INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF PURE MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 216 Minimal number of generators: 37 Number of equivalence classes of cusps: 24 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -3/1 -3/2 0/1 1/1 9/7 3/2 27/17 9/5 2/1 9/4 5/2 18/7 8/3 3/1 27/8 7/2 18/5 4/1 9/2 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 0/1 -5/1 -1/1 1/0 -9/2 -1/1 -4/1 -1/2 -7/2 -1/2 0/1 -3/1 -1/1 -1/3 -11/4 -1/2 0/1 -19/7 -1/1 0/1 -27/10 -1/1 -8/3 -1/2 -5/2 -1/2 -2/5 -12/5 -2/5 -19/8 -2/5 -3/8 -7/3 -3/8 -1/3 -9/4 -1/3 -2/1 -1/4 -9/5 -1/3 -1/5 -7/4 -1/4 0/1 -12/7 0/1 -5/3 -1/3 -1/4 -18/11 -1/4 -13/8 -1/4 -4/17 -8/5 -1/4 -3/2 -1/5 -10/7 -1/6 -27/19 -1/5 -1/7 -17/12 -1/6 0/1 -7/5 -1/5 -1/6 -18/13 -1/6 -11/8 -1/6 0/1 -4/3 -1/6 -9/7 -1/5 -1/7 -5/4 -1/6 0/1 -6/5 0/1 -7/6 -1/6 0/1 -1/1 -1/7 0/1 0/1 0/1 1/1 0/1 1/7 6/5 0/1 5/4 0/1 1/6 9/7 1/7 1/5 4/3 1/6 7/5 1/6 1/5 3/2 1/5 11/7 1/5 1/4 19/12 2/9 1/4 27/17 1/5 3/13 8/5 1/4 5/3 1/4 1/3 12/7 0/1 19/11 0/1 1/5 7/4 0/1 1/4 9/5 1/5 1/3 2/1 1/4 9/4 1/3 7/3 1/3 3/8 12/5 2/5 5/2 2/5 1/2 18/7 1/2 13/5 1/2 3/5 8/3 1/2 3/1 1/3 1/1 10/3 1/2 27/8 1/1 17/5 0/1 1/1 7/2 0/1 1/2 18/5 1/2 11/3 1/2 1/1 4/1 1/2 9/2 1/1 5/1 1/1 1/0 6/1 0/1 7/1 1/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(19,126,-8,-53) (-6/1,1/0) -> (-12/5,-19/8) Hyperbolic Matrix(17,90,10,53) (-6/1,-5/1) -> (5/3,12/7) Hyperbolic Matrix(19,90,4,19) (-5/1,-9/2) -> (9/2,5/1) Hyperbolic Matrix(17,72,4,17) (-9/2,-4/1) -> (4/1,9/2) Hyperbolic Matrix(19,72,-14,-53) (-4/1,-7/2) -> (-11/8,-4/3) Hyperbolic Matrix(17,54,-6,-19) (-7/2,-3/1) -> (-3/1,-11/4) Parabolic Matrix(125,342,72,197) (-11/4,-19/7) -> (19/11,7/4) Hyperbolic Matrix(359,972,106,287) (-19/7,-27/10) -> (27/8,17/5) Hyperbolic Matrix(181,486,54,145) (-27/10,-8/3) -> (10/3,27/8) Hyperbolic Matrix(55,144,-34,-89) (-8/3,-5/2) -> (-13/8,-8/5) Hyperbolic Matrix(37,90,30,73) (-5/2,-12/5) -> (6/5,5/4) Hyperbolic Matrix(145,342,92,217) (-19/8,-7/3) -> (11/7,19/12) Hyperbolic Matrix(55,126,24,55) (-7/3,-9/4) -> (9/4,7/3) Hyperbolic Matrix(17,36,8,17) (-9/4,-2/1) -> (2/1,9/4) Hyperbolic Matrix(19,36,10,19) (-2/1,-9/5) -> (9/5,2/1) Hyperbolic Matrix(71,126,40,71) (-9/5,-7/4) -> (7/4,9/5) Hyperbolic Matrix(73,126,-62,-107) (-7/4,-12/7) -> (-6/5,-7/6) Hyperbolic Matrix(53,90,10,17) (-12/7,-5/3) -> (5/1,6/1) Hyperbolic Matrix(197,324,76,125) (-5/3,-18/11) -> (18/7,13/5) Hyperbolic Matrix(199,324,78,127) (-18/11,-13/8) -> (5/2,18/7) Hyperbolic Matrix(35,54,-24,-37) (-8/5,-3/2) -> (-3/2,-10/7) Parabolic Matrix(341,486,214,305) (-10/7,-27/19) -> (27/17,8/5) Hyperbolic Matrix(685,972,432,613) (-27/19,-17/12) -> (19/12,27/17) Hyperbolic Matrix(89,126,12,17) (-17/12,-7/5) -> (7/1,1/0) Hyperbolic Matrix(233,324,64,89) (-7/5,-18/13) -> (18/5,11/3) Hyperbolic Matrix(235,324,66,91) (-18/13,-11/8) -> (7/2,18/5) Hyperbolic Matrix(55,72,42,55) (-4/3,-9/7) -> (9/7,4/3) Hyperbolic Matrix(71,90,56,71) (-9/7,-5/4) -> (5/4,9/7) Hyperbolic Matrix(73,90,30,37) (-5/4,-6/5) -> (12/5,5/2) Hyperbolic Matrix(109,126,32,37) (-7/6,-1/1) -> (17/5,7/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(107,-126,62,-73) (1/1,6/5) -> (12/7,19/11) Hyperbolic Matrix(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(53,-126,8,-19) (7/3,12/5) -> (6/1,7/1) Hyperbolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(19,126,-8,-53) -> Matrix(3,2,-8,-5) Matrix(17,90,10,53) -> Matrix(1,0,4,1) Matrix(19,90,4,19) -> Matrix(1,2,0,1) Matrix(17,72,4,17) -> Matrix(3,2,4,3) Matrix(19,72,-14,-53) -> Matrix(1,0,-4,1) Matrix(17,54,-6,-19) -> Matrix(1,0,0,1) Matrix(125,342,72,197) -> Matrix(1,0,6,1) Matrix(359,972,106,287) -> Matrix(1,0,2,1) Matrix(181,486,54,145) -> Matrix(3,2,4,3) Matrix(55,144,-34,-89) -> Matrix(3,2,-14,-9) Matrix(37,90,30,73) -> Matrix(5,2,32,13) Matrix(145,342,92,217) -> Matrix(11,4,52,19) Matrix(55,126,24,55) -> Matrix(17,6,48,17) Matrix(17,36,8,17) -> Matrix(7,2,24,7) Matrix(19,36,10,19) -> Matrix(1,0,8,1) Matrix(71,126,40,71) -> Matrix(1,0,8,1) Matrix(73,126,-62,-107) -> Matrix(1,0,-2,1) Matrix(53,90,10,17) -> Matrix(1,0,4,1) Matrix(197,324,76,125) -> Matrix(15,4,26,7) Matrix(199,324,78,127) -> Matrix(25,6,54,13) Matrix(35,54,-24,-37) -> Matrix(9,2,-50,-11) Matrix(341,486,214,305) -> Matrix(13,2,58,9) Matrix(685,972,432,613) -> Matrix(13,2,58,9) Matrix(89,126,12,17) -> Matrix(1,0,6,1) Matrix(233,324,64,89) -> Matrix(11,2,16,3) Matrix(235,324,66,91) -> Matrix(1,0,8,1) Matrix(55,72,42,55) -> Matrix(1,0,12,1) Matrix(71,90,56,71) -> Matrix(1,0,12,1) Matrix(73,90,30,37) -> Matrix(13,2,32,5) Matrix(109,126,32,37) -> Matrix(1,0,8,1) Matrix(1,0,2,1) -> Matrix(1,0,14,1) Matrix(107,-126,62,-73) -> Matrix(1,0,-2,1) Matrix(53,-72,14,-19) -> Matrix(1,0,-4,1) Matrix(37,-54,24,-35) -> Matrix(11,-2,50,-9) Matrix(89,-144,34,-55) -> Matrix(9,-2,14,-3) Matrix(53,-126,8,-19) -> Matrix(5,-2,8,-3) Matrix(19,-54,6,-17) -> Matrix(1,0,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 9 Degree of the the map X: 9 Degree of the the map Y: 36 Permutation triple for Y: ((2,6,23,13,4,3,12,24,7)(5,18,16,15,10,9,27,8,19)(11,31,36,26,21,20,35,34,14)(17,22,29)(25,28,33); (1,4,16,33,36,31,17,5,2)(3,10,11)(6,21,32,14,13,28,27,9,22)(7,26,8)(12,29,20,19,30,15,34,25,24); (1,2,8,28,34,35,29,9,3)(4,14,15)(5,20,6)(7,25,16,18,17,12,11,32,21)(10,30,19,26,33,13,23,22,31)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0 DeckMod(f) is trivial. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PGL(2,Z) OF THE GROUP OF EXTENDED MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 0/1 7 1 1/1 (0/1,1/7) 0 18 6/5 0/1 1 3 5/4 (0/1,1/6) 0 18 9/7 (0/1,1/6) 0 2 4/3 1/6 1 9 7/5 (1/6,1/5) 0 18 3/2 1/5 2 6 11/7 (1/5,1/4) 0 18 19/12 (2/9,1/4) 0 18 27/17 (2/9,1/4) 0 2 8/5 1/4 1 9 5/3 (1/4,1/3) 0 18 12/7 0/1 1 3 19/11 (0/1,1/5) 0 18 7/4 (0/1,1/4) 0 18 9/5 (0/1,1/4) 0 2 2/1 1/4 1 9 9/4 1/3 8 2 7/3 (1/3,3/8) 0 18 12/5 2/5 1 3 5/2 (2/5,1/2) 0 18 18/7 1/2 5 1 13/5 (1/2,3/5) 0 18 8/3 1/2 1 9 3/1 0 6 10/3 1/2 1 9 27/8 1/1 2 2 17/5 (0/1,1/1) 0 18 7/2 (0/1,1/2) 0 18 18/5 1/2 1 1 11/3 (1/2,1/1) 0 18 4/1 1/2 1 9 9/2 1/1 4 2 5/1 (1/1,1/0) 0 18 6/1 0/1 1 3 7/1 (1/1,1/0) 0 18 1/0 (0/1,1/0) 0 18 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(107,-126,62,-73) (1/1,6/5) -> (12/7,19/11) Hyperbolic Matrix(73,-90,30,-37) (6/5,5/4) -> (12/5,5/2) Glide Reflection Matrix(71,-90,56,-71) (5/4,9/7) -> (5/4,9/7) Reflection Matrix(55,-72,42,-55) (9/7,4/3) -> (9/7,4/3) Reflection Matrix(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(91,-144,12,-19) (11/7,19/12) -> (7/1,1/0) Glide Reflection Matrix(647,-1026,408,-647) (19/12,27/17) -> (19/12,27/17) Reflection Matrix(271,-432,170,-271) (27/17,8/5) -> (27/17,8/5) Reflection Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(53,-90,10,-17) (5/3,12/7) -> (5/1,6/1) Glide Reflection Matrix(145,-252,42,-73) (19/11,7/4) -> (17/5,7/2) Glide Reflection Matrix(71,-126,40,-71) (7/4,9/5) -> (7/4,9/5) Reflection Matrix(19,-36,10,-19) (9/5,2/1) -> (9/5,2/1) Reflection Matrix(17,-36,8,-17) (2/1,9/4) -> (2/1,9/4) Reflection Matrix(55,-126,24,-55) (9/4,7/3) -> (9/4,7/3) Reflection Matrix(53,-126,8,-19) (7/3,12/5) -> (6/1,7/1) Hyperbolic Matrix(71,-180,28,-71) (5/2,18/7) -> (5/2,18/7) Reflection Matrix(181,-468,70,-181) (18/7,13/5) -> (18/7,13/5) Reflection Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(161,-540,48,-161) (10/3,27/8) -> (10/3,27/8) Reflection Matrix(271,-918,80,-271) (27/8,17/5) -> (27/8,17/5) Reflection Matrix(71,-252,20,-71) (7/2,18/5) -> (7/2,18/5) Reflection Matrix(109,-396,30,-109) (18/5,11/3) -> (18/5,11/3) Reflection Matrix(17,-72,4,-17) (4/1,9/2) -> (4/1,9/2) Reflection Matrix(19,-90,4,-19) (9/2,5/1) -> (9/2,5/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,2,-1) -> Matrix(1,0,14,-1) (0/1,1/1) -> (0/1,1/7) Matrix(107,-126,62,-73) -> Matrix(1,0,-2,1) 0/1 Matrix(73,-90,30,-37) -> Matrix(13,-2,32,-5) Matrix(71,-90,56,-71) -> Matrix(1,0,12,-1) (5/4,9/7) -> (0/1,1/6) Matrix(55,-72,42,-55) -> Matrix(1,0,12,-1) (9/7,4/3) -> (0/1,1/6) Matrix(53,-72,14,-19) -> Matrix(1,0,-4,1) 0/1 Matrix(37,-54,24,-35) -> Matrix(11,-2,50,-9) 1/5 Matrix(91,-144,12,-19) -> Matrix(9,-2,4,-1) Matrix(647,-1026,408,-647) -> Matrix(17,-4,72,-17) (19/12,27/17) -> (2/9,1/4) Matrix(271,-432,170,-271) -> Matrix(17,-4,72,-17) (27/17,8/5) -> (2/9,1/4) Matrix(89,-144,34,-55) -> Matrix(9,-2,14,-3) Matrix(53,-90,10,-17) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(145,-252,42,-73) -> Matrix(1,0,6,-1) *** -> (0/1,1/3) Matrix(71,-126,40,-71) -> Matrix(1,0,8,-1) (7/4,9/5) -> (0/1,1/4) Matrix(19,-36,10,-19) -> Matrix(1,0,8,-1) (9/5,2/1) -> (0/1,1/4) Matrix(17,-36,8,-17) -> Matrix(7,-2,24,-7) (2/1,9/4) -> (1/4,1/3) Matrix(55,-126,24,-55) -> Matrix(17,-6,48,-17) (9/4,7/3) -> (1/3,3/8) Matrix(53,-126,8,-19) -> Matrix(5,-2,8,-3) 1/2 Matrix(71,-180,28,-71) -> Matrix(9,-4,20,-9) (5/2,18/7) -> (2/5,1/2) Matrix(181,-468,70,-181) -> Matrix(11,-6,20,-11) (18/7,13/5) -> (1/2,3/5) Matrix(19,-54,6,-17) -> Matrix(1,0,0,1) Matrix(161,-540,48,-161) -> Matrix(3,-2,4,-3) (10/3,27/8) -> (1/2,1/1) Matrix(271,-918,80,-271) -> Matrix(1,0,2,-1) (27/8,17/5) -> (0/1,1/1) Matrix(71,-252,20,-71) -> Matrix(1,0,4,-1) (7/2,18/5) -> (0/1,1/2) Matrix(109,-396,30,-109) -> Matrix(3,-2,4,-3) (18/5,11/3) -> (1/2,1/1) Matrix(17,-72,4,-17) -> Matrix(3,-2,4,-3) (4/1,9/2) -> (1/2,1/1) Matrix(19,-90,4,-19) -> Matrix(-1,2,0,1) (9/2,5/1) -> (1/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.