== A. the cover phi_0 of Theorem 1.1 ==
PASS p = 3: I = [[0,-1],[1,0]], J = [[1,1],[1,-1]]
PASS Q_8 = <I,J> has order 8, I^2 = J^2 = -1, IJ = -JI
PASS Q_8 lies in SL(2,3)
PASS the six elements of order 4 of Q_8 have no eigenvalue in F_3 (no invariant line)
PASS -I = [[0,1],[2,0]], IJ = [[2,1],[1,1]]
PASS image lists: T = 120453786, U = 063174285, W = 138570624
PASS cycles: T = (0 1 2)(3 4 5)(6 7 8), U = (1 6 2 3)(4 7 8 5), W = (0 1 3 5)(2 8 4 7)
PASS [T,U] = t_(1,1), [T,W] = t_(2,2), and [T,U][T,W] = 1 (composition of maps, [p,q] = p q p^-1 q^-1)
PASS convention: maps with [p,q] = p^-1 q^-1 p q: the relation FAILS for the printed permutations
PASS convention: left-to-right products with [p,q] = p q p^-1 q^-1: the relation FAILS
PASS convention: left-to-right products with [p,q] = p^-1 q^-1 p q (GAP): the relation holds
PASS convention: the inverse permutations satisfy the relation for left-to-right products and p q p^-1 q^-1
PASS u^-1 x u (left to right) is the same permutation as u x u^-1 (maps)
PASS <T,U,W> has order 72 and equals {z -> Az+t : A in Q_8, t in F_3^2} = M_9
PASS M_9 is sharply 2-transitive on the 9 points
PASS element orders in M_9: 1, 9, 8, 54 elements of orders 1, 2, 3, 4
== B. the central extension G~ = E x| Q_8 (order 216) ==
PASS the product rule is defined on 216 elements (rows are permutations)
PASS the product rule is associative (all 216^3 triples)
PASS inverse: [t,s;A]^-1 = [-A^-1 t, -s; A^-1]
PASS the centre of G~ is Z = {[0,s;1]}, of order 3
PASS [t,s;A] -> (A|t) is a homomorphism G~ -> M_9 (all 216^2 products)
PASS it is onto M_9 with kernel Z
PASS E = {[t,s;1]} has order 27, exponent 3 and is not abelian
PASS [[t,0;1],[t',0;1]] = [0, omega(t,t'); 1]
PASS conjugation formula (7): g [a,0;1] g^-1 = [Aa, omega(c,Aa); 1] for g = [c,s;A]
PASS commutator formula (8): [[a,0;1], g] = [(1-A)a, -omega(c,Aa) - omega(a,Aa)/2; 1]
PASS element orders in G~: [(1, 1), (2, 9), (3, 26), (4, 54), (6, 18), (12, 108)]
PASS every element of order 1, 2 or 4 of M_9 has exactly one lift of the same order
== C. the lifting obstruction of phi_0 ==
PASS [x~,u~] = [(1,1),2;1] and [v~,w~] = [(2,2),0;1] for the lifts [e,0;1], [0,0;-I], [e,0;1], [e,0;IJ]
PASS kappa(phi_0) = [(0,0),2;1], not the identity
PASS kappa(phi_0) = 2 for all 81 choices of the lifts
== D. all homomorphisms pi_1(S_2) -> M_9; the main lemma for M_9 ==
PASS M_9 has 68 subgroups
PASS a subgroup of M_9 is transitive iff it contains V
PASS 62 intransitive subgroups: 46 two-groups meeting V trivially, 4 of order 3, 12 of order 6 with linear parts +-1
PASS number of homomorphisms pi_1(S_2) -> M_9: 1592136 = 72(4*72^2 + 36^2 + 9^2)
PASS epimorphisms: 1036800
PASS liftable epimorphisms: 414720, of which 25920 with J intransitive and 388800 with J transitive
PASS non-liftable epimorphisms: 622080, ALL with J = <x, u x u^-1, v, w v w^-1> transitive (Lemma 5.1 for M_9)
PASS 622080 = 1440 * |AGL(2,3)| = 8640 * |M_9|
PASS generating tuples with J intransitive: 19008 with J n V = 1 and 6912 with |J n V| = 3; x, v have the orders used in the proof
PASS J = <x, [x,u], v> (checked on every 97th homomorphism)
PASS homomorphisms that are not onto and do not lift: 321840 (the generation hypothesis is needed)
PASS example: x = 1, v = t_(1,0), w = t_(0,1): relation holds, J = <v> is intransitive, kappa = 1 != 0
== E. the orbit of phi_0 under the twist substitutions ==
PASS the ten substitutions A1,B1,C,B2,A2 and G1,...,G5 are automorphisms of the free group mapping the relator to a conjugate of itself
PASS [a, ba] = [a,b] and [ab, b] = [a,b] in the free group (handle moves)
PASS the moves on tuples are the precompositions of phi with the substitutions
PASS the orbit of phi_0 has 1440 classes modulo relabelling of the 9 points, for both sets of five substitutions
PASS on all 1440 classes J is transitive; |J| = 9, 18, 36, 72 on 24, 72, 576, 768 classes
PASS <x,v> is intransitive on 190 classes and <x, u x u^-1> on 240 classes (pairs of pants with disconnected preimage)
PASS 1440 * 9! = 522547200
== F. pairs of pants and the Galois closure for phi_0 ==
PASS <phi_0(a_1), phi_0(a_2)> = <T> has order 3 and three orbits
PASS (a_1 b_1^2, b_1, a_2, b_2) is mapped to a tuple satisfying the relation
PASS <x u^2, u (x u^2) u^-1> has order 6 and two orbits
PASS phi_0(pi_1 H_0) is the translation group V, of order 9 and index 8 in M_9
== G. the family Gamma_p = F_p^2 x| Q_8, p = 3 mod 4 ==
PASS p =  3: (al,be) = (1,1); Q_8 in SL(2,p), no eigenvalues for order 4; b = (be-al-1, 1-al-be)/2, omega(b,Jb) = al; kappa = -c1 - (1+al)/2 for all c1; image of order 72 = 8p^2
PASS p =  7: (al,be) = (2,3); Q_8 in SL(2,p), no eigenvalues for order 4; b = (be-al-1, 1-al-be)/2, omega(b,Jb) = al; kappa = -c1 - (1+al)/2 for all c1; image of order 392 = 8p^2
PASS p = 11: (al,be) = (1,3); Q_8 in SL(2,p), no eigenvalues for order 4; b = (be-al-1, 1-al-be)/2, omega(b,Jb) = al; kappa = -c1 - (1+al)/2 for all c1; image of order 968 = 8p^2
PASS p = 19: (al,be) = (1,6); Q_8 in SL(2,p), no eigenvalues for order 4; b = (be-al-1, 1-al-be)/2, omega(b,Jb) = al; kappa = -c1 - (1+al)/2 for all c1; image of order 2888 = 8p^2
PASS p = 23: (al,be) = (2,8); Q_8 in SL(2,p), no eigenvalues for order 4; b = (be-al-1, 1-al-be)/2, omega(b,Jb) = al; kappa = -c1 - (1+al)/2 for all c1; image of order 4232 = 8p^2
PASS p = 31: (al,be) = (4,13); Q_8 in SL(2,p), no eigenvalues for order 4; b = (be-al-1, 1-al-be)/2, omega(b,Jb) = al; kappa = -c1 - (1+al)/2 for all c1
PASS p = 43: (al,be) = (1,16); Q_8 in SL(2,p), no eigenvalues for order 4; b = (be-al-1, 1-al-be)/2, omega(b,Jb) = al; kappa = -c1 - (1+al)/2 for all c1
PASS p = 47: (al,be) = (2,18); Q_8 in SL(2,p), no eigenvalues for order 4; b = (be-al-1, 1-al-be)/2, omega(b,Jb) = al; kappa = -c1 - (1+al)/2 for all c1
PASS p = 59: (al,be) = (1,23); Q_8 in SL(2,p), no eigenvalues for order 4; b = (be-al-1, 1-al-be)/2, omega(b,Jb) = al; kappa = -c1 - (1+al)/2 for all c1
PASS p = 67: (al,be) = (1,20); Q_8 in SL(2,p), no eigenvalues for order 4; b = (be-al-1, 1-al-be)/2, omega(b,Jb) = al; kappa = -c1 - (1+al)/2 for all c1
PASS p = 71: (al,be) = (4,14); Q_8 in SL(2,p), no eigenvalues for order 4; b = (be-al-1, 1-al-be)/2, omega(b,Jb) = al; kappa = -c1 - (1+al)/2 for all c1
PASS p = 79: (al,be) = (4,33); Q_8 in SL(2,p), no eigenvalues for order 4; b = (be-al-1, 1-al-be)/2, omega(b,Jb) = al; kappa = -c1 - (1+al)/2 for all c1
PASS p = 83: (al,be) = (1,9); Q_8 in SL(2,p), no eigenvalues for order 4; b = (be-al-1, 1-al-be)/2, omega(b,Jb) = al; kappa = -c1 - (1+al)/2 for all c1
== H. p = 5 and p = 13: what fails for p = 1 mod 4 ==
PASS p = 5: the printed system satisfies the relation, generates Gamma_p (order 200), has kappa = 4 != 0, and J has order 10 (intransitive); the elements of order 4 have eigenvalues
PASS p = 5: the intermediate values printed in the proof of Proposition 6.1 ([x~,u~] = [(3,0),0;1], v~w~ = [(0,3),0;J], w~v~ = [(3,3),3;J], [v~,w~] = [(2,0),4;1])
PASS p = 13: the printed system satisfies the relation, generates Gamma_p (order 1352), has kappa = 3 != 0, and J has order 26 (intransitive); the elements of order 4 have eigenvalues
== I. identities (5) and (10), used for genus >= 3 (Proposition 1.3) ==
PASS rho_c rho_d = t_{2(c-d)}, t_d rho_c = rho_{c+d/2}, and t_alpha rho_e t_alpha^k = rho_{e+(1-k)alpha/2} (p = 3, 5, 7)
PASS [A,B] = (-1)^det(Abar,Bbar) for A, B in Q_8 (p = 3, 7)
== J. hypothesis (E) of Lemma 5.1 for the groups of Remark 6.3 ==
PASS hypothesis (E) holds for Q_16 < SL(2,7), Dic_3 < SL(2,11) and SL(2,3) < SL(2,11)
PASS hypothesis (E) fails for Q_8 < SL(2,5), Q_8 < SL(2,13), SL(2,3) < SL(2,5) and SL(2,3) < SL(2,7)
== K. the cyclic order (4) and the boundary curves of Lemma 2.2 and Lemma 2.3 (face tracing) ==
PASS with the cyclic order (4), the wedge of all 2g loops has one boundary curve, with word [a_1,b_1]...[a_g,b_g] (g = 1,...,4)
PASS the diagonal delta_i, placed as stated after (4), cuts the polygon into a_i b_i a_i^-1 b_i^-1 delta_i^-1 and the rest (g = 2, 3, 4)
PASS the boundary curves of P(a_i,a_j), P(a_i,b_i), H(a_i,b_i;a_j) and H(a_1,a_2,a_3) are as stated in Lemma 2.3 (numbers of boundary curves 3, 3, 4, 4, so the surfaces are planar)
PASS [a_1,b_1] a_2 (b_2 a_2 b_2^-1)^-1 is the relator: the fourth boundary curve of H_0 is b_2 a_2 b_2^-1 in pi_1(S_2)
TOTAL failures: 0   (time 5 s)
