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https://github.com/kalmarek/SmallHyperbolic
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Marek Kaluba
GitHub
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89fbf76f32
15
README.md
15
README.md
@ -1,11 +1,11 @@
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The repository contains code for running experiments for
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_Hyperbolic generalized triangle groups_ by
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[_Hyperbolic generalized triangle groups, property (T) and finite simple quotients_](https://arxiv.org/abs/2011.09276) by
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[Pierre-Emmanuel Caprace](https://perso.uclouvain.be/pierre-emmanuel.caprace/),
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[Marston Conder](https://www.math.auckland.ac.nz/~conder/),
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[Marek Kaluba](https://kalmar.faculty.wmi.amu.edu.pl/) and
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[Stefan Witzel](https://www.math.uni-bielefeld.de/~switzel/).
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There are two disjoint computations covered in this repository.
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There are three disjoint computations covered in this repository.
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## Eigenvalues computations for _PSL₂(p)_
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@ -138,3 +138,14 @@ One can perform those computations in bulk by e.g. calling
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make 2_4_4
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```
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to run all examples in `presentations_2_4_4.txt` in parallel.
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## Creating the tables of [arXiv:2011.09276](https://arxiv.org/abs/2011.09276)
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The scripts are located in `magma` directory and thoroughly commented.
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There are two files, one contains the core Magma code used to create the tables,
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the other one is a python script that augments the magma file by a list of
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hyperbolic words. To use it put both files in a common folder, run
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```bash
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python3 hyperbolic_words.py
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```
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and then load the resulting file `small_hyperbolic.magma` in Magma.
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27
magma/hyperbolic_words.py
Normal file
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magma/hyperbolic_words.py
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# This script augments creates the magma file small_hyperbolic.magma out of
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# the file small_hyperbolic.magma_template by filling in the lines
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# "hyp_words_a := ..." and "hyp_words_c := ..." with lists of all short words
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# representing short hyperbolic elements.
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from itertools import permutations,product
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import sys
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basic_words_a = ['abcb','abcabc','abcbabcb','abcabcabcb','abcbabcbabcb','abcabcabcabc']#,'abcabcabcbabcb']
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basic_words_c = ['acbc','cbca','acabcb','cabcba','acbcacbc','cbcacbca','acabcabcbc','cabcabcbca','acabcbacabcb','cabcbacabcba','acbcacbcacbc','cbcacbcacbca']
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def words(basic_word):
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for p in permutations('ab'):
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d = dict(zip('abc',p + ('c',)))
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subs_word = ''.join(d[i] for i in basic_word)
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for exps in product(('','^-1'), repeat=len(subs_word)):
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word = ' * '.join((c + e) for (c,e) in zip(subs_word,exps))
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yield word
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with open("small_hyperbolic.magma_template","r") as magma_template:
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with open("small_hyperbolic.magma","w") as magma:
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for line in magma_template:
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if 'hyp_words_a := ' in line:
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magma.write('hyp_words_a := [ %s ];\n' % ' , '.join('{ %s }' % ', '.join(words(basic_word)) for basic_word in basic_words_a))
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elif 'hyp_words_c := ' in line:
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magma.write('hyp_words_c := [ %s ];\n' % ' , '.join('{ %s }' % ', '.join(words(basic_word)) for basic_word in basic_words_c))
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else:
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magma.write(line)
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135
magma/small_hyperbolic.magma_template
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magma/small_hyperbolic.magma_template
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// Magma Functions related to the article
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//
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// Hyperbolic generalized triangle groups, property (T) and finite simple quotients
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// by Pierre-Emmanuel Caprace, Marston Conder, Marek Kaluba, Stefan Witzel
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//
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// The functions and procedures in this file were written by Pierre-Emmanuel Caprace and Stefan Witzel
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//
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// The file is working magma code except for the two lines definint hyp_words_a and hyp_words_c that need to be replaced using hyperbolic_words.py
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// It does not contain (all) the code that was used to create the tables but it is suited to reconstruct the tables.
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SetColumns(0);
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SetAutoColumns(false);
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SetVerbose("KBMAG",1);
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max_quotient_order:=5*10^7;
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// Procedure testing whether a trivalent generalized triangle group of half girth type (3,3,3) ("type A") or half girth type (2,4,4) ("type C") contains a copy of Z^2 generated by short elements.
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//
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// The arguments are an automatic group representing a generalized triangle group and the list hyp_words_a or hyp_words_c depening on type (these lists are the only part specific to trivalent triangle groups).
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//
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// The prints generators of a copy of Z^2 f one is found (so the group is not hyperbolic). If none is found the group may or may not be hyperbolic.
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find_flat := procedure(GA, ~hyp_words)
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printf "Commuting pair: ";
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for s in [1..#hyp_words] do
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for i in [1..s] do
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for x in hyp_words[i] do
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for y in hyp_words[i] do
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// Do x and y commute?
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if (x, y) ne GA!1 then
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continue;
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end if;
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// If so, do they span a cyclic group?
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num := GreatestCommonDivisor(#x,#y);
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ex := Round(#y/num);
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ey := Round(#x/num);
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if x^ex eq y^ey or x^ex eq y^(-ey) then
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continue;
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end if;
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// If not they span a copy of Z^2
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printf "%o, %o;\t", x, y;
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return;
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end for;
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end for;
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end for;
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end for;
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printf "\n";
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end procedure;
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// Auxiliary function determining the name of a simple quotient
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identify := function(Qs)
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succ, res := SimpleGroupName(Image(Qs[1]));
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if succ then
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return [*res,Order(Image(Qs[1])), #Qs*];
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else
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return [* [*<0,0,0>*], Order(Image(Qs[1])),#Qs*];
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end if;
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end function;
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// Procedure collecting information around the hyperbolicity of a generalized triangle group of half girth type (3,3,3) (type A) or (2,4,4) (type C).
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//
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// Arguments are a generalized triangle group (as a finitely presented group) and "A" or "C" describing the type.
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//
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// It will first try to find an automatic structure and prints whether it has found one (this should never fail).
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// Based on the automatic struture, it tries to prove the group hyperbolic and prints whether it manged.
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// If the group could not be verified to be hyperbolic, the procedure tries to find a copy of Z^2 and print, whether it has found one.
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//
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// Of course, if a group is expected to contain Z^2, it is reasonable to call find_flat before calling IsHyperbolic.
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hyperbolic := procedure(G, type)
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isa, GA := IsAutomaticGroup(G);
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printf "automatic: %o\n", isa;
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if not isa then
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return;
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end if;
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ish, GH := IsHyperbolic(GA: MaxTries := 20);
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printf "hyperbolic: %o\n", ish;
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if ish then
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return;
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end if;
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a := Generators(GA)[1];
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b := Generators(GA)[2];
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c := Generators(GA)[3];
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if type eq "A" then
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hyp_words_a := [ False ];
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find_flat(GA, ~hyp_words_a);
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elif type eq "C" then
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hyp_words_c := [ False ];
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find_flat(GA, ~hyp_words_c);
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end if;
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end procedure;
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// Procedure collecting information abound finite simple quotients of a finitely presented group
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//
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// Arguments are a finitely presented group and a bound on the order of quotients (such as max_quotient_order)
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//
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// Prints the abelianization, the L2 quotients, and the finite simple quotients that are not L2 quotients up to the given order.
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quotients := procedure(G,n)
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Q, mor := AbelianQuotient(G);;
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printf "rk(H_1): %o\n", #Generators(AbelianQuotient(G));
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printf "l2-quotients: %o\n", L2Quotients(G);
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printf "quotients: %o, %o\n", [identify(Qs) : Qs in SimpleQuotients(G,n:Family:="notPSL2",Limit:=0)], n;
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end procedure;
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// Function testing whether a generalized triangle group given in triangular presentation is virtually torsion free (VTF)
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//
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// Returns "true" if group is certified to be virtually torsion-free. The outpue "false" is inconclusive.
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//
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// The input is a quadruple consisting of the group presentation, followed by the triple of the orders of the vertex links, respectively corresponding to <a, b>, <b, c>, <c, a>
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VTF:=function(g, n1, n2, n3)
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o1:=n1*3/2; // Order group vx group generated by a, b
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o2:=n2*3/2; // Order group vx group generated by b, c
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o3:=n3*3/2; // Order group vx group generated by c, a
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// Test whether vertex groups inject in simple quotients
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t:=false;
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quot:=SimpleQuotientProcess(g, 6, 10^3, 1, 5*10^7:Limit:=1);
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while not t and not IsEmptySimpleQuotientProcess(quot) do
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f:=SimpleEpimorphisms(quot)[1];
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Im:=Parent(f(g.1));
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o_im_1:=#sub<Im|f(g.1), f(g.2)>;
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o_im_2:=#sub<Im|f(g.2), f(g.3)>;
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o_im_3:=#sub<Im|f(g.3), f(g.1)>;
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t:=o1 eq o_im_1 and o2 eq o_im_2 and o3 eq o_im_3;
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NextSimpleQuotient(~quot);
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end while;
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return t;
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end function;
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@ -1,102 +0,0 @@
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using Nemo
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using DelimitedFiles
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using LinearAlgebra
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include("src/nemo_utils.jl")
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const PRECISION = 256
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function parse_eval(expr_str, value, var_name)
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ex = Meta.parse(expr_str)
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svar = :($var_name)
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return @eval begin
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let $svar = $value
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$ex
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end
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end
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end
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function read_eval(fname, var_name, value)
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a = readdlm(fname, ',', String)
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a .= replace.(a, '/' => "//")
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return parse_eval.(a, value, var_name)
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end
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function load_discrete_repr(i, q = 109; CC = AcbField(PRECISION))
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ζ = root_of_unity(CC, (q + 1) ÷ 2)
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degree = q - 1
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ra = read_eval(
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"data/Discrete reps PSL(2, $q)/discrete_rep_$(i)_a.txt",
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:Z,
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ζ,
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)
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a = matrix(CC, [CC(s) for s in ra[1:degree, 1:degree]])
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rb = read_eval(
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"data/Discrete reps PSL(2, $q)/discrete_rep_$(i)_b.txt",
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:Z,
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ζ,
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)
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b = matrix(CC, [CC(s) for s in rb[1:degree, 1:degree]])
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@assert contains(det(a), 1)
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@assert contains(det(b), 1)
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return a, b
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end
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function load_principal_repr(i, q = 109; CC = AcbField(PRECISION))
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ζ = root_of_unity(CC, (q - 1) ÷ 2)
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degree = q + 1
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ra = read_eval(
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"data/Principal reps PSL(2, $q)/principal_rep_$(i)_a.txt",
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:zz,
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ζ,
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)
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a = matrix(CC, [CC(z) for z in ra[1:degree, 1:degree]])
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rb = read_eval(
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"data/Principal reps PSL(2, $q)/principal_rep_$(i)_b.txt",
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:zz,
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ζ,
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)
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b = matrix(CC, [CC(z) for z in rb[1:degree, 1:degree]])
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@assert contains(det(a), 1)
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@assert contains(det(b), 1)
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return a, b
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end
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if !isinteractive()
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for i = 0:27
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try
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a, b = load_principal_repr(i)
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adjacency = sum(a^i for i = 1:4) + sum(b^i for i = 1:4)
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@time ev = let evs = safe_eigvals(adjacency)
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_count_multiplicites(evs)
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end
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@info "Principal Series Representation $i" ev[1:2] ev[end]
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catch ex
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@error "Principal Series Representation $i failed" ex
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ex isa InterruptException && throw(ex)
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end
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end
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for i = 1:27
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try
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a, b = load_discrete_repr(i)
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adjacency = sum(a^i for i = 1:4) + sum(b^i for i = 1:4)
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@time ev = let evs = safe_eigvals(adjacency)
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_count_multiplicites(evs)
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end
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@info "Discrete Series Representation $i" ev[1:2] ev[end]
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catch ex
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@error "Discrete Series Representation $i : failed" ex
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ex isa InterruptException && rethrow(ex)
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end
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end
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end
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