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 -5/1 -4/1 -3/1 -7/3 -2/1 -1/1 -1/2 -1/3 0/1 1/4 1/3 1/2 3/5 2/3 1/1 4/3 5/3 2/1 3/1 11/3 4/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -1/3 -4/1 -1/3 0/1 -7/2 -1/2 -3/1 0/1 -8/3 -1/3 0/1 -5/2 -1/2 -7/3 -1/3 -2/1 0/1 -1/1 0/1 -2/3 0/1 -7/11 1/1 -5/8 1/0 -8/13 0/1 1/1 -3/5 0/1 -7/12 1/0 -4/7 0/1 1/1 -5/9 1/1 -1/2 1/0 -2/5 0/1 -3/8 1/0 -4/11 1/1 1/0 -1/3 1/0 -4/13 -1/1 1/0 -3/10 1/0 -2/7 -2/1 -1/4 -3/2 0/1 -1/1 0/1 1/4 -3/2 1/3 -1/1 3/8 -9/10 5/13 -8/9 2/5 -6/7 3/7 -4/5 1/2 -3/4 4/7 -5/7 -2/3 3/5 -2/3 8/13 -2/3 -7/11 5/8 -5/8 2/3 -2/3 7/10 -5/8 5/7 -3/5 3/4 -1/2 1/1 -1/2 5/4 -1/2 9/7 -1/2 13/10 -1/2 4/3 -1/2 -1/3 11/8 -1/2 7/5 -1/3 3/2 -1/2 8/5 -1/3 0/1 5/3 0/1 12/7 -1/1 0/1 7/4 -1/2 2/1 0/1 5/2 1/0 3/1 -1/1 7/2 -3/4 18/5 -8/11 11/3 -2/3 4/1 -1/1 -2/3 5/1 -2/3 6/1 -4/7 13/2 -1/2 7/1 -1/2 1/0 -1/2 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,20,4,27) (-5/1,1/0) -> (5/7,3/4) Hyperbolic Matrix(9,40,-16,-71) (-5/1,-4/1) -> (-4/7,-5/9) Hyperbolic Matrix(19,68,12,43) (-4/1,-7/2) -> (3/2,8/5) Hyperbolic Matrix(7,24,16,55) (-7/2,-3/1) -> (3/7,1/2) Hyperbolic Matrix(19,52,4,11) (-3/1,-8/3) -> (4/1,5/1) Hyperbolic Matrix(11,28,20,51) (-8/3,-5/2) -> (1/2,4/7) Hyperbolic Matrix(23,56,16,39) (-5/2,-7/3) -> (7/5,3/2) Hyperbolic Matrix(13,28,-20,-43) (-7/3,-2/1) -> (-2/3,-7/11) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(177,112,128,81) (-7/11,-5/8) -> (11/8,7/5) Hyperbolic Matrix(129,80,208,129) (-5/8,-8/13) -> (8/13,5/8) Hyperbolic Matrix(163,100,44,27) (-8/13,-3/5) -> (11/3,4/1) Hyperbolic Matrix(75,44,196,115) (-3/5,-7/12) -> (3/8,5/13) Hyperbolic Matrix(193,112,112,65) (-7/12,-4/7) -> (12/7,7/4) Hyperbolic Matrix(73,40,104,57) (-5/9,-1/2) -> (7/10,5/7) Hyperbolic Matrix(29,12,12,5) (-1/2,-2/5) -> (2/1,5/2) Hyperbolic Matrix(51,20,28,11) (-2/5,-3/8) -> (7/4,2/1) Hyperbolic Matrix(153,56,112,41) (-3/8,-4/11) -> (4/3,11/8) Hyperbolic Matrix(23,8,-72,-25) (-4/11,-1/3) -> (-1/3,-4/13) Parabolic Matrix(209,64,160,49) (-4/13,-3/10) -> (13/10,4/3) Hyperbolic Matrix(69,20,100,29) (-3/10,-2/7) -> (2/3,7/10) Hyperbolic Matrix(43,12,68,19) (-2/7,-1/4) -> (5/8,2/3) Hyperbolic Matrix(1,0,8,1) (-1/4,0/1) -> (0/1,1/4) Parabolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(289,-112,80,-31) (5/13,2/5) -> (18/5,11/3) Hyperbolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(61,-36,100,-59) (4/7,3/5) -> (3/5,8/13) Parabolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(35,-44,4,-5) (5/4,9/7) -> (7/1,1/0) Hyperbolic Matrix(161,-208,24,-31) (9/7,13/10) -> (13/2,7/1) Hyperbolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(77,-276,12,-43) (7/2,18/5) -> (6/1,13/2) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(3,20,4,27) -> Matrix(9,4,-16,-7) Matrix(9,40,-16,-71) -> Matrix(1,0,4,1) Matrix(19,68,12,43) -> Matrix(1,0,0,1) Matrix(7,24,16,55) -> Matrix(11,4,-14,-5) Matrix(19,52,4,11) -> Matrix(5,2,-8,-3) Matrix(11,28,20,51) -> Matrix(1,2,-2,-3) Matrix(23,56,16,39) -> Matrix(1,0,0,1) Matrix(13,28,-20,-43) -> Matrix(1,0,4,1) Matrix(3,4,-4,-5) -> Matrix(1,0,2,1) Matrix(177,112,128,81) -> Matrix(1,-2,-2,5) Matrix(129,80,208,129) -> Matrix(5,2,-8,-3) Matrix(163,100,44,27) -> Matrix(3,-2,-4,3) Matrix(75,44,196,115) -> Matrix(9,-8,-10,9) Matrix(193,112,112,65) -> Matrix(1,0,-2,1) Matrix(73,40,104,57) -> Matrix(5,-8,-8,13) Matrix(29,12,12,5) -> Matrix(1,0,0,1) Matrix(51,20,28,11) -> Matrix(1,0,-2,1) Matrix(153,56,112,41) -> Matrix(1,-2,-2,5) Matrix(23,8,-72,-25) -> Matrix(1,-2,0,1) Matrix(209,64,160,49) -> Matrix(1,0,-2,1) Matrix(69,20,100,29) -> Matrix(5,12,-8,-19) Matrix(43,12,68,19) -> Matrix(1,4,-2,-7) Matrix(1,0,8,1) -> Matrix(1,0,0,1) Matrix(13,-4,36,-11) -> Matrix(11,12,-12,-13) Matrix(289,-112,80,-31) -> Matrix(43,38,-60,-53) Matrix(67,-28,12,-5) -> Matrix(17,14,-28,-23) Matrix(61,-36,100,-59) -> Matrix(35,24,-54,-37) Matrix(9,-8,8,-7) -> Matrix(3,2,-8,-5) Matrix(35,-44,4,-5) -> Matrix(1,0,0,1) Matrix(161,-208,24,-31) -> Matrix(21,10,-40,-19) Matrix(61,-100,36,-59) -> Matrix(1,0,2,1) Matrix(13,-36,4,-11) -> Matrix(3,4,-4,-5) Matrix(77,-276,12,-43) -> Matrix(27,20,-50,-37) 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: 32 Permutation triple for Y: ((1,7,8,2)(3,10,29,11)(4,9,15,5)(6,19,24,20)(12,16,27,26)(13,21,28,14)(17,25,32,18)(22,30,31,23); (1,5,18,6)(2,3)(4,14)(7,23)(8,24,25,9)(10,27,31,28)(11,13,22,12)(15,16)(17,29)(19,26)(20,21)(30,32); (1,3,12,19,18,30,13,4)(2,9,14,31,32,24,26,10)(5,16,22,7,6,21,11,17)(8,23,27,15,25,29,28,20)) ----------------------------------------------------------------------- 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 -5/1 -1/3 2 2 -4/1 (-1/3,0/1) 0 8 -7/2 -1/2 1 8 -3/1 0/1 1 4 -8/3 (-1/3,0/1) 0 8 -5/2 -1/2 1 8 -7/3 -1/3 1 2 -2/1 0/1 2 8 -1/1 0/1 1 2 0/1 (-1/1,0/1) 0 8 1/3 -1/1 6 2 2/5 -6/7 2 8 3/7 -4/5 1 4 1/2 -3/4 1 8 4/7 (-5/7,-2/3) 0 8 3/5 -2/3 3 2 2/3 -2/3 2 8 5/7 -3/5 2 2 3/4 -1/2 1 8 1/1 -1/2 1 4 5/4 -1/2 1 8 9/7 -1/2 5 2 4/3 (-1/2,-1/3) 0 8 7/5 -1/3 1 2 3/2 -1/2 1 8 8/5 (-1/3,0/1) 0 8 5/3 0/1 1 2 2/1 0/1 2 8 3/1 -1/1 2 2 4/1 (-1/1,-2/3) 0 8 5/1 -2/3 1 4 6/1 -4/7 2 8 7/1 -1/2 5 2 1/0 -1/2 1 8 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,20,4,27) (-5/1,1/0) -> (5/7,3/4) Hyperbolic Matrix(9,40,-2,-9) (-5/1,-4/1) -> (-5/1,-4/1) Reflection Matrix(19,68,12,43) (-4/1,-7/2) -> (3/2,8/5) Hyperbolic Matrix(7,24,16,55) (-7/2,-3/1) -> (3/7,1/2) Hyperbolic Matrix(19,52,4,11) (-3/1,-8/3) -> (4/1,5/1) Hyperbolic Matrix(11,28,20,51) (-8/3,-5/2) -> (1/2,4/7) Hyperbolic Matrix(23,56,16,39) (-5/2,-7/3) -> (7/5,3/2) Hyperbolic Matrix(13,28,-6,-13) (-7/3,-2/1) -> (-7/3,-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(1,0,6,-1) (0/1,1/3) -> (0/1,1/3) Reflection Matrix(11,-4,30,-11) (1/3,2/5) -> (1/3,2/5) Reflection Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(41,-24,70,-41) (4/7,3/5) -> (4/7,3/5) Reflection Matrix(19,-12,30,-19) (3/5,2/3) -> (3/5,2/3) Reflection Matrix(29,-20,42,-29) (2/3,5/7) -> (2/3,5/7) Reflection Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(35,-44,4,-5) (5/4,9/7) -> (7/1,1/0) Hyperbolic Matrix(55,-72,42,-55) (9/7,4/3) -> (9/7,4/3) Reflection Matrix(41,-56,30,-41) (4/3,7/5) -> (4/3,7/5) Reflection Matrix(49,-80,30,-49) (8/5,5/3) -> (8/5,5/3) Reflection Matrix(11,-20,6,-11) (5/3,2/1) -> (5/3,2/1) Reflection Matrix(5,-12,2,-5) (2/1,3/1) -> (2/1,3/1) Reflection Matrix(7,-24,2,-7) (3/1,4/1) -> (3/1,4/1) Reflection Matrix(13,-84,2,-13) (6/1,7/1) -> (6/1,7/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(3,20,4,27) -> Matrix(9,4,-16,-7) -1/2 Matrix(9,40,-2,-9) -> Matrix(-1,0,6,1) (-5/1,-4/1) -> (-1/3,0/1) Matrix(19,68,12,43) -> Matrix(1,0,0,1) Matrix(7,24,16,55) -> Matrix(11,4,-14,-5) Matrix(19,52,4,11) -> Matrix(5,2,-8,-3) -1/2 Matrix(11,28,20,51) -> Matrix(1,2,-2,-3) -1/1 Matrix(23,56,16,39) -> Matrix(1,0,0,1) Matrix(13,28,-6,-13) -> Matrix(-1,0,6,1) (-7/3,-2/1) -> (-1/3,0/1) Matrix(3,4,-2,-3) -> Matrix(-1,0,4,1) (-2/1,-1/1) -> (-1/2,0/1) Matrix(-1,0,2,1) -> Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Matrix(1,0,6,-1) -> Matrix(-1,0,2,1) (0/1,1/3) -> (-1/1,0/1) Matrix(11,-4,30,-11) -> Matrix(13,12,-14,-13) (1/3,2/5) -> (-1/1,-6/7) Matrix(67,-28,12,-5) -> Matrix(17,14,-28,-23) Matrix(41,-24,70,-41) -> Matrix(29,20,-42,-29) (4/7,3/5) -> (-5/7,-2/3) Matrix(19,-12,30,-19) -> Matrix(7,4,-12,-7) (3/5,2/3) -> (-2/3,-1/2) Matrix(29,-20,42,-29) -> Matrix(19,12,-30,-19) (2/3,5/7) -> (-2/3,-3/5) Matrix(9,-8,8,-7) -> Matrix(3,2,-8,-5) -1/2 Matrix(35,-44,4,-5) -> Matrix(1,0,0,1) Matrix(55,-72,42,-55) -> Matrix(5,2,-12,-5) (9/7,4/3) -> (-1/2,-1/3) Matrix(41,-56,30,-41) -> Matrix(5,2,-12,-5) (4/3,7/5) -> (-1/2,-1/3) Matrix(49,-80,30,-49) -> Matrix(-1,0,6,1) (8/5,5/3) -> (-1/3,0/1) Matrix(11,-20,6,-11) -> Matrix(-1,0,4,1) (5/3,2/1) -> (-1/2,0/1) Matrix(5,-12,2,-5) -> Matrix(-1,0,2,1) (2/1,3/1) -> (-1/1,0/1) Matrix(7,-24,2,-7) -> Matrix(5,4,-6,-5) (3/1,4/1) -> (-1/1,-2/3) Matrix(13,-84,2,-13) -> Matrix(15,8,-28,-15) (6/1,7/1) -> (-4/7,-1/2) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.