Timeline for Fractal-like structures arising from the action of a group on $\mathbb{Z}^2$
Current License: CC BY-SA 3.0
26 events
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| Mar 22, 2014 at 23:31 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
I've updated some informations about $\Gamma$.
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| Mar 22, 2014 at 6:48 | comment | added | მამუკა ჯიბლაძე | On the afterthought though, that stabilizer still has some torsion - e. g. the cube of the above order 6 element | |
| Mar 22, 2014 at 6:39 | comment | added | მამუკა ჯიბლაძე | Oh yes you are right of course. That's interesting. The action of the quotient $\Gamma/\mathbb Z^2$ on the torus $(\mathbb R/2\mathbb Z)^2$ has an orbit $\{(0,0),(1,0),(0,1),(1,1)\}$ on which it acts as the full symmetric group; the stabilizer contains ($\mathbb Z^2$-residue classes of) things like $a^4$, $aba(bab)^{-1}$, etc. If it is torsion free, the picture might be completely clarified. | |
| Mar 21, 2014 at 23:26 | comment | added | Sebastien Palcoux | @მამუკაჯიბლაძე: By your first comment, the element $g_1:(m,n) \to (m+2,n)$ is in $\Gamma$. By symmetry, $g_2:(m,n) \to (m,n+2)$ is also in $\Gamma$. But $\langle g_1 , g_2 \rangle \simeq \mathbb{Z}^2$. Then $\mathbb{Z}^2$ is a subgroup of $\Gamma$, and so $\Gamma$ is not hyperbolic. But $\mathbb{Z}^2 \triangleleft \Gamma$, so perhaps $\Gamma / \mathbb{Z}^2$ is hyperbolic. | |
| Mar 21, 2014 at 21:19 | comment | added | Stefan Kohl♦ | @მამუკაჯიბლაძე: This happens to be the first generator of my example. | |
| Mar 21, 2014 at 21:00 | comment | added | მამუკა ჯიბლაძე | I don't know whether this is useful but your group also has some torsion: e. g. $$ b^2a^{-1}b^{-1}ab^{-2}a(m,n)=(m-n,m) $$ has order 6. | |
| Mar 21, 2014 at 20:52 | comment | added | Stefan Kohl♦ | @SébastienPalcoux: I think in your example you may be able to find a way to compute small parts of larger spheres separately -- what is needed for this is to know a computationally easy way to find the shortest path from a given point to $(0,0)$. -- Since your example doesn't involve divisions, this looks feasible (though I haven't tried). However I think having mere affine mappings as generators significantly limits the complexity of the structure of the images (though I don't know how and to what extent). | |
| Mar 21, 2014 at 18:31 | comment | added | Stefan Kohl♦ | @მამუკაჯიბლაძე: Indeed. -- The sizes of the spheres of radii $0, \dots, 8$ about the identity in this group are $1, 4, 12, 36, 108, 324, 944, 2716, 7619$, which shows a deviation from the free group starting from radius $r = 6$ (by $28$ at $r = 6$, by $200$ at $r = 7$ and by $1129$ at $r = 8$). In particular the relation you give is shortest possible. | |
| Mar 21, 2014 at 18:14 | comment | added | Sebastien Palcoux | @მამუკაჯიბლაძე: Very nice! So the topological dimension of the total space of $\Gamma$ is at least $2$. | |
| Mar 21, 2014 at 17:26 | comment | added | მამუკა ჯიბლაძე | In fact this particular group is not free: for example, $$ ba^{-1}bab^{-1}a(m,n)=a^{-1}bab^{-1}ab(m,n)=(m+2,n). $$ Still most likely it is hyperbolic... | |
| Mar 21, 2014 at 11:00 | comment | added | მამუკა ჯიბლაძე | This one vaguely resembles the s. c. "Farey sunbursts" obtained by joining consecutive $(m,n)$s corresponding to the $m/n$s in the Farey sequence... | |
| Mar 20, 2014 at 17:15 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
I've added the ball of radius 14 with a rainbow gradient.
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| Mar 20, 2014 at 0:39 | comment | added | Sebastien Palcoux | @StefanKohl: Thank you for the computation of the spheres of my example up to radius $18$. I am not convinced that "local" fractal-like structures (as for your example) will not appear for higher radii, because they are not yet revealed for your own example at radius even $20$. So the spheres of my example should be computed up to radius $30$ or $35$ for forming an opinion. The problem is that the cardinality of the spheres of my example increases much much faster, so perhaps they can't be computed at radius $35$ in a reasonable time, with a normal computer. | |
| Mar 19, 2014 at 1:23 | comment | added | Sebastien Palcoux | @StefanKohl: I think a reason why the spheres for this action of $\mathbb{F}_2$ give more and more fine structures is that the total space of $\mathbb{F}_2$ (its Cayley graph) is topological dimension $1$. One should obtain "thicker" spheres with (non-virtually cyclic) torsion-free hyperbolic groups whose total space is $n$-dim. with $n>1$, for example the surface group $\Gamma_2=\langle a_1,b_1,a_2,b_2 \vert [a_1,b_1][a_2,b_2]=e \rangle$ (is there a relevant injection of $\Gamma_2$ into $Sym(\mathbb{Z}^2)$?). | |
| Mar 19, 2014 at 1:16 | comment | added | Sebastien Palcoux | @StefanKohl: One should better see the global fractal structure of these entire spheres by applying a logarithmic contraction (for offsetting the explosive deformation). | |
| Mar 18, 2014 at 23:56 | comment | added | Stefan Kohl♦ | As to "renormalized limit" of the spheres: I don't know for which groups and in which precise sense such limits do exist. In any case I think one needs to be careful with the definitions here, otherwise one might easily get something unintuitive like the empty picture or a single point as the limit. As to $\Gamma < {\rm Sym}(\mathbb{Z}^3)$: yes, this would be interesting -- but good visualization in 3 dimensions (and high resolution!) is not so easy. | |
| Mar 18, 2014 at 23:41 | comment | added | Stefan Kohl♦ | The spheres in your action of $\mathbb{F}_2$ on $\mathbb{Z}^2$ are all star-shaped, with the star getting finer and finer as the radius grows, as far as I can tell. -- I have computed the spheres up to radius 18, see here. In particular the images are much different than those for the group my question is about. | |
| Mar 18, 2014 at 16:45 | comment | added | Sebastien Palcoux | @StefanKohl: These pictures are not defined as a non-escaping set of a complex function (because the group acts transitively...), nevertheless I ask myself if there is a way to define a transformation such that its non-escaping set is exactly the renormalized limit of the spheres. What do you think? Also, by taking $\Gamma<Sym(\mathbb{Z}^3)$ (non virtually cyclic) transitive hyperbolic, one should obtain nice 3D fractals! There is an expert in mathematical imagery: Jos Leys, he often works with Etienne Ghys. | |
| Mar 18, 2014 at 14:08 | comment | added | Sebastien Palcoux | @StefanKohl: I guess this action of $\mathbb{F}_2$ reveals also fractal-like structures, but I can't (yet) compute a sphere of radius large enough for seeing them. | |
| Mar 18, 2014 at 0:02 | comment | added | Stefan Kohl♦ | Nice picture! -- In fact, there are many diffferent possibilities how these pictures may look like for a particular group -- for some groups one just gets more-or-less random-looking ring-shaped pixel clouds, for others one gets various geometric patterns with varying complexity, and for some groups one observes fractal-like patterns like for the group my question is about. Some much simpler examples are shown here: gap-system.org/DevelopersPages/StefanKohl/rcwa/pictures.html | |
| Mar 17, 2014 at 23:52 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
Minor edit
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| Mar 17, 2014 at 23:15 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
I've added some data and a picture.
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| Mar 17, 2014 at 19:59 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
Minor edit
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| Mar 17, 2014 at 15:55 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
Minor edit
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| Mar 17, 2014 at 14:45 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
minor edits and links
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| Mar 17, 2014 at 14:32 | history | answered | Sebastien Palcoux | CC BY-SA 3.0 |
