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): 288 Minimal number of generators: 49 Number of equivalence classes of cusps: 32 Genus: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -5/1 -4/1 -7/2 -3/1 -8/3 -9/4 -2/1 -1/1 -8/13 -6/11 -1/2 -4/9 -2/7 0/1 1/4 3/7 1/2 2/3 1/1 6/5 11/9 4/3 3/2 2/1 7/3 5/2 8/3 3/1 4/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -2/1 0/1 -11/2 1/0 -5/1 -3/2 1/0 -9/2 -3/2 -4/1 -1/1 1/0 -7/2 -1/1 -10/3 -1/1 -2/3 -3/1 -1/2 1/0 -8/3 -1/1 -13/5 -3/4 -1/2 -18/7 -1/2 0/1 -5/2 -1/2 -12/5 0/1 1/0 -7/3 -1/2 1/0 -9/4 -1/1 -2/1 -1/1 0/1 -1/1 0/1 -2/3 0/1 1/1 -9/14 1/1 -7/11 1/2 1/0 -12/19 0/1 1/0 -5/8 1/2 -8/13 1/1 -11/18 1/0 -3/5 1/2 1/0 -10/17 2/3 1/1 -7/12 1/1 -4/7 1/1 1/0 -9/16 3/2 -5/9 3/2 1/0 -6/11 0/1 2/1 -1/2 1/0 -4/9 1/0 -3/7 -3/2 1/0 -5/12 -3/2 -2/5 -1/1 0/1 -1/3 -1/2 1/0 -2/7 0/1 -3/11 1/4 1/2 -1/4 1/0 -2/9 0/1 1/1 -1/5 -1/2 1/0 0/1 0/1 1/0 1/5 -1/2 1/0 1/4 0/1 3/11 1/4 1/2 5/18 1/2 2/7 0/1 1/2 3/10 1/2 1/3 1/2 1/0 4/11 1/0 3/8 1/0 5/13 -3/2 1/0 2/5 -1/1 0/1 5/12 -1/2 3/7 0/1 7/16 1/4 4/9 0/1 1/2 1/2 1/0 2/3 0/1 3/4 1/4 7/9 1/4 1/2 4/5 1/3 1/2 9/11 1/2 5/6 1/2 1/1 1/2 1/0 7/6 1/0 6/5 2/1 1/0 11/9 1/0 5/4 1/0 14/11 -2/1 0/1 9/7 -1/2 1/0 4/3 0/1 1/0 3/2 0/1 8/5 0/1 1/2 21/13 1/4 1/2 13/8 1/2 31/19 1/2 18/11 1/2 2/3 5/3 1/2 1/0 2/1 0/1 1/1 9/4 3/4 7/3 1/1 19/8 7/6 12/5 1/1 5/4 5/2 3/2 13/5 5/2 1/0 21/8 1/0 29/11 1/0 8/3 1/1 1/0 19/7 1/2 1/0 11/4 1/1 3/1 3/2 1/0 4/1 1/0 5/1 -1/2 1/0 6/1 0/1 1/1 7/1 3/2 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(11,72,-2,-13) (-6/1,1/0) -> (-6/1,-11/2) Parabolic Matrix(55,296,34,183) (-11/2,-5/1) -> (21/13,13/8) Hyperbolic Matrix(5,24,16,77) (-5/1,-9/2) -> (3/10,1/3) Hyperbolic Matrix(29,128,12,53) (-9/2,-4/1) -> (12/5,5/2) Hyperbolic Matrix(15,56,-26,-97) (-4/1,-7/2) -> (-7/12,-4/7) Hyperbolic Matrix(41,140,-70,-239) (-7/2,-10/3) -> (-10/17,-7/12) Hyperbolic Matrix(17,56,44,145) (-10/3,-3/1) -> (5/13,2/5) Hyperbolic Matrix(47,128,-18,-49) (-3/1,-8/3) -> (-8/3,-13/5) Parabolic Matrix(125,324,76,197) (-13/5,-18/7) -> (18/11,5/3) Hyperbolic Matrix(39,100,140,359) (-18/7,-5/2) -> (5/18,2/7) Hyperbolic Matrix(13,32,28,69) (-5/2,-12/5) -> (4/9,1/2) Hyperbolic Matrix(57,136,44,105) (-12/5,-7/3) -> (9/7,4/3) Hyperbolic Matrix(45,104,16,37) (-7/3,-9/4) -> (11/4,3/1) Hyperbolic Matrix(17,36,-26,-55) (-9/4,-2/1) -> (-2/3,-9/14) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(387,248,142,91) (-9/14,-7/11) -> (19/7,11/4) Hyperbolic Matrix(487,308,302,191) (-7/11,-12/19) -> (8/5,21/13) Hyperbolic Matrix(165,104,376,237) (-12/19,-5/8) -> (7/16,4/9) Hyperbolic Matrix(207,128,-338,-209) (-5/8,-8/13) -> (-8/13,-11/18) Parabolic Matrix(53,32,48,29) (-11/18,-3/5) -> (1/1,7/6) Hyperbolic Matrix(149,88,22,13) (-3/5,-10/17) -> (6/1,7/1) Hyperbolic Matrix(325,184,136,77) (-4/7,-9/16) -> (19/8,12/5) Hyperbolic Matrix(93,52,338,189) (-9/16,-5/9) -> (3/11,5/18) Hyperbolic Matrix(225,124,176,97) (-5/9,-6/11) -> (14/11,9/7) Hyperbolic Matrix(83,44,66,35) (-6/11,-1/2) -> (5/4,14/11) Hyperbolic Matrix(35,16,94,43) (-1/2,-4/9) -> (4/11,3/8) Hyperbolic Matrix(37,16,104,45) (-4/9,-3/7) -> (1/3,4/11) Hyperbolic Matrix(105,44,136,57) (-3/7,-5/12) -> (3/4,7/9) Hyperbolic Matrix(69,28,32,13) (-5/12,-2/5) -> (2/1,9/4) Hyperbolic Matrix(31,12,18,7) (-2/5,-1/3) -> (5/3,2/1) Hyperbolic Matrix(27,8,-98,-29) (-1/3,-2/7) -> (-2/7,-3/11) Parabolic Matrix(107,28,42,11) (-3/11,-1/4) -> (5/2,13/5) Hyperbolic Matrix(53,12,128,29) (-1/4,-2/9) -> (2/5,5/12) Hyperbolic Matrix(75,16,14,3) (-2/9,-1/5) -> (5/1,6/1) Hyperbolic Matrix(1,0,10,1) (-1/5,0/1) -> (0/1,1/5) Parabolic Matrix(17,-4,64,-15) (1/5,1/4) -> (1/4,3/11) Parabolic Matrix(109,-32,92,-27) (2/7,3/10) -> (7/6,6/5) Hyperbolic Matrix(377,-144,144,-55) (3/8,5/13) -> (13/5,21/8) Hyperbolic Matrix(85,-36,196,-83) (5/12,3/7) -> (3/7,7/16) Parabolic Matrix(13,-8,18,-11) (1/2,2/3) -> (2/3,3/4) Parabolic Matrix(167,-132,62,-49) (7/9,4/5) -> (8/3,19/7) Hyperbolic Matrix(233,-188,88,-71) (4/5,9/11) -> (29/11,8/3) Hyperbolic Matrix(329,-272,202,-167) (9/11,5/6) -> (13/8,31/19) Hyperbolic Matrix(43,-36,6,-5) (5/6,1/1) -> (7/1,1/0) Hyperbolic Matrix(317,-384,194,-235) (6/5,11/9) -> (31/19,18/11) Hyperbolic Matrix(305,-376,116,-143) (11/9,5/4) -> (21/8,29/11) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(85,-196,36,-83) (9/4,7/3) -> (7/3,19/8) Parabolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(11,72,-2,-13) -> Matrix(1,0,0,1) Matrix(55,296,34,183) -> Matrix(1,2,2,5) Matrix(5,24,16,77) -> Matrix(1,2,0,1) Matrix(29,128,12,53) -> Matrix(5,6,4,5) Matrix(15,56,-26,-97) -> Matrix(1,2,0,1) Matrix(41,140,-70,-239) -> Matrix(5,4,6,5) Matrix(17,56,44,145) -> Matrix(3,2,-2,-1) Matrix(47,128,-18,-49) -> Matrix(1,2,-2,-3) Matrix(125,324,76,197) -> Matrix(3,2,4,3) Matrix(39,100,140,359) -> Matrix(1,0,4,1) Matrix(13,32,28,69) -> Matrix(1,0,2,1) Matrix(57,136,44,105) -> Matrix(1,0,0,1) Matrix(45,104,16,37) -> Matrix(1,2,0,1) Matrix(17,36,-26,-55) -> Matrix(1,0,2,1) Matrix(3,4,-4,-5) -> Matrix(1,0,2,1) Matrix(387,248,142,91) -> Matrix(1,0,0,1) Matrix(487,308,302,191) -> Matrix(1,0,2,1) Matrix(165,104,376,237) -> Matrix(1,0,2,1) Matrix(207,128,-338,-209) -> Matrix(3,-2,2,-1) Matrix(53,32,48,29) -> Matrix(1,0,0,1) Matrix(149,88,22,13) -> Matrix(3,-2,2,-1) Matrix(325,184,136,77) -> Matrix(5,-4,4,-3) Matrix(93,52,338,189) -> Matrix(1,-2,4,-7) Matrix(225,124,176,97) -> Matrix(1,-2,0,1) Matrix(83,44,66,35) -> Matrix(1,-2,0,1) Matrix(35,16,94,43) -> Matrix(1,0,0,1) Matrix(37,16,104,45) -> Matrix(1,2,0,1) Matrix(105,44,136,57) -> Matrix(1,2,2,5) Matrix(69,28,32,13) -> Matrix(1,0,2,1) Matrix(31,12,18,7) -> Matrix(1,0,2,1) Matrix(27,8,-98,-29) -> Matrix(1,0,4,1) Matrix(107,28,42,11) -> Matrix(3,-2,2,-1) Matrix(53,12,128,29) -> Matrix(1,0,-2,1) Matrix(75,16,14,3) -> Matrix(1,0,0,1) Matrix(1,0,10,1) -> Matrix(1,0,0,1) Matrix(17,-4,64,-15) -> Matrix(1,0,4,1) Matrix(109,-32,92,-27) -> Matrix(3,-2,2,-1) Matrix(377,-144,144,-55) -> Matrix(1,4,0,1) Matrix(85,-36,196,-83) -> Matrix(1,0,6,1) Matrix(13,-8,18,-11) -> Matrix(1,0,4,1) Matrix(167,-132,62,-49) -> Matrix(1,0,-2,1) Matrix(233,-188,88,-71) -> Matrix(1,0,-2,1) Matrix(329,-272,202,-167) -> Matrix(3,-2,8,-5) Matrix(43,-36,6,-5) -> Matrix(3,-2,2,-1) Matrix(317,-384,194,-235) -> Matrix(1,-4,2,-7) Matrix(305,-376,116,-143) -> Matrix(1,4,0,1) Matrix(25,-36,16,-23) -> Matrix(1,0,2,1) Matrix(85,-196,36,-83) -> Matrix(11,-10,10,-9) Matrix(9,-32,2,-7) -> Matrix(1,-2,0,1) 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): 11 Degree of the the map X: 11 Degree of the the map Y: 48 Permutation triple for Y: ((1,7,22,8,2)(3,10,30,31,11)(4,15,25,9,5)(12,28,27,40,35)(14,38,18,46,39)(16,42,47,32,17)(19,23,37,34,20)(21,45,44,41,48); (1,5,17,45,29,10,28,37,18,6)(2,3)(4,13,23,22,48,27,43,32,31,14)(7,20,26,25,38,30,42,36,35,21)(8,24,39,34,12,11,33,41,16,9)(15,40)(19,47)(44,46); (1,3,12,36,42,41,46,37,13,4)(2,9,26,20,39,44,17,43,27,10)(5,16)(6,18,25,40,48,33,11,32,19,7)(8,23,47,30,29,45,35,15,14,24)(21,22)(28,34)(31,38)) ----------------------------------------------------------------------- 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 -6/1 0 2 -5/1 0 10 -9/2 -3/2 1 10 -4/1 (-1/1,1/0) 0 10 -7/2 -1/1 3 2 -10/3 (-1/1,-2/3) 0 10 -3/1 0 10 -8/3 -1/1 2 2 -13/5 0 10 -5/2 -1/2 1 10 -12/5 (0/1,1/0) 0 10 -7/3 0 10 -9/4 -1/1 1 2 -2/1 (-1/1,0/1) 0 10 -1/1 0/1 1 2 0/1 (0/1,1/0) 0 10 1/4 0/1 2 2 2/7 (0/1,1/2) 0 10 3/10 1/2 1 10 1/3 0 10 3/8 1/0 1 10 5/13 0 10 2/5 (-1/1,0/1) 0 10 3/7 0/1 3 2 4/9 (0/1,1/2) 0 10 1/2 1/0 1 10 2/3 0/1 4 2 3/4 1/4 1 10 4/5 (1/3,1/2) 0 10 9/11 1/2 5 2 5/6 1/2 1 10 1/1 0 10 7/6 1/0 1 10 6/5 (2/1,1/0) 0 10 11/9 1/0 5 2 5/4 1/0 1 10 14/11 0 2 9/7 0 10 4/3 (0/1,1/0) 0 10 3/2 0/1 1 2 2/1 (0/1,1/1) 0 10 7/3 1/1 5 2 12/5 (1/1,5/4) 0 10 5/2 3/2 1 10 8/3 (1/1,1/0) 0 10 11/4 1/1 1 2 3/1 0 10 4/1 1/0 2 2 5/1 0 10 1/0 1/0 1 10 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,44,4,35) (-6/1,1/0) -> (5/4,14/11) Glide Reflection Matrix(23,124,18,97) (-6/1,-5/1) -> (14/11,9/7) Glide Reflection Matrix(5,24,16,77) (-5/1,-9/2) -> (3/10,1/3) Hyperbolic Matrix(29,128,12,53) (-9/2,-4/1) -> (12/5,5/2) Hyperbolic Matrix(15,56,-4,-15) (-4/1,-7/2) -> (-4/1,-7/2) Reflection Matrix(41,140,-12,-41) (-7/2,-10/3) -> (-7/2,-10/3) Reflection Matrix(17,56,44,145) (-10/3,-3/1) -> (5/13,2/5) Hyperbolic Matrix(47,128,-18,-49) (-3/1,-8/3) -> (-8/3,-13/5) Parabolic Matrix(37,96,32,83) (-13/5,-5/2) -> (1/1,7/6) Glide Reflection Matrix(13,32,28,69) (-5/2,-12/5) -> (4/9,1/2) Hyperbolic Matrix(57,136,44,105) (-12/5,-7/3) -> (9/7,4/3) Hyperbolic Matrix(45,104,16,37) (-7/3,-9/4) -> (11/4,3/1) Hyperbolic Matrix(17,36,-8,-17) (-9/4,-2/1) -> (-9/4,-2/1) Reflection Matrix(3,4,-2,-3) (-2/1,-1/1) -> (-2/1,-1/1) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,8,-1) (0/1,1/4) -> (0/1,1/4) Reflection Matrix(15,-4,56,-15) (1/4,2/7) -> (1/4,2/7) Reflection Matrix(109,-32,92,-27) (2/7,3/10) -> (7/6,6/5) Hyperbolic Matrix(43,-16,8,-3) (1/3,3/8) -> (5/1,1/0) Glide Reflection Matrix(73,-28,86,-33) (3/8,5/13) -> (5/6,1/1) Glide Reflection Matrix(29,-12,70,-29) (2/5,3/7) -> (2/5,3/7) Reflection Matrix(55,-24,126,-55) (3/7,4/9) -> (3/7,4/9) Reflection Matrix(13,-8,18,-11) (1/2,2/3) -> (2/3,3/4) Parabolic Matrix(57,-44,22,-17) (3/4,4/5) -> (5/2,8/3) Glide Reflection Matrix(89,-72,110,-89) (4/5,9/11) -> (4/5,9/11) Reflection Matrix(121,-100,98,-81) (9/11,5/6) -> (11/9,5/4) Glide Reflection Matrix(109,-132,90,-109) (6/5,11/9) -> (6/5,11/9) Reflection Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(7,-12,4,-7) (3/2,2/1) -> (3/2,2/1) Reflection Matrix(13,-28,6,-13) (2/1,7/3) -> (2/1,7/3) Reflection Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(65,-176,24,-65) (8/3,11/4) -> (8/3,11/4) Reflection Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(5,44,4,35) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(23,124,18,97) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(5,24,16,77) -> Matrix(1,2,0,1) 1/0 Matrix(29,128,12,53) -> Matrix(5,6,4,5) Matrix(15,56,-4,-15) -> Matrix(1,2,0,-1) (-4/1,-7/2) -> (-1/1,1/0) Matrix(41,140,-12,-41) -> Matrix(5,4,-6,-5) (-7/2,-10/3) -> (-1/1,-2/3) Matrix(17,56,44,145) -> Matrix(3,2,-2,-1) -1/1 Matrix(47,128,-18,-49) -> Matrix(1,2,-2,-3) -1/1 Matrix(37,96,32,83) -> Matrix(3,2,2,1) Matrix(13,32,28,69) -> Matrix(1,0,2,1) 0/1 Matrix(57,136,44,105) -> Matrix(1,0,0,1) Matrix(45,104,16,37) -> Matrix(1,2,0,1) 1/0 Matrix(17,36,-8,-17) -> Matrix(-1,0,2,1) (-9/4,-2/1) -> (-1/1,0/1) Matrix(3,4,-2,-3) -> Matrix(-1,0,2,1) (-2/1,-1/1) -> (-1/1,0/1) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(1,0,8,-1) -> Matrix(1,0,0,-1) (0/1,1/4) -> (0/1,1/0) Matrix(15,-4,56,-15) -> Matrix(1,0,4,-1) (1/4,2/7) -> (0/1,1/2) Matrix(109,-32,92,-27) -> Matrix(3,-2,2,-1) 1/1 Matrix(43,-16,8,-3) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(73,-28,86,-33) -> Matrix(1,2,2,3) Matrix(29,-12,70,-29) -> Matrix(-1,0,2,1) (2/5,3/7) -> (-1/1,0/1) Matrix(55,-24,126,-55) -> Matrix(1,0,4,-1) (3/7,4/9) -> (0/1,1/2) Matrix(13,-8,18,-11) -> Matrix(1,0,4,1) 0/1 Matrix(57,-44,22,-17) -> Matrix(5,-2,2,-1) Matrix(89,-72,110,-89) -> Matrix(5,-2,12,-5) (4/5,9/11) -> (1/3,1/2) Matrix(121,-100,98,-81) -> Matrix(3,-2,-2,1) Matrix(109,-132,90,-109) -> Matrix(-1,4,0,1) (6/5,11/9) -> (2/1,1/0) Matrix(17,-24,12,-17) -> Matrix(1,0,0,-1) (4/3,3/2) -> (0/1,1/0) Matrix(7,-12,4,-7) -> Matrix(1,0,2,-1) (3/2,2/1) -> (0/1,1/1) Matrix(13,-28,6,-13) -> Matrix(1,0,2,-1) (2/1,7/3) -> (0/1,1/1) Matrix(71,-168,30,-71) -> Matrix(9,-10,8,-9) (7/3,12/5) -> (1/1,5/4) Matrix(65,-176,24,-65) -> Matrix(-1,2,0,1) (8/3,11/4) -> (1/1,1/0) Matrix(9,-32,2,-7) -> Matrix(1,-2,0,1) 1/0 ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.