Вложение графа Петерсена на торе, часть 1

считаю класс смежности подгруппы Гекке
для вложения графа Петерсена в Риманову поверхность с родом 1

TODO: это же какая-то подгруппа модулярной группы? у неё есть какие-то свойства (например, арифметичность)?

TODO
  у октаэдра есть генератор вида
  x*y^-1*x*y*x*y*x^-1*y^-2*x^-1
  особенность, что тут два раза подряд y^-1 (в виде y^-2)
  в генераторах вложения графа Петерсена я такого не заметил, (x или x^-1) чередуются с (y или y^-1)
  интересно проверить для других вложений

f:=FreeGroup("x", "y");
H3:=f/[f.1^2,f.2^3];
hom:=GroupHomomorphismByImages(H3,Group(
(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30),
(1,3,5)(4,7,9)(10,11,13)(15,14,17)(6,16,19)(2,21,23)(8,27,25)(12,29,24)(18,22,28)(20,30,26)),
GeneratorsOfGroup(H3),
[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30),
(1,3,5)(4,7,9)(10,11,13)(15,14,17)(6,16,19)(2,21,23)(8,27,25)(12,29,24)(18,22,28)(20,30,26)]);

petersen_group:=PreImage(hom,Stabilizer(Image(hom),1));
iso:=IsomorphismFpGroup(petersen_group);

[ <[ [ 1, 1 ] ]|(y*x)^2*y*(x^-1*y^-1)^2*x^-1>,
  <[ [ 2, 1 ] ]|y*x*y^-1*x*y*x^-1*y^-1*x^-1*y*x^-1>,
  <[ [ 3, 1 ] ]|y^-1*x*y*x*y^-1*x^-1*y*x^-1*y^-1*x^-1>,
  <[ [ 4, 1 ] ]|(y^-1*x)^2*(y*x^-1)^3>,
  <[ [ 5, 1 ] ]|(y*x)^2*y^-1*x*(y*x^-1)^2*y>,
  <[ [ 6, 1 ] ]|y*(x*y^-1)^3*x^-1*y^-1*x^-1*y>
 ] -> [ F1, F2, F3, F4, F5, F6 ]


x =
0 -1
1 0

y =
0 -1
1 1

(y*x)^2*y*(x^-1*y^-1)^2*x^-1
F1 = (y@x)@(y@x)@y@(xi@yi)@(xi@yi)@xi
 1 -2
-3  7

y*x*y^-1*x*y*x^-1*y^-1*x^-1*y*x^-1
F2 = y@x@yi@x@y@xi@yi@xi@y@xi
 5 -3
-8  5

y^-1*x*y*x*y^-1*x^-1*y*x^-1*y^-1*x^-1
F3 = yi@x@y@x@yi@xi@y@xi@yi@xi
-5  8
 3 -5

(y^-1*x)^2*(y*x^-1)^3
F4 = (yi@x)@(yi@x)@(y@xi)@(y@xi)@(y@xi)
 7 -2
-3  1

(y*x)^2*y^-1*x*(y*x^-1)^2*y
F5 = (y@x)@(y@x)@yi@x@(y@xi)@(y@xi)@y
-1 -4
 3 11

y*(x*y^-1)^3*x^-1*y^-1*x^-1*y
F6 = y@(x@yi)@(x@yi)@(x@yi)@xi@yi@xi@y
 4  5
-5 -6



f:=FreeGroup("x", "y");
H3:=f/[f.1^2,f.2^3];
hom:=GroupHomomorphismByImages(H3,Group(
(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30),
(1,19,18)(2,3,29)(4,5,26)(6,7,22)(8,9,28)(10,11,30)(12,13,24)(14,15,25)(16,17,27)(20,23,21)),
GeneratorsOfGroup(H3),
[(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30),
(1,19,18)(2,3,29)(4,5,26)(6,7,22)(8,9,28)(10,11,30)(12,13,24)(14,15,25)(16,17,27)(20,23,21)]);

petersen_group:=PreImage(hom,Stabilizer(Image(hom),1));
iso:=IsomorphismFpGroup(petersen_group);

[ <[ [ 1, 1 ] ]|(y*x)^2*(y*x^-1)^3>,
  <[ [ 2, 1 ] ]|y*x*y^-1*x*y*(x^-1*y^-1)^2*x^-1>,
  <[ [ 3, 1 ] ]|y^-1*x*y*x*(y^-1*x^-1)^2*y*x^-1>,
  <[ [ 4, 1 ] ]|(y^-1*x)^2*(y*x^-1)^2*y^-1*x^-1>,
  <[ [ 5, 1 ] ]|(y*x)^2*y^-1*x*(y*x^-1)^2*y>,
  <[ [ 6, 1 ] ]|y*(x*y^-1)^3*x^-1*y^-1*x^-1*y> ]
-> [ F1, F2, F3, F4, F5, F6 ]


F1 = (y@x)@(y@x)@(y@xi)@(y@xi)@(y@xi)
 1 0
-5 1
F2 = y@x@yi@x@y@(xi@yi)@(xi@yi)@xi
-2 5
3 -8
F3 = yi@x@y@x@(yi@xi)@(yi@xi)@y@xi
-7  5
 4 -3
F4 = (yi@x)@(yi@x)@(y@xi)@(y@xi)@yi@xi
-5  7
 2 -3
F5 = (y@x)@(y@x)@yi@x@(y@xi)@(y@xi)@y
-1 -4
 3 11
F6 = y@(x@yi)@(x@yi)@(x@yi)@xi@yi@xi@y
 4  5
-5 -6



пояснение к алгоритму из статьи
He Y-H and Read J (2015) Hecke Groups, Dessins d’Enfants, and the Archimedean Solids. Front. Phys. 3:91. doi: 10.3389/fphy.2015.00091

We can find the generators for a representative of all the conjugacy classes of subgroups of interest using GAP [6, 31].

1. First, we use the permutation data σ_0, σ_1 obtained from each of the Schreier coset graphs (in turn obtained from each of the dessins) to find the group homomorphism by images between the relevant Hecke group and a representative of the conjugacy class of subgroups of interest. 
2. We then use this to define the representative in question.
3. Finally, we use the GAP command IsomorphismFpGroup(G), which returns an isomorphism from the given representative to a finitely presented group isomorphic to that representative. This function first chooses a set of generators of the representative and then computes a presentation in terms of these generators.