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 -5/1 -4/1 -3/1 -5/2 -2/1 -3/2 -4/3 -1/1 -4/5 -2/3 -1/2 -2/5 0/1 1/3 2/5 1/2 2/3 4/5 5/6 1/1 6/5 4/3 3/2 5/3 2/1 12/5 5/2 3/1 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 1/0 -5/1 -2/1 -1/1 1/0 -4/1 -1/1 -11/3 -1/2 1/0 -7/2 -1/2 1/0 -3/1 -2/1 0/1 -8/3 1/0 -5/2 0/1 -2/1 1/0 -7/4 -2/1 -12/7 -2/1 -5/3 -3/2 1/0 -8/5 -1/1 -3/2 -1/1 -7/5 0/1 1/1 1/0 -18/13 1/0 -11/8 0/1 -4/3 1/0 -5/4 -3/2 1/0 -11/9 -3/2 1/0 -6/5 -1/1 -1/1 -1/1 1/1 -6/7 -1/1 -5/6 -1/2 -4/5 -1/2 1/0 -3/4 -1/1 -8/11 -1/1 -5/7 -1/1 -1/2 0/1 -2/3 -1/2 1/0 -7/11 -1/1 -1/2 0/1 -12/19 0/1 -5/8 0/1 -3/5 -2/1 0/1 -7/12 -3/2 -4/7 -1/1 -5/9 -3/4 -1/2 -6/11 -1/2 -1/2 -1/2 1/0 -4/9 -1/2 -3/7 -2/5 0/1 -5/12 -1/4 -2/5 0/1 -7/18 1/2 -12/31 0/1 -5/13 0/1 1/3 1/2 -3/8 1/1 -4/11 1/0 -1/3 -1/2 1/0 0/1 0/1 1/3 1/2 1/0 4/11 1/1 3/8 1/1 2/5 1/2 1/0 5/12 1/2 3/7 0/1 2/1 1/2 0/1 4/7 1/2 1/0 7/12 1/0 3/5 0/1 2/3 5/8 1/2 1/0 2/3 1/2 1/0 7/10 1/2 1/0 5/7 1/3 1/1 3/4 1/1 4/5 1/1 5/6 1/0 1/1 0/1 1/1 1/0 7/6 1/0 6/5 1/1 5/4 2/1 4/3 1/0 11/8 1/2 1/0 7/5 -1/1 1/1 17/12 1/2 10/7 1/2 1/0 3/2 1/1 8/5 3/2 1/0 13/8 2/1 18/11 1/0 5/3 3/2 1/0 2/1 1/0 7/3 -1/2 1/0 12/5 0/1 29/12 1/4 17/7 1/3 1/1 5/2 1/2 1/0 8/3 1/0 11/4 0/1 3/1 0/1 2/1 7/2 2/1 18/5 3/1 11/3 7/2 1/0 4/1 1/0 9/2 -1/1 5/1 -1/1 1/1 11/2 1/2 1/0 6/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,12,0,1) (-6/1,1/0) -> (6/1,1/0) Parabolic Matrix(25,132,-18,-95) (-6/1,-5/1) -> (-7/5,-18/13) Hyperbolic Matrix(13,60,-18,-83) (-5/1,-4/1) -> (-8/11,-5/7) Hyperbolic Matrix(13,48,36,133) (-4/1,-11/3) -> (1/3,4/11) Hyperbolic Matrix(37,132,-30,-107) (-11/3,-7/2) -> (-5/4,-11/9) Hyperbolic Matrix(11,36,18,59) (-7/2,-3/1) -> (3/5,5/8) Hyperbolic Matrix(13,36,-30,-83) (-3/1,-8/3) -> (-4/9,-3/7) Hyperbolic Matrix(23,60,18,47) (-8/3,-5/2) -> (5/4,4/3) Hyperbolic Matrix(11,24,-6,-13) (-5/2,-2/1) -> (-2/1,-7/4) Parabolic Matrix(83,144,-132,-229) (-7/4,-12/7) -> (-12/19,-5/8) Hyperbolic Matrix(85,144,36,61) (-12/7,-5/3) -> (7/3,12/5) Hyperbolic Matrix(37,60,-66,-107) (-5/3,-8/5) -> (-4/7,-5/9) Hyperbolic Matrix(23,36,30,47) (-8/5,-3/2) -> (3/4,4/5) Hyperbolic Matrix(25,36,-66,-95) (-3/2,-7/5) -> (-5/13,-3/8) Hyperbolic Matrix(313,432,192,265) (-18/13,-11/8) -> (13/8,18/11) Hyperbolic Matrix(97,132,36,49) (-11/8,-4/3) -> (8/3,11/4) Hyperbolic Matrix(47,60,18,23) (-4/3,-5/4) -> (5/2,8/3) Hyperbolic Matrix(217,264,60,73) (-11/9,-6/5) -> (18/5,11/3) Hyperbolic Matrix(11,12,-12,-13) (-6/5,-1/1) -> (-1/1,-6/7) Parabolic Matrix(85,72,72,61) (-6/7,-5/6) -> (7/6,6/5) Hyperbolic Matrix(59,48,102,83) (-5/6,-4/5) -> (4/7,7/12) Hyperbolic Matrix(47,36,30,23) (-4/5,-3/4) -> (3/2,8/5) Hyperbolic Matrix(49,36,132,97) (-3/4,-8/11) -> (4/11,3/8) Hyperbolic Matrix(35,24,-54,-37) (-5/7,-2/3) -> (-2/3,-7/11) Parabolic Matrix(227,144,-588,-373) (-7/11,-12/19) -> (-12/31,-5/13) Hyperbolic Matrix(59,36,18,11) (-5/8,-3/5) -> (3/1,7/2) Hyperbolic Matrix(61,36,144,85) (-3/5,-7/12) -> (5/12,3/7) Hyperbolic Matrix(83,48,102,59) (-7/12,-4/7) -> (4/5,5/6) Hyperbolic Matrix(217,120,132,73) (-5/9,-6/11) -> (18/11,5/3) Hyperbolic Matrix(133,72,24,13) (-6/11,-1/2) -> (11/2,6/1) Hyperbolic Matrix(107,48,78,35) (-1/2,-4/9) -> (4/3,11/8) Hyperbolic Matrix(85,36,144,61) (-3/7,-5/12) -> (7/12,3/5) Hyperbolic Matrix(59,24,-150,-61) (-5/12,-2/5) -> (-2/5,-7/18) Parabolic Matrix(1115,432,462,179) (-7/18,-12/31) -> (12/5,29/12) Hyperbolic Matrix(131,48,30,11) (-3/8,-4/11) -> (4/1,9/2) Hyperbolic Matrix(133,48,36,13) (-4/11,-1/3) -> (11/3,4/1) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(95,-36,66,-25) (3/8,2/5) -> (10/7,3/2) Hyperbolic Matrix(205,-84,144,-59) (2/5,5/12) -> (17/12,10/7) Hyperbolic Matrix(83,-36,30,-13) (3/7,1/2) -> (11/4,3/1) Hyperbolic Matrix(107,-60,66,-37) (1/2,4/7) -> (8/5,13/8) Hyperbolic Matrix(37,-24,54,-35) (5/8,2/3) -> (2/3,7/10) Parabolic Matrix(205,-144,84,-59) (7/10,5/7) -> (17/7,5/2) Hyperbolic Matrix(83,-60,18,-13) (5/7,3/4) -> (9/2,5/1) Hyperbolic Matrix(13,-12,12,-11) (5/6,1/1) -> (1/1,7/6) Parabolic Matrix(107,-132,30,-37) (6/5,5/4) -> (7/2,18/5) Hyperbolic Matrix(95,-132,18,-25) (11/8,7/5) -> (5/1,11/2) Hyperbolic Matrix(349,-492,144,-203) (7/5,17/12) -> (29/12,17/7) Hyperbolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,2,0,1) Matrix(25,132,-18,-95) -> Matrix(1,2,0,1) Matrix(13,60,-18,-83) -> Matrix(1,2,-2,-3) Matrix(13,48,36,133) -> Matrix(1,0,2,1) Matrix(37,132,-30,-107) -> Matrix(3,2,-2,-1) Matrix(11,36,18,59) -> Matrix(1,0,2,1) Matrix(13,36,-30,-83) -> Matrix(1,0,-2,1) Matrix(23,60,18,47) -> Matrix(1,2,0,1) Matrix(11,24,-6,-13) -> Matrix(1,-2,0,1) Matrix(83,144,-132,-229) -> Matrix(1,2,0,1) Matrix(85,144,36,61) -> Matrix(1,2,-2,-3) Matrix(37,60,-66,-107) -> Matrix(3,4,-4,-5) Matrix(23,36,30,47) -> Matrix(1,0,2,1) Matrix(25,36,-66,-95) -> Matrix(1,0,2,1) Matrix(313,432,192,265) -> Matrix(1,2,0,1) Matrix(97,132,36,49) -> Matrix(1,0,0,1) Matrix(47,60,18,23) -> Matrix(1,2,0,1) Matrix(217,264,60,73) -> Matrix(7,10,2,3) Matrix(11,12,-12,-13) -> Matrix(1,0,0,1) Matrix(85,72,72,61) -> Matrix(1,0,2,1) Matrix(59,48,102,83) -> Matrix(1,0,2,1) Matrix(47,36,30,23) -> Matrix(1,2,0,1) Matrix(49,36,132,97) -> Matrix(1,0,2,1) Matrix(35,24,-54,-37) -> Matrix(1,0,0,1) Matrix(227,144,-588,-373) -> Matrix(1,0,4,1) Matrix(59,36,18,11) -> Matrix(1,2,0,1) Matrix(61,36,144,85) -> Matrix(1,2,0,1) Matrix(83,48,102,59) -> Matrix(3,4,2,3) Matrix(217,120,132,73) -> Matrix(1,0,2,1) Matrix(133,72,24,13) -> Matrix(1,0,2,1) Matrix(107,48,78,35) -> Matrix(1,0,2,1) Matrix(85,36,144,61) -> Matrix(1,0,4,1) Matrix(59,24,-150,-61) -> Matrix(1,0,6,1) Matrix(1115,432,462,179) -> Matrix(1,0,2,1) Matrix(131,48,30,11) -> Matrix(1,-2,0,1) Matrix(133,48,36,13) -> Matrix(1,4,0,1) Matrix(1,0,6,1) -> Matrix(1,0,2,1) Matrix(95,-36,66,-25) -> Matrix(1,0,0,1) Matrix(205,-84,144,-59) -> Matrix(1,0,0,1) Matrix(83,-36,30,-13) -> Matrix(1,0,0,1) Matrix(107,-60,66,-37) -> Matrix(3,-2,2,-1) Matrix(37,-24,54,-35) -> Matrix(1,0,0,1) Matrix(205,-144,84,-59) -> Matrix(1,0,0,1) Matrix(83,-60,18,-13) -> Matrix(1,0,-2,1) Matrix(13,-12,12,-11) -> Matrix(1,0,0,1) Matrix(107,-132,30,-37) -> Matrix(5,-8,2,-3) Matrix(95,-132,18,-25) -> Matrix(1,0,0,1) Matrix(349,-492,144,-203) -> Matrix(1,0,2,1) Matrix(13,-24,6,-11) -> 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): 10 Degree of the the map X: 10 Degree of the the map Y: 48 Permutation triple for Y: ((1,7,2)(3,12,13)(4,18,5)(6,23,24)(8,29,30)(9,34,10)(11,17,38)(14,40,35)(15,28,16)(19,25,36)(20,31,21)(22,33,32)(26,46,27)(37,47,43)(39,44,48)(41,45,42); (1,5,21,47,22,6)(2,10,36,37,11,3)(4,16,9,33,26,17)(7,27,40,43,28,8)(12,32,45,19,18,39)(13,31,30,25,24,14)(15,42,20,46,44,23)(29,38,41,35,34,48); (1,3,14,41,15,4)(2,8,31,42,32,9)(5,19,10,35,27,20)(6,25,45,38,26,7)(11,29,28,23,22,12)(13,39,34,16,43,21)(17,37,40,24,44,18)(30,48,46,33,47,36)) ----------------------------------------------------------------------- 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 -2/1 1/0 1 3 -5/3 (-3/2,1/0) 0 6 -8/5 -1/1 4 3 -3/2 -1/1 1 6 -4/3 1/0 1 3 -5/4 0 6 -1/1 0 6 -2/3 0 3 -3/5 0 6 -7/12 -3/2 1 6 -4/7 -1/1 4 3 -1/2 0 6 -3/7 0 6 -5/12 -1/4 1 6 -2/5 0/1 3 3 -3/8 1/1 1 6 -1/3 (-1/2,1/0) 0 6 0/1 0/1 1 3 1/3 (1/2,1/0) 0 6 2/5 (1/2,1/0) 0 3 5/12 1/2 1 6 3/7 0 6 1/2 0/1 1 6 4/7 (1/2,1/0) 0 3 7/12 1/0 1 6 3/5 0 6 2/3 0 3 5/7 0 6 3/4 1/1 1 6 4/5 1/1 4 3 5/6 1/0 1 6 1/1 0 6 7/6 1/0 1 6 6/5 1/1 3 3 5/4 2/1 1 6 4/3 1/0 1 3 7/5 0 6 3/2 1/1 1 6 8/5 (3/2,1/0) 0 3 5/3 (3/2,1/0) 0 6 2/1 1/0 1 3 7/3 (-1/2,1/0) 0 6 12/5 0/1 4 3 5/2 0 6 8/3 1/0 1 3 3/1 0 6 7/2 2/1 1 6 11/3 (7/2,1/0) 0 6 4/1 1/0 3 3 1/0 1/0 1 6 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,-1) (-2/1,1/0) -> (-2/1,1/0) Reflection Matrix(11,20,-6,-11) (-2/1,-5/3) -> (-2/1,-5/3) Reflection Matrix(71,116,30,49) (-5/3,-8/5) -> (7/3,12/5) Glide Reflection Matrix(23,36,30,47) (-8/5,-3/2) -> (3/4,4/5) Hyperbolic Matrix(23,32,18,25) (-3/2,-4/3) -> (5/4,4/3) Glide Reflection Matrix(47,60,18,23) (-4/3,-5/4) -> (5/2,8/3) Hyperbolic Matrix(13,16,-30,-37) (-5/4,-1/1) -> (-1/2,-3/7) Glide Reflection Matrix(11,8,18,13) (-1/1,-2/3) -> (3/5,2/3) Glide Reflection Matrix(25,16,36,23) (-2/3,-3/5) -> (2/3,5/7) Glide Reflection Matrix(61,36,144,85) (-3/5,-7/12) -> (5/12,3/7) Hyperbolic Matrix(83,48,102,59) (-7/12,-4/7) -> (4/5,5/6) Hyperbolic Matrix(59,32,24,13) (-4/7,-1/2) -> (12/5,5/2) Glide Reflection Matrix(85,36,144,61) (-3/7,-5/12) -> (7/12,3/5) Hyperbolic Matrix(107,44,90,37) (-5/12,-2/5) -> (7/6,6/5) Glide Reflection Matrix(73,28,60,23) (-2/5,-3/8) -> (6/5,5/4) Glide Reflection Matrix(109,40,30,11) (-3/8,-1/3) -> (7/2,11/3) Glide Reflection Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(11,-4,30,-11) (1/3,2/5) -> (1/3,2/5) Reflection Matrix(49,-20,120,-49) (2/5,5/12) -> (2/5,5/12) Reflection Matrix(35,-16,24,-11) (3/7,1/2) -> (7/5,3/2) Glide Reflection Matrix(37,-20,24,-13) (1/2,4/7) -> (3/2,8/5) Glide Reflection Matrix(97,-56,168,-97) (4/7,7/12) -> (4/7,7/12) Reflection Matrix(61,-44,18,-13) (5/7,3/4) -> (3/1,7/2) Glide Reflection Matrix(13,-12,12,-11) (5/6,1/1) -> (1/1,7/6) Parabolic Matrix(49,-68,18,-25) (4/3,7/5) -> (8/3,3/1) Glide Reflection Matrix(49,-80,30,-49) (8/5,5/3) -> (8/5,5/3) Reflection Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(23,-88,6,-23) (11/3,4/1) -> (11/3,4/1) Reflection Matrix(-1,8,0,1) (4/1,1/0) -> (4/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,4,0,-1) -> Matrix(1,1,0,-1) (-2/1,1/0) -> (-1/2,1/0) Matrix(11,20,-6,-11) -> Matrix(1,3,0,-1) (-2/1,-5/3) -> (-3/2,1/0) Matrix(71,116,30,49) -> Matrix(1,1,-2,-3) Matrix(23,36,30,47) -> Matrix(1,0,2,1) 0/1 Matrix(23,32,18,25) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(47,60,18,23) -> Matrix(1,2,0,1) 1/0 Matrix(13,16,-30,-37) -> Matrix(1,1,-2,-3) Matrix(11,8,18,13) -> Matrix(1,1,2,1) Matrix(25,16,36,23) -> Matrix(1,1,2,1) Matrix(61,36,144,85) -> Matrix(1,2,0,1) 1/0 Matrix(83,48,102,59) -> Matrix(3,4,2,3) Matrix(59,32,24,13) -> Matrix(1,1,2,1) Matrix(85,36,144,61) -> Matrix(1,0,4,1) 0/1 Matrix(107,44,90,37) -> Matrix(3,1,4,1) Matrix(73,28,60,23) -> Matrix(3,-1,2,-1) Matrix(109,40,30,11) -> Matrix(-1,3,0,1) *** -> (3/2,1/0) Matrix(1,0,6,1) -> Matrix(1,0,2,1) 0/1 Matrix(11,-4,30,-11) -> Matrix(-1,1,0,1) (1/3,2/5) -> (1/2,1/0) Matrix(49,-20,120,-49) -> Matrix(-1,1,0,1) (2/5,5/12) -> (1/2,1/0) Matrix(35,-16,24,-11) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(37,-20,24,-13) -> Matrix(3,-1,2,-1) Matrix(97,-56,168,-97) -> Matrix(-1,1,0,1) (4/7,7/12) -> (1/2,1/0) Matrix(61,-44,18,-13) -> Matrix(3,-1,2,-1) Matrix(13,-12,12,-11) -> Matrix(1,0,0,1) Matrix(49,-68,18,-25) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(49,-80,30,-49) -> Matrix(-1,3,0,1) (8/5,5/3) -> (3/2,1/0) Matrix(13,-24,6,-11) -> Matrix(1,-2,0,1) 1/0 Matrix(23,-88,6,-23) -> Matrix(-1,7,0,1) (11/3,4/1) -> (7/2,1/0) Matrix(-1,8,0,1) -> Matrix(-1,1,0,1) (4/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.