Looking for dense subgroups of SU(2) with finite presentations Text

Ben

Jul6-04, 02:47 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no,scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nHi,\n\nI need to find a dense subgroup of SU(2) with a finite presentation.\nSpecifically, I need the matrices that form the generators and the\npresentation. I take a finite presentation to be a finite set of\ngenerators (abstract symbols) followed by a finite set of words in\nthose generators that are equal to the identity. For example, Z/2 is\npresented as\n\n<a|a^2=I>\n\nbut it is a finite group with only two elements. I am looking for\ndense subgroups of SU(2), those that approximate every element of the\ncontinuum of elements to any accuracy. If you know of even one, or\nyou have a reference, please post a follow-up.\n\nBen Sprott\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hi,

I need to find a dense subgroup of SU(2) with a finite presentation.
Specifically, I need the matrices that form the generators and the
presentation. I take a finite presentation to be a finite set of
generators (abstract symbols) followed by a finite set of words in
those generators that are equal to the identity. For example, Z/2 is
presented as

<a|a^2=I>

but it is a finite group with only two elements. I am looking for
dense subgroups of SU(2), those that approximate every element of the
continuum of elements to any accuracy. If you know of even one, or
you have a reference, please post a follow-up.

Ben Sprott

Robert C. Helling

Jul9-04, 08:35 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no,scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nOn 6 Jul 2004 14:47:34 -0400, Ben <btsprott@sciborg.uwaterloo.ca> wrote:\n>\n> Hi,\n>\n> I need to find a dense subgroup of SU(2) with a finite presentation.\n> Specifically, I need the matrices that form the generators and the\n> presentation. I take a finite presentation to be a finite set of\n> generators (abstract symbols) followed by a finite set of words in\n> those generators that are equal to the identity.\n\nDo you know of any other example where this works for a Lie group? For\nexample take U(1) with elements g(x) = exp(2 pi x). Then you get dense\nsubgroups generated by g(x) for irrational x (or serveral of\nthose). However, there are no relations and the subgroup you generate\nis isomorphic to Z. There are only relations with "irrational powers"\nbut I don\'t know how to define those. If you get this to work, do a\nsimilar thing for SU(2) (or SO(3)) by taking an irrational rotation\naround the x-axis and another one around the y-axis. They generate a\ndense subset.\n\nRobert\n\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO\nRobert C. Helling Department of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nprint "Just another Phone: +44/1223/766870\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On 6 Jul 2004 14:47:34 -0400, Ben <btsprott@sciborg.uwaterloo.ca> wrote:
>
> Hi,
>
> I need to find a dense subgroup of SU(2) with a finite presentation.
> Specifically, I need the matrices that form the generators and the
> presentation. I take a finite presentation to be a finite set of
> generators (abstract symbols) followed by a finite set of words in
> those generators that are equal to the identity.

Do you know of any other example where this works for a Lie group? For
example take U(1) with elements g(x) = \exp(2 \pi x). Then you get dense
subgroups generated by g(x) for irrational x (or serveral of
those). However, there are no relations and the subgroup you generate
is isomorphic to Z. There are only relations with "irrational powers"
but I don't know how to define those. If you get this to work, do a
similar thing for SU(2) (or SO(3)) by taking an irrational rotation
around the x-axis and another one around the y-axis. They generate a
dense subset.

Robert

--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling

Pierre Asselin

Jul9-04, 10:57 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no,scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Ben <btsprott@sciborg.uwaterloo.ca> wrote:\n\n> I need to find a dense subgroup of SU(2) with a finite presentation.\n> Specifically, I need the matrices that form the generators and the\n> presentation. I take a finite presentation to be a finite set of\n> generators (abstract symbols) followed by a finite set of words in\n> those generators that are equal to the identity. For example, Z/2 is\n> presented as\n\n> <a|a^2=I>\n\n> but it is a finite group with only two elements. I am looking for\n> dense subgroups of SU(2), those that approximate every element of the\n> continuum of elements to any accuracy. If you know of even one, or\n> you have a reference, please post a follow-up.\n\nHow about the one that\'s used in the Banach-Tarski paradox ?\n\nA= rotation by arcos(1/3) around z\nB= rotation by arcos(1/3) around y\n\nThese two generate a free group: *no* word in A and B is the identity.\nIt seems to me that the group must be dense, because you can write any\nrotation as R_z(\\phi)R_y(\\theta)R_z(\\psi) in terms of its Euler angles,\nand you can approximate each factor by an appropriate power of A or B.\n(You do this in the spin-1/2 representation of course.)\n\nFor a proof that the group is free, see chapter 2 in Stan Wagon\'s book,\n"The Banach-Tarski Paradox", Cambridge University press 1985, ISBN\n0-521-45704-1. Or Google for Banach-Tarski, you never know.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Ben <btsprott@sciborg.uwaterloo.ca> wrote:

> I need to find a dense subgroup of SU(2) with a finite presentation.
> Specifically, I need the matrices that form the generators and the
> presentation. I take a finite presentation to be a finite set of
> generators (abstract symbols) followed by a finite set of words in
> those generators that are equal to the identity. For example, Z/2 is
> presented as

> <a|a^2=I>

> but it is a finite group with only two elements. I am looking for
> dense subgroups of SU(2), those that approximate every element of the
> continuum of elements to any accuracy. If you know of even one, or
> you have a reference, please post a follow-up.

How about the one that's used in the Banach-Tarski paradox ?

A= rotation by arcos(1/3) around z
B= rotation by arcos(1/3) around y

These two generate a free group: *no* word in A and B is the identity.
It seems to me that the group must be dense, because you can write any
rotation as R_z(\phi)R_y(\theta)R_z(\psi) in terms of its Euler angles,
and you can approximate each factor by an appropriate power of A or B.
(You do this in the spin-1/2 representation of course.)

For a proof that the group is free, see chapter 2 in Stan Wagon's book,
"The Banach-Tarski Paradox", Cambridge University press 1985, ISBN
0-521-45704-1. Or Google for Banach-Tarski, you never know.