Translation distance in the curve complex - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T17:35:31Z http://mathoverflow.net/feeds/question/108372 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108372/translation-distance-in-the-curve-complex Translation distance in the curve complex Igor Rivin 2012-09-28T19:06:47Z 2012-12-17T16:17:58Z <p>Given a map $\psi: S\rightarrow S,$ for $S$ a closed surface, is there any algorithm to compute its translation distance in the curve complex? I should say that I mostly care about checking that the translation distance is/is not very small. That is, if the algorithm can pick among the possibilities: translation distance is 0, 1, 2, 3, many, then I am happy...</p> <p>I know there are algorithms for computing distances IN the curve complex, but this is not quite the same...</p> http://mathoverflow.net/questions/108372/translation-distance-in-the-curve-complex/108377#108377 Answer by Lee Mosher for Translation distance in the curve complex Lee Mosher 2012-09-28T20:43:19Z 2012-09-28T20:43:19Z <p>In the case that $\psi$ is pseudo-Anosov, the best one can do in general, as far as I know, is to get upper and lower bounds which are linear in translation length. These come from train track considerations. Assuming you have an invariant train track $T$ for $\psi$ in your hands (obtained by some algorithmic method of currently unknowable complexity as per my comment), factor it into a sequence of train track splits $$T=T_0, T_1, \ldots, T_k = \psi(T)$$ then using the method in the Masur-Minsky paper "Geometry of the curve comples I: hyperbolicity", one can algorithmically break the split sequence into blocks $$T_0 = T_{m_0}, ..., T_{m_1}, ..., T_{m_a}=T_k$$ such that the diameter of the subsequence from $T_{m_i}$ to $T_{m_{i+1}}$ has a certain constant upper bound and the diameter from $T_{m_i}$ to $T_{m_j}$ has a certain lower bound which is a constant times $|i-j|$. The material needed to do this is described in the section of their paper entitled "the nested train track argument".</p> <p>Other than that, Shackleton's paper "An acylindricity theorem for the mapping class group" contains some algorithmic detail, but not enough to answer your question. </p> http://mathoverflow.net/questions/108372/translation-distance-in-the-curve-complex/108382#108382 Answer by Richard Kent for Translation distance in the curve complex Richard Kent 2012-09-29T01:36:02Z 2012-09-29T01:36:02Z <p>In the braid group, Ko and Lee have given a polynomial time test of reducibility using the Garside structure. (See <a href="http://arxiv.org/abs/math/0610746" rel="nofollow">http://arxiv.org/abs/math/0610746</a>)</p> http://mathoverflow.net/questions/108372/translation-distance-in-the-curve-complex/108389#108389 Answer by Agol for Translation distance in the curve complex Agol 2012-09-29T05:56:29Z 2012-09-29T05:56:29Z <p>I don't know an algorithm, but here's a possible approach. As Richard and Lee have observed, one may assume that $\psi$ is pseudo-Anosov. In that case, the mapping torus $T_\psi$ is a hyperbolic 3-manifold fibering over $S^1$, with fiber $S$. There is a short exact sequence $\pi_1(S)\to\pi_1(T_\psi)\to \mathbb{Z}$. </p> <p>Here's a characterization of the translation length on the curve complex in terms of the topology of $T_\psi$. The fiber $S$ represents a homology class $[S]\in H_2(T_\psi)$. Let $\Sigma \looparrowright T_\psi$ be an immersed connected surface, such that $[\Sigma]=k[S]$ and such that $\chi(\Sigma)=k\chi(S)$. Moreover, assume that the composite map $\pi_1(\Sigma)\to \pi_1(T_\psi) \to \mathbb{Z}$ is non-trivial, so that $\Sigma$ is not homotopic to a finite-sheeted cover of $S$. Let $K(\psi)$ be the minimal such $k$, and let $D(\psi)$ be the minimal such $k$ so that the surface has only double curves of intersection. Clearly $K(\psi)\leq D(\psi)$. </p> <p><strong>Claim:</strong> The curve complex translation distance of $\psi$ is $=D(\psi)$. </p> <p>One direction: Let $k$ be the translation length of $\psi$. There exists a sequence of non-separating curves $c_1,c_2,\ldots, c_k \subset S$, such that $\psi(c_1)=c_k$, and $c_i \cap c_{i+1}=\emptyset$. One creates a surface $\Sigma\subset T_\psi$ by taking $k$ copies of $S$, $S_1 \sqcup \cdots \sqcup S_k \subset T_\psi$ in circular order. Cut out annular neighborhoods of $c_i, c_{i+1}$ inside $S_i$, and insert cross annuli between $S_{i-1}$ and $S_i$ (taking indices $\mod k$) between the 4 copies of $c_i$. This construction generalizes a construction of <a href="http://www.springerlink.com/content/u0v2725568212002/" rel="nofollow">Cooper-Long-Reid</a>. One can see that the resulting surface has the properties above. </p> <p>Conversely, if one has such an immersed surface with only double curves, one may cut and paste the self-intersection curves to get $k$ parallel copies of $S$. The cross cut curves gives a sequence of closed curves in $S$, which one can prove using the homology condition forms a closed loop in the curve complex $\mod \psi$. </p> <p>I don't know yet how to make this criterion into an algorithm. I think there is an algorithm to compute $K(\psi)$. For a given genus $g$, <a href="http://www.sciencedirect.com/science/article/pii/0040938394000557" rel="nofollow">Canary proved</a> that there are only finitely many homotopy classes of immersed surfaces of genus $g$. I think this proof can be made effective, and should give one a method to compute $K(\psi)$. This would at least give an algorithmic lower bound, since $K(\psi)\leq D(\psi)$. Also, there is a constant $0&lt; c_S &lt;1$ such that $D(\psi)\leq c_S K(\psi)$ (this may be proved using hyperbolic geometry techniques). </p> <p>One could try to algorithmically to construct all surfaces realizing $K(\psi)$, and then try to homotope them to have only double curves of intersection, e.g. using normal surfaces. However, there is a result of <a href="http://www.ams.org/mathscinet-getitem?mr=899054" rel="nofollow">Gulliver-Scott</a> that an immersed surface with only double curves of intersection might have a minimal area representative which has triple points. So I don't know yet how to make an algorithm by computing $D(\psi)$ using this approach. </p> http://mathoverflow.net/questions/108372/translation-distance-in-the-curve-complex/116618#116618 Answer by unknown (google) for Translation distance in the curve complex unknown (google) 2012-12-17T16:17:58Z 2012-12-17T16:17:58Z <p>Please specify what translation length" is here. Is it $\lim_{n\rightarrow\infty} \frac{1}{n} d(x,\psi^n x)$, or, $\min_x d(x,\psi x)$; or something else that you have in mind?</p>