mirror of
https://github.com/kalmarek/SmallHyperbolic
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136 lines
5.2 KiB
Plaintext
136 lines
5.2 KiB
Plaintext
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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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