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<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLor </script> <title>GAP (Congruence) - Chapter 4: Farey symbols for congruence subgroups</title> <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> <script src="manual.js" type="text/javascript"></script> <script type="text/javascript">overwriteStyle();</script> </head> <body class="chap4" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top< <div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.ht <p id="mathjaxlink" class="pcenter"><a href="chap4.html">[MathJax off]</a></p> <p><a id="X831C60277F7D80B2" name="X831C60277F7D80B2"></a></p> <div class="ChapSects"><a href="chap4_mj.html#X831C60277F7D80B2">4 <span class="Heading">F <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8594896287DCFE8 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8790C1498107A39 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X78779BDF7A1DB4A <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7F792846795E3A6 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7905B050800E441 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X79C44528864044C </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X80EED3418340810 </div></div> </div> <h3>4 <span class="Heading">Farey symbols for congruence subgroups</span></h3> <p>The package <strong class="pkg">Congruence</strong> provides functions to construct Farey sy <p>The development of an algorithm to determine the Farey symbol for a subgroup G of a finite index <p><a id="X7F43DB8B803F313F" name="X7F43DB8B803F313F"></a></p> <h4>4.1 <span class="Heading">Computation of the Farey symbol for a finite index subgroup</span></ <p><a id="X8594896287DCFE8D" name="X8594896287DCFE8D"></a></p> <h5>4.1-1 FareySymbol</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>For a subgroup of a finite index G, this attribute stores one of the Farey symbols correspondin <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">FareySymbol(PrincipalCongruence [ infinity, 0, 1/4, 1/3, 3/8, 2/5, 1/2, 3/5, 5/8, 2/3, 3/4, 1, 5/4, 4/3, 11/8, 7/5, 3/2, 8/5, 13/8, 5/3, 7/4, 2, 9/4, 7/3, 19/8, 12/5, 5/2, 13/5, 21/8, 8/3, 11/4, 3, 13/4, 10/3, 27/8, 17/5, 7/2, 18/5, 29/8, 11/3, 15/4, 4, 17/4, 13/3, 9/2, 14/3, 19/4, 5, 21/4, 16/3, 11/2, 17/3, 23/4, 6, 25/4, 19/3, 13/2, 20/3, 27/4, 7, 29/4, 22/3, 15/2, 23/3, 31/4, 8, infinity ] [ 1, 17, 10, 26, 32, 18, 19, 27, 30, 5, 2, 2, 13, 28, 26, 20, 21, 29, 27, 7, 3, 3, 16, 31, 28, 22, 23, 33, 29, 9, 4, 4, 5, 30, 31, 24, 25, 32, 33, 12, 6, 6, 7, 19, 18, 15, 8, 8, 9, 21, 20, 10, 11, 11, 12, 23, 22, 13, 14, 14, 15, 25, 24, 16, 17, 1 ] <span class="GAPprompt">gap></span> <span class="GAPinput">FareySymbol(CongruenceSubgroupG [ infinity, 0, 1/5, 1/4, 2/7, 3/10, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1, infinity ] [ 1, 3, 4, 6, 7, 7, 5, 2, 2, 3, 6, 4, 5, 1 ] </pre></div> <p><a id="X80AE179D869BEE90" name="X80AE179D869BEE90"></a></p> <h4>4.2 <span class="Heading">Computation of generators of a finite index subgroup from its Fare <p>If <var class="Arg">fs</var> is the Farey symbol for a group <span class="SimpleMath">\(G\)</span> <p>for each even interval <span class="SimpleMath">\([x_i, x_{i+1}]\)</span>, take the matrix</p> <pre class="normal"> A= [a_{i+1} b_{i+1} + a_i b_i -a_i^2 - a_{i+1}^2 ] [b_i^2 +b_{i+1}^2 -a_{i+1} b_{i+1} - a_i b_i] </pre> <p>for each odd interval <span class="SimpleMath">\([x_j,x_{j+1}]\)</span>, take the matrix</p> <pre class="normal"> B= [a_{j+1} b_{j+1} + a_j b_{j+1} + a_j b_j -a_j^2 - a_j a_{j+1} -a_{j+1}^2] [ b_j^2 + b_j b_{j+1} + b_{j+1}^2 -a_{j+1} b_{j+1} - a_{j+1} b_j - a_j b_j] </pre> <p>for each pair of free intervals <span class="SimpleMath">\([x_k,x_{k+1}]\)</span> and <span cl <pre class="normal"> C= [a_{s+1} b_{k+1} + a_s b_k -a_s a_k - a_{s+1} a_{k+1}] [b_s b_k- b_{s+1} b_{k+1}c -a_{k+1} b_{s+1} - a_k b_s] </pre> <p><a id="X8790C1498107A39A" name="X8790C1498107A39A"></a></p> <h5>4.2-1 MatrixByEvenInterval</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns the matrix corresponding to the even interval i in the generalized Farey sequence <var <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">H:=CongruenceSubgroupGamma0(5); < <congruence subgroup CongruenceSubgroupGamma_0(5) in SL_2(Z)> <span class="GAPprompt">gap></span> <span class="GAPinput">fs:=FareySymbol(H);</span> [ infinity, 0, 1/2, 1, infinity ] [ 1, "even", "even", 1 ] <span class="GAPprompt">gap></span> <span class="GAPinput">gfs:=GeneralizedFareySequence(f [ infinity, 0, 1/2, 1, infinity ] <span class="GAPprompt">gap></span> <span class="GAPinput">MatrixByEvenInterval(gfs,2); </sp [ [ 2, -1 ], [ 5, -2 ] ] </pre></div> <p><a id="X78779BDF7A1DB4AE" name="X78779BDF7A1DB4AE"></a></p> <h5>4.2-2 MatrixByOddInterval</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns the matrix corresponding to the odd interval i in the generalized Farey sequence <var c <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">fs_oo:=FareySymbolByData([infin <span class="GAPprompt">gap></span> <span class="GAPinput">gfs_oo:=GeneralizedFareySequenc [ infinity, 0, infinity ] <span class="GAPprompt">gap></span> <span class="GAPinput">MatrixByOddInterval(gfs_oo,1);</ [ [ -1, -1 ], [ 1, 0 ] ] </pre></div> <p><a id="X7F792846795E3A63" name="X7F792846795E3A63"></a></p> <h5>4.2-3 MatrixByFreePairOfIntervals</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns the matrix corresponding to the pair of free intervals k and kp in the generalized Fare <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">fs_free:=FareySymbolByData([inf <span class="GAPprompt">gap></span> <span class="GAPinput">gfs_free:=GeneralizedFareySeque <span class="GAPprompt">gap></span> <span class="GAPinput">MatrixByFreePairOfIntervals(gfs [ [ 3, -2 ], [ 2, -1 ] ] </pre></div> <p><a id="X7905B050800E4416" name="X7905B050800E4416"></a></p> <h5>4.2-4 GeneratorsByFareySymbol</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns a set of matrices constructed as above.</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">fs_eo:=FareySymbolByData([infin <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsByFareySymbol(last); </ [ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 0, -1 ], [ 1, -1 ] ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsByFareySymbol(fs); </sp [ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 2, -1 ], [ 5, -2 ] ], [ [ 3, -2 ], [ 5, -3 ] ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsByFareySymbol(fs_oo);< [ [ [ -1, -1 ], [ 1, 0 ] ], [ [ 0, -1 ], [ 1, -1 ] ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsByFareySymbol(fs_free [ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 3, -2 ], [ 2, -1 ] ] ] </pre></div> <p><a id="X79C44528864044C5" name="X79C44528864044C5"></a></p> <h5>4.2-5 GeneratorsOfGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns a set of generators for the finite index group G in <span class="SimpleMath">\(SL_2(Z) <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">G:=PrincipalCongruenceSubgroup( <principal congruence subgroup of level 2 in SL_2(Z)> <span class="GAPprompt">gap></span> <span class="GAPinput">FareySymbol(G);</span> [ infinity, 0, 1, 2, infinity ] [ 2, 1, 1, 2 ] <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup(G);</span> #I Using the Congruence package for GeneratorsOfGroup ... [ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 3, -2 ], [ 2, -1 ] ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">H:=CongruenceSubgroupGamma0(5); < <congruence subgroup CongruenceSubgroupGamma_0(5) in SL_2(Z)> <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup(H);</span> #I Using the Congruence package for GeneratorsOfGroup ... [ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 2, -1 ], [ 5, -2 ] ], [ [ 3, -2 ], [ 5, -3 ] ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">I:=IntersectionOfCongruenceSubg <intersection of congruence subgroups of resulting level 6 in SL_2(Z)> <span class="GAPprompt">gap></span> <span class="GAPinput">FareySymbol(I);</span> [ infinity, 0, 1/3, 1/2, 2/3, 1, 4/3, 3/2, 5/3, 2, infinity ] [ 1, 5, 4, 3, 2, 2, 3, 4, 5, 1 ] <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup(I); </span> #I Using the Congruence package for GeneratorsOfGroup ... [ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 11, -2 ], [ 6, -1 ] ], [ [ 19, -8 ], [ 12, -5 ] ], [ [ 17, -10 ], [ 12, -7 ] ], [ [ 7, -6 ], [ 6, -5 ] ] ] </pre></div> <p><a id="X7C5AB1D786207745" name="X7C5AB1D786207745"></a></p> <h4>4.3 <span class="Heading">Other properties derived from Farey symbols</span></h4> <p><a id="X80EED34183408106" name="X80EED34183408106"></a></p> <h5>4.3-1 IndexInPSL2ZByFareySymbol</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>By Proposition 7.2 in [Kulkarni], for the Farey symbol with underlying generalized Farey seq <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">IndexInPSL2ZByFareySymbol(fs);</ 6 <span class="GAPprompt">gap></span> <span class="GAPinput">IndexInPSL2ZByFareySymbol(fs_oo 2 <span class="GAPprompt">gap></span> <span class="GAPinput">IndexInPSL2ZByFareySymbol(fs_fr 6 </pre></div> <div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.ht <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top< <hr /> <p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GA </body> </html>
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