and $d$. </p> <p>Or, is it possible that the octagon is weird(?), for example, some of its edges intersect each other not at a vertex but at an middle of an edge? (in this case, the boundary of the octagon has self intersection.)</p> <p>(I might have heard that sometimes the 'weird case' occurs but I'm not sure.)</p> <p>Thanks in advance.</p> http://mathoverflow.net/questions/98561/embedding-again Embedding Again KENSO 2012-06-01T10:26:39Z 2012-06-01T14:48:43Z <p>Let $S=[(x,y)\in\mathbb{H}^{2}:0&lt; x&lt; 2\pi]$ where $\mathbb{H}^{2}$ is a hyperplane with standard metric. I.e., a strip whose boundary components are geodesics, both approaching a common infinite point.</p> <p>My simple question is about whether it can be isometrically embedded in $\mathbb{R}^{3}$.</p> <p>I don't remember exactly but I read some articles about an isometric embedding from a portion of $\mathbb{H}^{2}$ into $\mathbb{R}^{3}$. For example, it might have been proved that infinite polygons of some type can be isometrically embedded but I could not catch the meaning of the type the author said about. Or, I read in another article that every(?) equidistant strip in $\mathbb{H}^{2}$ can be embedded in $\mathbb{R}^{3}$. If this is true, the answer to my question would be positive.</p> <p>Is there anyone who know about this content precisely?</p> <p>How about $T=[(x,y)\in\mathbb{H}^{2}:0&lt;\sqrt{{x}^{2}+{y}^{2}}&lt;{e}^{2\pi}]$? This strip also has geodesic boundary components but both components are approaching different infinite point.</p> http://mathoverflow.net/questions/98184/a-question-about-embedding-hyperbolic-space-onto-pseudosphere A question about embedding hyperbolic space onto pseudosphere KENSO 2012-05-28T10:00:39Z 2012-05-30T04:06:43Z <p>I have a difficulty with hyperbolic geometry. </p> <p>Let $\mathbb{H}^{2}$ be a 2-dimensional hyperbolic plane.</p> <p>(i.e., upper half plane in $\mathbb{R}^{2}$ with a metric $\frac{ds}{y}$)</p> <p>(or, upper half plane in $\mathbb{C}$ with a metric $\frac{|dz|}{\textrm{Im}(z)}$ ) </p> <p>You may have heard about pseudosphere in $\mathbb{R}^{3}$. Let's denote the half-pseudosphere by $P$</p> <p>This can be obtained by glueing both side($x=0$ and $x=2\pi$ parts) of </p> <p>$\left[(x,y)\in\mathbb{H}^{2} : 0\leq x\leq 2\pi, y>1 \right]$</p> <p>Denoting this quotient space by $A$(Note that $A$ is homeomorphic to a cylinder), we can now get a "globally isometrically embedding" map $\rho:A \rightarrow \mathbb{R}^{3}$ with $\rho(A)=P$</p> <p>To be specific, $\rho(x,y) = (t-\tanh(t), \textrm{sech}(t)\ \cos(x), \textrm{sech}(t)\sin(x))$ where $t=\textrm{arccosh}(y)$</p> <p>Now the question I have is following </p> <p>: Is there another possible (global and isometric)embedding $\rho$ from $A$ into $\mathbb{R}^{3}$ ? </p> <p>Actually I'm interested in $\rho(A)$ and by calculating, one can easily find that if $\rho(A)$ is a "surface of revolution", then it should be $P$(up to isometry of $\mathbb{R}^3$"</p> <p>Thus I'm looking for $\rho(A)$ which is different from a surface of revolution.</p> <p>Any idea?</p> http://mathoverflow.net/questions/100130/a-question-about-hyperbolic-double-torus Comment by KENSO KENSO 2012-06-21T05:57:23Z 2012-06-21T05:57:23Z @Misha: Can you explain a bit more about the relation of torsion-freeness and discreteness of $\Gamma$? http://mathoverflow.net/questions/100130/a-question-about-hyperbolic-double-torus/100137#100137 Comment by KENSO KENSO 2012-06-21T05:39:49Z 2012-06-21T05:39:49Z @Misha: I see. Can you give me URL address for your paper mentioned above? http://mathoverflow.net/questions/100130/a-question-about-hyperbolic-double-torus/100137#100137 Comment by KENSO KENSO 2012-06-20T19:29:13Z 2012-06-20T19:29:13Z Sorry for additional questions. What is the definition of nonelementary representation?? http://mathoverflow.net/questions/100130/a-question-about-hyperbolic-double-torus/100137#100137 Comment by KENSO KENSO 2012-06-20T19:01:42Z 2012-06-20T19:01:42Z @Misha and Lee: I've got your points.......... ;_; http://mathoverflow.net/questions/100130/a-question-about-hyperbolic-double-torus Comment by KENSO KENSO 2012-06-20T18:42:57Z 2012-06-20T18:42:57Z To sum, my question is whether $\Sigma_{2}$ can be obtained from 'a geodesic octagon' in $\mathbb{H}^{2}$ by USING the generators $a$~$d$ of $\Gamma$ to side-pair, if $\Sigma_{2}=\mathbb{H}^{2}/ \Gamma$. http://mathoverflow.net/questions/100130/a-question-about-hyperbolic-double-torus/100137#100137 Comment by KENSO KENSO 2012-06-20T18:34:22Z 2012-06-20T18:34:22Z @Lee: I can't catch your point. So what do you mean by &quot;to arrange&quot;? Anyway, do you mean that nonabelian $\Gamma$ whose generators are all hyperbolic can be indiscrete? http://mathoverflow.net/questions/100130/a-question-about-hyperbolic-double-torus/100137#100137 Comment by KENSO KENSO 2012-06-20T18:32:45Z 2012-06-20T18:32:45Z @Misha: That was the fact that I used to prove the &quot;easy&quot; part. http://mathoverflow.net/questions/100130/a-question-about-hyperbolic-double-torus Comment by KENSO KENSO 2012-06-20T18:14:09Z 2012-06-20T18:14:09Z @Misha: If I'm not mistaken, I was confusing you because I did not mention that my $\Sigma_{2}$ is smooth. For easy part, I think $\Gamma$ does not have any elliptic elements(candidates of finite order) since it is the deck transformation group for $\Sigma$. For the &quot;fundamental domain polygon&quot;, I meant that polygon whose edges are all geodesics in $\mathbb{H}^{2}$, which makes $a$~$d$ be the side-pairings to get $\Sigma_{2}$. Since $\Sigma_{2}$ should be smooth, the angle sum should be $2\pi$. http://mathoverflow.net/questions/100130/a-question-about-hyperbolic-double-torus/100137#100137 Comment by KENSO KENSO 2012-06-20T16:36:01Z 2012-06-20T16:36:01Z @Lee: I see. So the space given by side-pairing of a fundamental domain(polygon) always equals to the space given by a quotient space over the group generated by side-pairing maps, right? And what about my comment for your last paragraph? something false? http://mathoverflow.net/questions/100130/a-question-about-hyperbolic-double-torus/100137#100137 Comment by KENSO KENSO 2012-06-20T15:54:09Z 2012-06-20T15:54:09Z @Lee: Hmm... for the last paragraph of your comment, I'm gonna tell you that there's a theorem(maybe due to Nielsen) which says that if $\Gamma\subset\textrm{Isom}^{+}(\mathbb{H}^{2})$ which consists only of the identity and hyperbolic elements is NONabelian, then $\Gamma$ is discrete. So I'm not sure about your comment, especially about the 'continuous perturbation'. For the first paragraph, you approach this problem more roughly. (I would think more about &quot;When you straighten the four circles α,β,γ,δ in this new hyperbolic structure,&quot;......) http://mathoverflow.net/questions/98561/embedding-again Comment by KENSO KENSO 2012-06-01T15:35:44Z 2012-06-01T15:35:44Z @O'Rourke: Yeah, pablo seems to give a nice answer. But I think the pictures there are not that helpful to me actually... Thanks. http://mathoverflow.net/questions/98561/embedding-again/98574#98574 Comment by KENSO KENSO 2012-06-01T15:31:50Z 2012-06-01T15:31:50Z @Bonnot: I see what you mean, especially the type ${M}_{1}$ of polygon. And sorry that there is a typo in my question: $T=[(x,y)\in\mathbb{H}^{2}:1&lt;\sqrt{{x}^{2}+{y}^{2}}&lt;{e}^{2\pi}]$. If I'm not mistaken, $T$ is also able to be embedded in ${E}^{3}$, right? Then one more question. What is the definition of polygons in $\mathbb{H}^{2}$ in general? Do they allow the ideal parts to be an edge?? http://mathoverflow.net/questions/98184/a-question-about-embedding-hyperbolic-space-onto-pseudosphere/98186#98186 Comment by KENSO KENSO 2012-05-31T09:46:52Z 2012-05-31T09:46:52Z @Bryant: However, one can find that the strip which is a part of a sphere cannot be isometrically twisted, observing that a surface with constant positive curvature should be on a sphere. I think the situation of a pseudosphere-strip is not much different from that of this sphere-strip. So if $A$ is embedded onto a subset which is not a pseudosphere, then we should be able to twist the pseudosphere to make the subset but I don't believe this is not possible. http://mathoverflow.net/questions/98184/a-question-about-embedding-hyperbolic-space-onto-pseudosphere/98186#98186 Comment by KENSO KENSO 2012-05-31T09:40:07Z 2012-05-31T09:40:07Z @Bryant: I can't clarify the anticipated &quot;evidence&quot; so I've come here. But the intuition underlying my thought is following. Consider 3 tiny strips(surfaces of revolution) in $\mathbb{R}^{3}$, homeomorphic to a cylinder, each of which is a part of a cylinder, a sphere, and a pseudosphere, respectively. Now let me isometrically twist them. It is easy to imagine that the strip which is a part of a cylinder is well twisted. For example, if the projecting image of the original strip toward its meridian direction is a circle, it can be a ellipse after isometrically twisting it. http://mathoverflow.net/questions/98184/a-question-about-embedding-hyperbolic-space-onto-pseudosphere Comment by KENSO KENSO 2012-05-30T05:37:49Z 2012-05-30T05:37:49Z @Agol: I think Dini's surface is even not homeomorphic to a cylinder.