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): 192 Minimal number of generators: 33 Number of equivalence classes of cusps: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -2/5 -1/3 -1/5 0/1 1/5 2/7 1/3 1/2 3/5 5/7 4/5 1/1 13/11 7/5 3/2 2/1 8/3 3/1 7/2 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/5 -3/7 1/4 -2/5 0/1 1/5 1/4 -5/13 1/4 -3/8 0/1 1/4 -4/11 0/1 1/5 -1/3 1/4 -3/10 0/1 1/4 -2/7 1/4 1/3 -1/4 1/3 -3/13 1/2 -2/9 0/1 1/3 -3/14 0/1 1/3 -1/5 1/4 1/2 -1/6 0/1 1/2 0/1 0/1 1/3 1/6 0/1 1/2 1/5 0/1 1/4 0/1 1/4 3/11 1/4 2/7 0/1 1/4 1/3 1/3 1/4 1/2 4/11 0/1 1/4 1/3 7/19 1/4 3/8 0/1 1/3 2/5 0/1 1/3 1/2 1/3 4/7 1/3 2/5 7/12 2/5 1/2 3/5 1/2 5/8 0/1 1/3 2/3 1/3 1/2 5/7 1/2 8/11 1/2 1/1 3/4 0/1 1/2 7/9 1/4 1/2 4/5 0/1 1/3 1/2 9/11 1/4 1/2 5/6 0/1 1/3 1/1 1/2 7/6 2/3 1/1 13/11 1/1 6/5 0/1 1/1 5/4 1/1 4/3 0/1 1/1 7/5 0/1 10/7 0/1 1/3 3/2 0/1 1/2 2/1 0/1 1/2 1/1 5/2 0/1 1/2 13/5 0/1 21/8 0/1 1/4 8/3 0/1 1/3 3/1 1/2 10/3 2/3 1/1 37/11 1/1 27/8 0/1 1/1 17/5 1/2 7/2 1/1 4/1 1/2 1/1 9/2 2/3 1/1 5/1 1/1 11/2 0/1 1/1 6/1 0/1 1/1 1/0 0/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(41,18,66,29) (-1/2,-3/7) -> (3/5,5/8) Hyperbolic Matrix(39,16,-100,-41) (-3/7,-2/5) -> (-2/5,-5/13) Parabolic Matrix(115,44,196,75) (-5/13,-3/8) -> (7/12,3/5) Hyperbolic Matrix(113,42,78,29) (-3/8,-4/11) -> (10/7,3/2) Hyperbolic Matrix(39,14,-170,-61) (-4/11,-1/3) -> (-3/13,-2/9) Hyperbolic Matrix(33,10,122,37) (-1/3,-3/10) -> (1/4,3/11) Hyperbolic Matrix(101,30,138,41) (-3/10,-2/7) -> (8/11,3/4) Hyperbolic Matrix(65,18,18,5) (-2/7,-1/4) -> (7/2,4/1) Hyperbolic Matrix(159,38,46,11) (-1/4,-3/13) -> (17/5,7/2) Hyperbolic Matrix(183,40,32,7) (-2/9,-3/14) -> (11/2,6/1) Hyperbolic Matrix(151,32,184,39) (-3/14,-1/5) -> (9/11,5/6) Hyperbolic Matrix(57,10,74,13) (-1/5,-1/6) -> (3/4,7/9) Hyperbolic Matrix(1,0,12,1) (-1/6,0/1) -> (0/1,1/6) Parabolic Matrix(183,-34,70,-13) (1/6,1/5) -> (13/5,21/8) Hyperbolic Matrix(77,-18,30,-7) (1/5,1/4) -> (5/2,13/5) Hyperbolic Matrix(93,-26,254,-71) (3/11,2/7) -> (4/11,7/19) Hyperbolic Matrix(19,-6,54,-17) (2/7,1/3) -> (1/3,4/11) Parabolic Matrix(649,-240,192,-71) (7/19,3/8) -> (27/8,17/5) Hyperbolic Matrix(53,-20,8,-3) (3/8,2/5) -> (6/1,1/0) Hyperbolic Matrix(33,-14,26,-11) (2/5,1/2) -> (5/4,4/3) Hyperbolic Matrix(47,-26,38,-21) (1/2,4/7) -> (6/5,5/4) Hyperbolic Matrix(243,-140,92,-53) (4/7,7/12) -> (21/8,8/3) Hyperbolic Matrix(59,-38,14,-9) (5/8,2/3) -> (4/1,9/2) Hyperbolic Matrix(71,-50,98,-69) (2/3,5/7) -> (5/7,8/11) Parabolic Matrix(81,-64,100,-79) (7/9,4/5) -> (4/5,9/11) Parabolic Matrix(13,-12,12,-11) (5/6,1/1) -> (1/1,7/6) Parabolic Matrix(519,-610,154,-181) (7/6,13/11) -> (37/11,27/8) Hyperbolic Matrix(295,-352,88,-105) (13/11,6/5) -> (10/3,37/11) Hyperbolic Matrix(71,-98,50,-69) (4/3,7/5) -> (7/5,10/7) Parabolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(21,-100,4,-19) (9/2,5/1) -> (5/1,11/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(41,18,66,29) -> Matrix(1,0,-2,1) Matrix(39,16,-100,-41) -> Matrix(1,0,0,1) Matrix(115,44,196,75) -> Matrix(7,-2,18,-5) Matrix(113,42,78,29) -> Matrix(1,0,-2,1) Matrix(39,14,-170,-61) -> Matrix(1,0,-2,1) Matrix(33,10,122,37) -> Matrix(1,0,0,1) Matrix(101,30,138,41) -> Matrix(1,0,-2,1) Matrix(65,18,18,5) -> Matrix(1,0,-2,1) Matrix(159,38,46,11) -> Matrix(5,-2,8,-3) Matrix(183,40,32,7) -> Matrix(1,0,-2,1) Matrix(151,32,184,39) -> Matrix(1,0,0,1) Matrix(57,10,74,13) -> Matrix(1,0,0,1) Matrix(1,0,12,1) -> Matrix(1,0,0,1) Matrix(183,-34,70,-13) -> Matrix(1,0,2,1) Matrix(77,-18,30,-7) -> Matrix(1,0,-2,1) Matrix(93,-26,254,-71) -> Matrix(1,0,0,1) Matrix(19,-6,54,-17) -> Matrix(1,0,0,1) Matrix(649,-240,192,-71) -> Matrix(1,0,-2,1) Matrix(53,-20,8,-3) -> Matrix(1,0,-2,1) Matrix(33,-14,26,-11) -> Matrix(1,0,-2,1) Matrix(47,-26,38,-21) -> Matrix(5,-2,8,-3) Matrix(243,-140,92,-53) -> Matrix(5,-2,18,-7) Matrix(59,-38,14,-9) -> Matrix(5,-2,8,-3) Matrix(71,-50,98,-69) -> Matrix(5,-2,8,-3) Matrix(81,-64,100,-79) -> Matrix(1,0,0,1) Matrix(13,-12,12,-11) -> Matrix(5,-2,8,-3) Matrix(519,-610,154,-181) -> Matrix(3,-2,2,-1) Matrix(295,-352,88,-105) -> Matrix(1,-2,2,-3) Matrix(71,-98,50,-69) -> Matrix(1,0,2,1) Matrix(9,-16,4,-7) -> Matrix(1,0,0,1) Matrix(19,-54,6,-17) -> Matrix(5,-2,8,-3) Matrix(21,-100,4,-19) -> Matrix(3,-2,2,-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): 5 Degree of the the map X: 5 Degree of the the map Y: 32 Permutation triple for Y: ((1,6,20,21,7,2)(3,4)(5,15,23,10,9,16)(8,25)(11,27,32,29,17,28)(12,24,22,26,18,13)(14,30)(19,31); (1,4,13,29,14,5)(3,10,11)(6,18,19)(7,23,31,27,24,8)(9,26)(16,25,17)(20,28)(21,30,22); (1,2,8,16,26,30,29,32,31,18,9,3)(4,11,20,19,23,15,14,21,28,25,24,12)(5,17,13,6)(7,22,27,10)) ----------------------------------------------------------------------- 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 -1/1 0/1 2 1 0/1 (0/1,1/3) 0 12 1/5 0/1 2 2 1/4 (0/1,1/4) 0 12 2/7 0 4 1/3 0 3 4/11 0 4 3/8 (0/1,1/3) 0 12 2/5 (0/1,1/3) 0 12 1/2 1/3 1 4 4/7 (1/3,2/5) 0 12 3/5 1/2 1 3 2/3 (1/3,1/2) 0 12 5/7 1/2 1 1 3/4 (0/1,1/2) 0 12 4/5 0 4 5/6 (0/1,1/3) 0 12 1/1 1/2 1 6 7/6 (2/3,1/1) 0 12 13/11 1/1 2 1 6/5 (0/1,1/1) 0 12 5/4 1/1 1 4 4/3 (0/1,1/1) 0 12 7/5 0/1 1 1 3/2 (0/1,1/2) 0 12 2/1 0 4 5/2 (0/1,1/2) 0 12 13/5 0/1 2 2 8/3 (0/1,1/3) 0 12 3/1 1/2 1 3 4/1 (1/2,1/1) 0 12 5/1 1/1 1 2 6/1 (0/1,1/1) 0 12 1/0 (0/1,1/1) 0 12 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,10,-1) (0/1,1/5) -> (0/1,1/5) Reflection Matrix(77,-18,30,-7) (1/5,1/4) -> (5/2,13/5) Hyperbolic Matrix(37,-10,48,-13) (1/4,2/7) -> (3/4,4/5) Glide Reflection Matrix(19,-6,54,-17) (2/7,1/3) -> (1/3,4/11) Parabolic Matrix(87,-32,106,-39) (4/11,3/8) -> (4/5,5/6) Glide Reflection Matrix(53,-20,8,-3) (3/8,2/5) -> (6/1,1/0) Hyperbolic Matrix(33,-14,26,-11) (2/5,1/2) -> (5/4,4/3) Hyperbolic Matrix(47,-26,38,-21) (1/2,4/7) -> (6/5,5/4) Hyperbolic Matrix(61,-36,22,-13) (4/7,3/5) -> (8/3,3/1) Glide Reflection Matrix(29,-18,8,-5) (3/5,2/3) -> (3/1,4/1) Glide Reflection Matrix(29,-20,42,-29) (2/3,5/7) -> (2/3,5/7) Reflection Matrix(41,-30,56,-41) (5/7,3/4) -> (5/7,3/4) Reflection Matrix(13,-12,12,-11) (5/6,1/1) -> (1/1,7/6) Parabolic Matrix(155,-182,132,-155) (7/6,13/11) -> (7/6,13/11) Reflection Matrix(131,-156,110,-131) (13/11,6/5) -> (13/11,6/5) Reflection Matrix(41,-56,30,-41) (4/3,7/5) -> (4/3,7/5) Reflection Matrix(29,-42,20,-29) (7/5,3/2) -> (7/5,3/2) Reflection Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(79,-208,30,-79) (13/5,8/3) -> (13/5,8/3) Reflection Matrix(9,-40,2,-9) (4/1,5/1) -> (4/1,5/1) Reflection Matrix(11,-60,2,-11) (5/1,6/1) -> (5/1,6/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,2,-1) (-1/1,1/0) -> (0/1,1/1) Matrix(-1,0,2,1) -> Matrix(1,0,6,-1) (-1/1,0/1) -> (0/1,1/3) Matrix(1,0,10,-1) -> Matrix(1,0,6,-1) (0/1,1/5) -> (0/1,1/3) Matrix(77,-18,30,-7) -> Matrix(1,0,-2,1) 0/1 Matrix(37,-10,48,-13) -> Matrix(1,0,6,-1) *** -> (0/1,1/3) Matrix(19,-6,54,-17) -> Matrix(1,0,0,1) Matrix(87,-32,106,-39) -> Matrix(1,0,6,-1) *** -> (0/1,1/3) Matrix(53,-20,8,-3) -> Matrix(1,0,-2,1) 0/1 Matrix(33,-14,26,-11) -> Matrix(1,0,-2,1) 0/1 Matrix(47,-26,38,-21) -> Matrix(5,-2,8,-3) 1/2 Matrix(61,-36,22,-13) -> Matrix(5,-2,12,-5) *** -> (1/3,1/2) Matrix(29,-18,8,-5) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(29,-20,42,-29) -> Matrix(5,-2,12,-5) (2/3,5/7) -> (1/3,1/2) Matrix(41,-30,56,-41) -> Matrix(1,0,4,-1) (5/7,3/4) -> (0/1,1/2) Matrix(13,-12,12,-11) -> Matrix(5,-2,8,-3) 1/2 Matrix(155,-182,132,-155) -> Matrix(5,-4,6,-5) (7/6,13/11) -> (2/3,1/1) Matrix(131,-156,110,-131) -> Matrix(1,0,2,-1) (13/11,6/5) -> (0/1,1/1) Matrix(41,-56,30,-41) -> Matrix(1,0,2,-1) (4/3,7/5) -> (0/1,1/1) Matrix(29,-42,20,-29) -> Matrix(1,0,4,-1) (7/5,3/2) -> (0/1,1/2) Matrix(9,-16,4,-7) -> Matrix(1,0,0,1) Matrix(79,-208,30,-79) -> Matrix(1,0,6,-1) (13/5,8/3) -> (0/1,1/3) Matrix(9,-40,2,-9) -> Matrix(3,-2,4,-3) (4/1,5/1) -> (1/2,1/1) Matrix(11,-60,2,-11) -> Matrix(1,0,2,-1) (5/1,6/1) -> (0/1,1/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.