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: 20 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/1 -1/1 0/1 1/2 2/3 3/4 1/1 6/5 5/4 4/3 3/2 2/1 9/4 12/5 5/2 3/1 4/1 9/2 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 -5/1 -1/2 -9/2 -1/2 -4/1 0/1 -3/1 -1/1 -8/3 0/1 -5/2 -1/2 -12/5 0/1 -7/3 1/0 -9/4 -1/1 -2/1 -1/1 -3/2 -1/2 -4/3 0/1 -5/4 -2/3 -1/2 -6/5 -1/2 -1/1 -1/2 -6/7 -1/3 -5/6 -1/2 -4/5 -2/5 -3/4 -1/3 -2/3 -1/3 -9/14 -3/10 -7/11 -3/10 -12/19 -2/7 -5/8 -2/7 -1/4 -8/13 0/1 -3/5 -1/3 -4/7 -2/7 -9/16 -3/11 -5/9 -1/4 -6/11 -1/4 -1/2 -1/4 0/1 0/1 1/2 1/2 3/5 1/1 5/8 1/2 2/3 7/11 3/4 2/3 1/1 3/4 1/1 4/5 2/1 1/1 1/0 6/5 1/0 5/4 -2/1 1/0 4/3 0/1 3/2 1/0 2/1 -1/1 9/4 -1/1 7/3 -1/2 19/8 -1/4 0/1 12/5 0/1 5/2 1/0 13/5 1/0 8/3 0/1 11/4 -2/1 1/0 3/1 -1/1 7/2 -1/2 18/5 -1/2 11/3 -1/2 15/4 -1/3 4/1 0/1 9/2 1/0 14/3 -1/1 5/1 1/0 11/2 1/0 6/1 -1/1 1/0 -1/1 0/1 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(7,36,-8,-41) (-6/1,-5/1) -> (-1/1,-6/7) Hyperbolic Matrix(23,108,-36,-169) (-5/1,-9/2) -> (-9/14,-7/11) Hyperbolic Matrix(17,72,4,17) (-9/2,-4/1) -> (4/1,9/2) Hyperbolic Matrix(7,24,-12,-41) (-4/1,-3/1) -> (-3/5,-4/7) Hyperbolic Matrix(17,48,-28,-79) (-3/1,-8/3) -> (-8/13,-3/5) Hyperbolic Matrix(23,60,-28,-73) (-8/3,-5/2) -> (-5/6,-4/5) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(71,168,-112,-265) (-12/5,-7/3) -> (-7/11,-12/19) Hyperbolic Matrix(47,108,-84,-193) (-7/3,-9/4) -> (-9/16,-5/9) Hyperbolic Matrix(17,36,8,17) (-9/4,-2/1) -> (2/1,9/4) Hyperbolic Matrix(7,12,4,7) (-2/1,-3/2) -> (3/2,2/1) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(47,60,-76,-97) (-4/3,-5/4) -> (-5/8,-8/13) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(31,36,-56,-65) (-6/5,-1/1) -> (-5/9,-6/11) Hyperbolic Matrix(113,96,20,17) (-6/7,-5/6) -> (11/2,6/1) Hyperbolic Matrix(31,24,40,31) (-4/5,-3/4) -> (3/4,4/5) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(223,144,48,31) (-2/3,-9/14) -> (9/2,14/3) Hyperbolic Matrix(457,288,192,121) (-12/19,-5/8) -> (19/8,12/5) Hyperbolic Matrix(169,96,44,25) (-4/7,-9/16) -> (15/4,4/1) Hyperbolic Matrix(113,60,32,17) (-6/11,-1/2) -> (7/2,18/5) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(41,-24,12,-7) (1/2,3/5) -> (3/1,7/2) Hyperbolic Matrix(79,-48,28,-17) (3/5,5/8) -> (11/4,3/1) Hyperbolic Matrix(265,-168,112,-71) (5/8,7/11) -> (7/3,19/8) Hyperbolic Matrix(169,-108,36,-23) (7/11,2/3) -> (14/3,5/1) Hyperbolic Matrix(73,-60,28,-23) (4/5,1/1) -> (13/5,8/3) Hyperbolic Matrix(73,-84,20,-23) (1/1,6/5) -> (18/5,11/3) Hyperbolic Matrix(65,-84,24,-31) (5/4,4/3) -> (8/3,11/4) Hyperbolic Matrix(89,-204,24,-55) (9/4,7/3) -> (11/3,15/4) Hyperbolic Matrix(65,-168,12,-31) (5/2,13/5) -> (5/1,11/2) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,0,0,1) Matrix(7,36,-8,-41) -> Matrix(3,2,-8,-5) Matrix(23,108,-36,-169) -> Matrix(1,2,-4,-7) Matrix(17,72,4,17) -> Matrix(1,0,2,1) Matrix(7,24,-12,-41) -> Matrix(1,2,-4,-7) Matrix(17,48,-28,-79) -> Matrix(1,0,-2,1) Matrix(23,60,-28,-73) -> Matrix(3,2,-8,-5) Matrix(49,120,20,49) -> Matrix(1,0,2,1) Matrix(71,168,-112,-265) -> Matrix(3,-2,-10,7) Matrix(47,108,-84,-193) -> Matrix(1,4,-4,-15) Matrix(17,36,8,17) -> Matrix(1,0,0,1) Matrix(7,12,4,7) -> Matrix(3,2,-2,-1) Matrix(17,24,12,17) -> Matrix(1,0,2,1) Matrix(47,60,-76,-97) -> Matrix(1,0,-2,1) Matrix(49,60,40,49) -> Matrix(7,4,-2,-1) Matrix(31,36,-56,-65) -> Matrix(9,4,-34,-15) Matrix(113,96,20,17) -> Matrix(1,0,2,1) Matrix(31,24,40,31) -> Matrix(11,4,8,3) Matrix(17,12,24,17) -> Matrix(1,0,4,1) Matrix(223,144,48,31) -> Matrix(13,4,-10,-3) Matrix(457,288,192,121) -> Matrix(7,2,-32,-9) Matrix(169,96,44,25) -> Matrix(7,2,-32,-9) Matrix(113,60,32,17) -> Matrix(1,0,2,1) Matrix(1,0,4,1) -> Matrix(1,0,6,1) Matrix(41,-24,12,-7) -> Matrix(3,-2,-4,3) Matrix(79,-48,28,-17) -> Matrix(1,0,-2,1) Matrix(265,-168,112,-71) -> Matrix(3,-2,-10,7) Matrix(169,-108,36,-23) -> Matrix(3,-2,-4,3) Matrix(73,-60,28,-23) -> Matrix(1,-2,0,1) Matrix(73,-84,20,-23) -> Matrix(1,-2,-2,5) Matrix(65,-84,24,-31) -> Matrix(1,0,0,1) Matrix(89,-204,24,-55) -> Matrix(3,2,-8,-5) Matrix(65,-168,12,-31) -> Matrix(1,0,0,1) 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: 32 Permutation triple for Y: ((1,2)(3,5,4,15,24,10)(6,8,7,21,29,18)(9,20,22,17,14,16)(11,19,30,13,12,26)(23,31)(25,32)(27,28); (1,5,16,31,30,15,32,29,12,28,17,6)(2,8,22,27,26,21,25,24,19,23,9,3)(4,13,18,14)(7,11,10,20); (1,3,11,27,12,4)(2,6,13,31,19,7)(5,9)(8,17)(10,25,15,14,28,22)(16,18,32,21,20,23)(24,30)(26,29)) ----------------------------------------------------------------------- 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 0/1 3 2 1/2 1/2 2 12 3/5 1/1 1 4 5/8 (1/2,2/3) 0 12 7/11 3/4 1 12 2/3 1/1 3 6 3/4 1/1 4 4 4/5 2/1 1 6 1/1 1/0 1 12 6/5 1/0 5 2 5/4 (-2/1,1/0) 0 12 4/3 0/1 1 6 3/2 1/0 2 4 2/1 -1/1 1 6 9/4 -1/1 4 4 7/3 -1/2 1 12 19/8 (-1/4,0/1) 0 12 12/5 0/1 4 2 5/2 1/0 2 12 13/5 1/0 1 12 8/3 0/1 1 6 11/4 (-2/1,1/0) 0 12 3/1 -1/1 1 4 7/2 -1/2 2 12 18/5 -1/2 5 2 11/3 -1/2 1 12 15/4 -1/3 4 4 4/1 0/1 3 6 9/2 1/0 2 4 14/3 -1/1 3 6 5/1 1/0 1 12 11/2 1/0 2 12 6/1 -1/1 1 2 1/0 (-1/1,0/1) 0 12 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(41,-24,12,-7) (1/2,3/5) -> (3/1,7/2) Hyperbolic Matrix(79,-48,28,-17) (3/5,5/8) -> (11/4,3/1) Hyperbolic Matrix(265,-168,112,-71) (5/8,7/11) -> (7/3,19/8) Hyperbolic Matrix(169,-108,36,-23) (7/11,2/3) -> (14/3,5/1) Hyperbolic Matrix(17,-12,24,-17) (2/3,3/4) -> (2/3,3/4) Reflection Matrix(31,-24,40,-31) (3/4,4/5) -> (3/4,4/5) Reflection Matrix(73,-60,28,-23) (4/5,1/1) -> (13/5,8/3) Hyperbolic Matrix(73,-84,20,-23) (1/1,6/5) -> (18/5,11/3) Hyperbolic Matrix(49,-60,40,-49) (6/5,5/4) -> (6/5,5/4) Reflection Matrix(65,-84,24,-31) (5/4,4/3) -> (8/3,11/4) Hyperbolic 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(17,-36,8,-17) (2/1,9/4) -> (2/1,9/4) Reflection Matrix(89,-204,24,-55) (9/4,7/3) -> (11/3,15/4) Hyperbolic Matrix(191,-456,80,-191) (19/8,12/5) -> (19/8,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(65,-168,12,-31) (5/2,13/5) -> (5/1,11/2) Hyperbolic Matrix(71,-252,20,-71) (7/2,18/5) -> (7/2,18/5) Reflection Matrix(31,-120,8,-31) (15/4,4/1) -> (15/4,4/1) Reflection Matrix(17,-72,4,-17) (4/1,9/2) -> (4/1,9/2) Reflection Matrix(55,-252,12,-55) (9/2,14/3) -> (9/2,14/3) Reflection Matrix(23,-132,4,-23) (11/2,6/1) -> (11/2,6/1) Reflection Matrix(-1,12,0,1) (6/1,1/0) -> (6/1,1/0) 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(1,0,4,-1) -> Matrix(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Matrix(41,-24,12,-7) -> Matrix(3,-2,-4,3) Matrix(79,-48,28,-17) -> Matrix(1,0,-2,1) 0/1 Matrix(265,-168,112,-71) -> Matrix(3,-2,-10,7) Matrix(169,-108,36,-23) -> Matrix(3,-2,-4,3) Matrix(17,-12,24,-17) -> Matrix(1,0,2,-1) (2/3,3/4) -> (0/1,1/1) Matrix(31,-24,40,-31) -> Matrix(3,-4,2,-3) (3/4,4/5) -> (1/1,2/1) Matrix(73,-60,28,-23) -> Matrix(1,-2,0,1) 1/0 Matrix(73,-84,20,-23) -> Matrix(1,-2,-2,5) Matrix(49,-60,40,-49) -> Matrix(1,4,0,-1) (6/5,5/4) -> (-2/1,1/0) Matrix(65,-84,24,-31) -> Matrix(1,0,0,1) Matrix(17,-24,12,-17) -> Matrix(1,0,0,-1) (4/3,3/2) -> (0/1,1/0) Matrix(7,-12,4,-7) -> Matrix(1,2,0,-1) (3/2,2/1) -> (-1/1,1/0) Matrix(17,-36,8,-17) -> Matrix(-1,0,2,1) (2/1,9/4) -> (-1/1,0/1) Matrix(89,-204,24,-55) -> Matrix(3,2,-8,-5) -1/2 Matrix(191,-456,80,-191) -> Matrix(-1,0,8,1) (19/8,12/5) -> (-1/4,0/1) Matrix(49,-120,20,-49) -> Matrix(1,0,0,-1) (12/5,5/2) -> (0/1,1/0) Matrix(65,-168,12,-31) -> Matrix(1,0,0,1) Matrix(71,-252,20,-71) -> Matrix(3,2,-4,-3) (7/2,18/5) -> (-1/1,-1/2) Matrix(31,-120,8,-31) -> Matrix(-1,0,6,1) (15/4,4/1) -> (-1/3,0/1) Matrix(17,-72,4,-17) -> Matrix(1,0,0,-1) (4/1,9/2) -> (0/1,1/0) Matrix(55,-252,12,-55) -> Matrix(1,2,0,-1) (9/2,14/3) -> (-1/1,1/0) Matrix(23,-132,4,-23) -> Matrix(1,2,0,-1) (11/2,6/1) -> (-1/1,1/0) Matrix(-1,12,0,1) -> Matrix(-1,0,2,1) (6/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.