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: 30 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/7 -1/2 -3/7 -15/56 -1/4 -3/13 -1/6 -1/7 0/1 1/5 3/14 1/4 3/11 2/7 1/3 5/14 2/5 3/7 1/2 5/9 4/7 13/21 9/14 2/3 5/7 11/14 5/6 6/7 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/0 -6/7 0/1 -5/6 1/0 -4/5 -1/1 0/1 -11/14 0/1 -7/9 0/1 1/2 -10/13 0/1 1/1 -3/4 1/0 -5/7 0/1 -7/10 1/2 -9/13 0/1 1/1 -2/3 0/1 1/0 -9/14 0/1 -7/11 0/1 1/3 -12/19 0/1 1/1 -5/8 1/2 -3/5 1/1 1/0 -4/7 1/0 -5/9 0/1 1/0 -6/11 0/1 1/0 -7/13 0/1 1/1 -1/2 1/0 -3/7 1/0 -5/12 1/0 -7/17 -3/1 -2/1 -2/5 -2/1 -1/1 -5/13 -1/1 0/1 -8/21 0/1 -3/8 1/0 -10/27 -1/1 0/1 -7/19 0/1 1/0 -4/11 0/1 1/0 -5/14 1/0 -1/3 -1/1 1/0 -2/7 -1/1 -3/11 -1/1 0/1 -7/26 1/0 -11/41 -2/1 -1/1 -15/56 -1/1 -4/15 -1/1 0/1 -1/4 1/0 -3/13 -2/1 -1/1 -2/9 -4/3 -1/1 -3/14 -1/1 -1/5 -1/1 -1/2 -1/6 1/0 -1/7 -1/1 -1/8 -1/2 0/1 -1/1 0/1 1/5 -1/1 1/0 3/14 -1/1 2/9 -1/1 -4/5 1/4 -1/2 4/15 -1/1 0/1 3/11 -1/1 0/1 2/7 -1/1 1/3 -1/1 -1/2 5/14 -1/2 4/11 -1/2 0/1 7/19 -1/2 0/1 3/8 -1/2 8/21 0/1 5/13 -1/1 0/1 7/18 1/0 2/5 -1/1 -2/3 3/7 -1/2 4/9 -1/3 0/1 5/11 -1/1 0/1 6/13 -1/1 0/1 1/2 -1/2 6/11 -1/2 0/1 5/9 -1/2 0/1 4/7 -1/2 3/5 -1/2 -1/3 8/13 -1/3 0/1 13/21 -1/3 18/29 -1/3 -2/7 5/8 -1/4 7/11 -1/5 0/1 9/14 0/1 2/3 -1/2 0/1 5/7 0/1 8/11 0/1 1/0 19/26 1/0 30/41 -2/1 -1/1 41/56 -1/1 11/15 -1/1 0/1 3/4 -1/2 10/13 -1/3 0/1 17/22 -1/4 7/9 -1/4 0/1 11/14 0/1 4/5 -1/1 0/1 9/11 -1/3 0/1 5/6 -1/2 6/7 0/1 1/1 -1/2 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(13,12,14,13) (-1/1,-6/7) -> (6/7,1/1) Hyperbolic Matrix(69,58,182,153) (-6/7,-5/6) -> (3/8,8/21) Hyperbolic Matrix(97,80,-154,-127) (-5/6,-4/5) -> (-12/19,-5/8) Hyperbolic Matrix(111,88,140,111) (-4/5,-11/14) -> (11/14,4/5) Hyperbolic Matrix(197,154,252,197) (-11/14,-7/9) -> (7/9,11/14) Hyperbolic Matrix(181,140,-490,-379) (-7/9,-10/13) -> (-10/27,-7/19) Hyperbolic Matrix(29,22,112,85) (-10/13,-3/4) -> (1/4,4/15) Hyperbolic Matrix(69,50,-98,-71) (-3/4,-5/7) -> (-5/7,-7/10) Parabolic Matrix(141,98,364,253) (-7/10,-9/13) -> (5/13,7/18) Hyperbolic Matrix(99,68,-182,-125) (-9/13,-2/3) -> (-6/11,-7/13) Hyperbolic Matrix(55,36,84,55) (-2/3,-9/14) -> (9/14,2/3) Hyperbolic Matrix(197,126,308,197) (-9/14,-7/11) -> (7/11,9/14) Hyperbolic Matrix(139,88,308,195) (-7/11,-12/19) -> (4/9,5/11) Hyperbolic Matrix(13,8,-70,-43) (-5/8,-3/5) -> (-1/5,-1/6) Hyperbolic Matrix(41,24,70,41) (-3/5,-4/7) -> (4/7,3/5) Hyperbolic Matrix(71,40,126,71) (-4/7,-5/9) -> (5/9,4/7) Hyperbolic Matrix(113,62,308,169) (-5/9,-6/11) -> (4/11,7/19) Hyperbolic Matrix(113,60,-420,-223) (-7/13,-1/2) -> (-7/26,-11/41) Hyperbolic Matrix(41,18,-98,-43) (-1/2,-3/7) -> (-3/7,-5/12) Parabolic Matrix(155,64,-574,-237) (-5/12,-7/17) -> (-3/11,-7/26) Hyperbolic Matrix(113,46,140,57) (-7/17,-2/5) -> (4/5,9/11) Hyperbolic Matrix(41,16,-182,-71) (-2/5,-5/13) -> (-3/13,-2/9) Hyperbolic Matrix(209,80,546,209) (-5/13,-8/21) -> (8/21,5/13) Hyperbolic Matrix(153,58,182,69) (-8/21,-3/8) -> (5/6,6/7) Hyperbolic Matrix(279,104,448,167) (-3/8,-10/27) -> (18/29,5/8) Hyperbolic Matrix(169,62,308,113) (-7/19,-4/11) -> (6/11,5/9) Hyperbolic Matrix(111,40,308,111) (-4/11,-5/14) -> (5/14,4/11) Hyperbolic Matrix(29,10,84,29) (-5/14,-1/3) -> (1/3,5/14) Hyperbolic Matrix(13,4,42,13) (-1/3,-2/7) -> (2/7,1/3) Hyperbolic Matrix(43,12,154,43) (-2/7,-3/11) -> (3/11,2/7) Hyperbolic Matrix(2297,616,3136,841) (-11/41,-15/56) -> (41/56,11/15) Hyperbolic Matrix(2295,614,3136,839) (-15/56,-4/15) -> (30/41,41/56) Hyperbolic Matrix(85,22,112,29) (-4/15,-1/4) -> (3/4,10/13) Hyperbolic Matrix(83,20,112,27) (-1/4,-3/13) -> (11/15,3/4) Hyperbolic Matrix(55,12,252,55) (-2/9,-3/14) -> (3/14,2/9) Hyperbolic Matrix(29,6,140,29) (-3/14,-1/5) -> (1/5,3/14) Hyperbolic Matrix(13,2,-98,-15) (-1/6,-1/7) -> (-1/7,-1/8) Parabolic Matrix(97,10,126,13) (-1/8,0/1) -> (10/13,17/22) Hyperbolic Matrix(43,-8,70,-13) (0/1,1/5) -> (3/5,8/13) Hyperbolic Matrix(71,-16,182,-41) (2/9,1/4) -> (7/18,2/5) Hyperbolic Matrix(141,-38,308,-83) (4/15,3/11) -> (5/11,6/13) Hyperbolic Matrix(379,-140,490,-181) (7/19,3/8) -> (17/22,7/9) Hyperbolic Matrix(43,-18,98,-41) (2/5,3/7) -> (3/7,4/9) Parabolic Matrix(307,-144,420,-197) (6/13,1/2) -> (19/26,30/41) Hyperbolic Matrix(225,-122,308,-167) (1/2,6/11) -> (8/11,19/26) Hyperbolic Matrix(547,-338,882,-545) (8/13,13/21) -> (13/21,18/29) Parabolic Matrix(127,-80,154,-97) (5/8,7/11) -> (9/11,5/6) Hyperbolic Matrix(71,-50,98,-69) (2/3,5/7) -> (5/7,8/11) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(13,12,14,13) -> Matrix(1,0,-2,1) Matrix(69,58,182,153) -> Matrix(1,0,-2,1) Matrix(97,80,-154,-127) -> Matrix(1,0,2,1) Matrix(111,88,140,111) -> Matrix(1,0,0,1) Matrix(197,154,252,197) -> Matrix(1,0,-6,1) Matrix(181,140,-490,-379) -> Matrix(1,0,-2,1) Matrix(29,22,112,85) -> Matrix(1,0,-2,1) Matrix(69,50,-98,-71) -> Matrix(1,0,2,1) Matrix(141,98,364,253) -> Matrix(1,0,-2,1) Matrix(99,68,-182,-125) -> Matrix(1,0,0,1) Matrix(55,36,84,55) -> Matrix(1,0,-2,1) Matrix(197,126,308,197) -> Matrix(1,0,-8,1) Matrix(139,88,308,195) -> Matrix(1,0,-4,1) Matrix(13,8,-70,-43) -> Matrix(1,0,-2,1) Matrix(41,24,70,41) -> Matrix(1,-2,-2,5) Matrix(71,40,126,71) -> Matrix(1,0,-2,1) Matrix(113,62,308,169) -> Matrix(1,0,-2,1) Matrix(113,60,-420,-223) -> Matrix(1,-2,0,1) Matrix(41,18,-98,-43) -> Matrix(1,-2,0,1) Matrix(155,64,-574,-237) -> Matrix(1,2,0,1) Matrix(113,46,140,57) -> Matrix(1,2,-2,-3) Matrix(41,16,-182,-71) -> Matrix(3,2,-2,-1) Matrix(209,80,546,209) -> Matrix(1,0,0,1) Matrix(153,58,182,69) -> Matrix(1,0,-2,1) Matrix(279,104,448,167) -> Matrix(1,2,-4,-7) Matrix(169,62,308,113) -> Matrix(1,0,-2,1) Matrix(111,40,308,111) -> Matrix(1,0,-2,1) Matrix(29,10,84,29) -> Matrix(1,2,-2,-3) Matrix(13,4,42,13) -> Matrix(1,2,-2,-3) Matrix(43,12,154,43) -> Matrix(1,0,0,1) Matrix(2297,616,3136,841) -> Matrix(1,2,-2,-3) Matrix(2295,614,3136,839) -> Matrix(3,2,-2,-1) Matrix(85,22,112,29) -> Matrix(1,0,-2,1) Matrix(83,20,112,27) -> Matrix(1,2,-2,-3) Matrix(55,12,252,55) -> Matrix(7,8,-8,-9) Matrix(29,6,140,29) -> Matrix(3,2,-2,-1) Matrix(13,2,-98,-15) -> Matrix(1,2,-2,-3) Matrix(97,10,126,13) -> Matrix(1,0,-2,1) Matrix(43,-8,70,-13) -> Matrix(1,0,-2,1) Matrix(71,-16,182,-41) -> Matrix(3,2,-2,-1) Matrix(141,-38,308,-83) -> Matrix(1,0,0,1) Matrix(379,-140,490,-181) -> Matrix(1,0,-2,1) Matrix(43,-18,98,-41) -> Matrix(3,2,-8,-5) Matrix(307,-144,420,-197) -> Matrix(3,2,-2,-1) Matrix(225,-122,308,-167) -> Matrix(1,0,2,1) Matrix(547,-338,882,-545) -> Matrix(5,2,-18,-7) Matrix(127,-80,154,-97) -> Matrix(1,0,2,1) Matrix(71,-50,98,-69) -> Matrix(1,0,2,1) 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): 10 Degree of the the map X: 10 Degree of the the map Y: 48 Permutation triple for Y: ((1,4,16,34,45,41,47,48,44,31,43,17,5,2)(3,10,32,35,20,40,39,46,25,8,7,24,33,11)(6,21,18,42,37,14,13,36,27,26,30,9,29,22)(12,23)(15,19)(28,38); (1,2,8,27,28,9,3)(4,14,40,20,6,5,15)(7,12,11,16,42,18,17)(19,44,29,33,24,36,41)(21,38,37,46,48,47,32)(23,39,31,30,26,45,35); (2,6,22,44,46,23,7)(3,12,35,47,36,13,4)(5,18,32,10,9,31,19)(8,25,37,16,15,41,26)(11,29,28,21,20,45,34)(14,38,27,24,17,43,39)) ----------------------------------------------------------------------- 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 (-1/1,0/1) 0 14 1/5 (-1/1,1/0) 0 14 3/14 -1/1 5 1 2/9 (-1/1,-4/5) 0 14 1/4 -1/2 1 7 4/15 (-1/1,0/1) 0 14 3/11 (-1/1,0/1) 0 14 2/7 -1/1 1 2 1/3 (-1/1,-1/2) 0 14 5/14 -1/2 1 1 4/11 (-1/2,0/1) 0 14 7/19 (-1/2,0/1) 0 14 3/8 -1/2 1 7 8/21 0/1 1 2 5/13 (-1/1,0/1) 0 14 7/18 1/0 1 7 2/5 (-1/1,-2/3) 0 14 3/7 -1/2 2 2 4/9 (-1/3,0/1) 0 14 5/11 (-1/1,0/1) 0 14 6/13 (-1/1,0/1) 0 14 1/2 -1/2 1 7 6/11 (-1/2,0/1) 0 14 5/9 (-1/2,0/1) 0 14 4/7 -1/2 1 2 3/5 (-1/2,-1/3) 0 14 8/13 (-1/3,0/1) 0 14 13/21 -1/3 2 2 18/29 (-1/3,-2/7) 0 14 5/8 -1/4 1 7 7/11 (-1/5,0/1) 0 14 9/14 0/1 3 1 2/3 (-1/2,0/1) 0 14 5/7 0/1 2 2 8/11 (0/1,1/0) 0 14 19/26 1/0 1 7 30/41 (-2/1,-1/1) 0 14 41/56 -1/1 2 1 11/15 (-1/1,0/1) 0 14 3/4 -1/2 1 7 10/13 (-1/3,0/1) 0 14 17/22 -1/4 1 7 7/9 (-1/4,0/1) 0 14 11/14 0/1 3 1 4/5 (-1/1,0/1) 0 14 9/11 (-1/3,0/1) 0 14 5/6 -1/2 1 7 6/7 0/1 1 2 1/1 (-1/2,0/1) 0 14 1/0 0/1 1 1 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(43,-8,70,-13) (0/1,1/5) -> (3/5,8/13) Hyperbolic Matrix(29,-6,140,-29) (1/5,3/14) -> (1/5,3/14) Reflection Matrix(55,-12,252,-55) (3/14,2/9) -> (3/14,2/9) Reflection Matrix(71,-16,182,-41) (2/9,1/4) -> (7/18,2/5) Hyperbolic Matrix(85,-22,112,-29) (1/4,4/15) -> (3/4,10/13) Glide Reflection Matrix(141,-38,308,-83) (4/15,3/11) -> (5/11,6/13) Hyperbolic Matrix(43,-12,154,-43) (3/11,2/7) -> (3/11,2/7) Reflection Matrix(13,-4,42,-13) (2/7,1/3) -> (2/7,1/3) Reflection Matrix(29,-10,84,-29) (1/3,5/14) -> (1/3,5/14) Reflection Matrix(111,-40,308,-111) (5/14,4/11) -> (5/14,4/11) Reflection Matrix(169,-62,308,-113) (4/11,7/19) -> (6/11,5/9) Glide Reflection Matrix(379,-140,490,-181) (7/19,3/8) -> (17/22,7/9) Hyperbolic Matrix(153,-58,182,-69) (3/8,8/21) -> (5/6,6/7) Glide Reflection Matrix(209,-80,546,-209) (8/21,5/13) -> (8/21,5/13) Reflection Matrix(237,-92,322,-125) (5/13,7/18) -> (11/15,3/4) Glide Reflection Matrix(43,-18,98,-41) (2/5,3/7) -> (3/7,4/9) Parabolic Matrix(125,-56,154,-69) (4/9,5/11) -> (4/5,9/11) Glide Reflection Matrix(307,-144,420,-197) (6/13,1/2) -> (19/26,30/41) Hyperbolic Matrix(225,-122,308,-167) (1/2,6/11) -> (8/11,19/26) Hyperbolic Matrix(71,-40,126,-71) (5/9,4/7) -> (5/9,4/7) Reflection Matrix(41,-24,70,-41) (4/7,3/5) -> (4/7,3/5) Reflection Matrix(547,-338,882,-545) (8/13,13/21) -> (13/21,18/29) Parabolic Matrix(573,-356,742,-461) (18/29,5/8) -> (10/13,17/22) Glide Reflection Matrix(127,-80,154,-97) (5/8,7/11) -> (9/11,5/6) Hyperbolic Matrix(197,-126,308,-197) (7/11,9/14) -> (7/11,9/14) Reflection Matrix(55,-36,84,-55) (9/14,2/3) -> (9/14,2/3) Reflection Matrix(71,-50,98,-69) (2/3,5/7) -> (5/7,8/11) Parabolic Matrix(3361,-2460,4592,-3361) (30/41,41/56) -> (30/41,41/56) Reflection Matrix(1231,-902,1680,-1231) (41/56,11/15) -> (41/56,11/15) Reflection Matrix(197,-154,252,-197) (7/9,11/14) -> (7/9,11/14) Reflection Matrix(111,-88,140,-111) (11/14,4/5) -> (11/14,4/5) Reflection Matrix(13,-12,14,-13) (6/7,1/1) -> (6/7,1/1) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(43,-8,70,-13) -> Matrix(1,0,-2,1) 0/1 Matrix(29,-6,140,-29) -> Matrix(1,2,0,-1) (1/5,3/14) -> (-1/1,1/0) Matrix(55,-12,252,-55) -> Matrix(9,8,-10,-9) (3/14,2/9) -> (-1/1,-4/5) Matrix(71,-16,182,-41) -> Matrix(3,2,-2,-1) -1/1 Matrix(85,-22,112,-29) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(141,-38,308,-83) -> Matrix(1,0,0,1) Matrix(43,-12,154,-43) -> Matrix(-1,0,2,1) (3/11,2/7) -> (-1/1,0/1) Matrix(13,-4,42,-13) -> Matrix(3,2,-4,-3) (2/7,1/3) -> (-1/1,-1/2) Matrix(29,-10,84,-29) -> Matrix(3,2,-4,-3) (1/3,5/14) -> (-1/1,-1/2) Matrix(111,-40,308,-111) -> Matrix(-1,0,4,1) (5/14,4/11) -> (-1/2,0/1) Matrix(169,-62,308,-113) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(379,-140,490,-181) -> Matrix(1,0,-2,1) 0/1 Matrix(153,-58,182,-69) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(209,-80,546,-209) -> Matrix(-1,0,2,1) (8/21,5/13) -> (-1/1,0/1) Matrix(237,-92,322,-125) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(43,-18,98,-41) -> Matrix(3,2,-8,-5) -1/2 Matrix(125,-56,154,-69) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(307,-144,420,-197) -> Matrix(3,2,-2,-1) -1/1 Matrix(225,-122,308,-167) -> Matrix(1,0,2,1) 0/1 Matrix(71,-40,126,-71) -> Matrix(-1,0,4,1) (5/9,4/7) -> (-1/2,0/1) Matrix(41,-24,70,-41) -> Matrix(5,2,-12,-5) (4/7,3/5) -> (-1/2,-1/3) Matrix(547,-338,882,-545) -> Matrix(5,2,-18,-7) -1/3 Matrix(573,-356,742,-461) -> Matrix(7,2,-24,-7) *** -> (-1/3,-1/4) Matrix(127,-80,154,-97) -> Matrix(1,0,2,1) 0/1 Matrix(197,-126,308,-197) -> Matrix(-1,0,10,1) (7/11,9/14) -> (-1/5,0/1) Matrix(55,-36,84,-55) -> Matrix(-1,0,4,1) (9/14,2/3) -> (-1/2,0/1) Matrix(71,-50,98,-69) -> Matrix(1,0,2,1) 0/1 Matrix(3361,-2460,4592,-3361) -> Matrix(3,4,-2,-3) (30/41,41/56) -> (-2/1,-1/1) Matrix(1231,-902,1680,-1231) -> Matrix(-1,0,2,1) (41/56,11/15) -> (-1/1,0/1) Matrix(197,-154,252,-197) -> Matrix(-1,0,8,1) (7/9,11/14) -> (-1/4,0/1) Matrix(111,-88,140,-111) -> Matrix(-1,0,2,1) (11/14,4/5) -> (-1/1,0/1) Matrix(13,-12,14,-13) -> Matrix(-1,0,4,1) (6/7,1/1) -> (-1/2,0/1) Matrix(-1,2,0,1) -> Matrix(-1,0,4,1) (1/1,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.