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: 12 Genus: 11 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 3/2 21/13 2/1 7/3 21/8 3/1 7/2 14/3 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -2/5 -5/1 -1/2 -4/1 -1/3 -1/4 -7/2 -1/3 0/1 -3/1 -1/3 -8/3 -1/3 -1/4 -21/8 -2/7 0/1 -13/5 -1/4 -18/7 0/1 -5/2 -1/3 -2/7 -12/5 -4/15 -7/3 -1/4 -2/1 -1/4 -1/5 -7/4 -1/5 0/1 -12/7 0/1 -5/3 -1/4 -13/8 -4/19 -1/5 -21/13 -1/5 -8/5 -1/5 -3/16 -11/7 -1/6 -14/9 -1/5 -1/6 -3/2 -2/11 0/1 -7/5 -1/6 -4/3 -1/6 -1/7 -9/7 -1/7 -14/11 -1/6 -1/7 -19/15 -1/6 -5/4 -2/13 -1/7 -11/9 -1/8 -17/14 -1/7 -2/15 -6/5 0/1 -1/1 -1/8 0/1 0/1 1/1 1/6 6/5 0/1 5/4 1/5 2/9 4/3 1/5 1/4 7/5 1/4 3/2 0/1 2/7 8/5 3/10 1/3 21/13 1/3 13/8 1/3 4/11 18/11 2/5 5/3 1/2 12/7 0/1 7/4 0/1 1/3 2/1 1/3 1/2 7/3 1/2 12/5 4/7 5/2 2/3 1/1 13/5 1/2 21/8 0/1 2/3 8/3 1/2 1/1 11/4 0/1 1/1 14/5 1/2 1/1 3/1 1/1 7/2 0/1 1/1 4/1 1/2 1/1 9/2 0/1 2/1 14/3 1/1 1/0 19/4 1/1 2/1 5/1 1/0 11/2 0/1 1/1 17/3 1/0 6/1 2/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,42,2,17) (-6/1,1/0) -> (12/5,5/2) Hyperbolic Matrix(31,168,-12,-65) (-6/1,-5/1) -> (-13/5,-18/7) Hyperbolic Matrix(19,84,-12,-53) (-5/1,-4/1) -> (-8/5,-11/7) Hyperbolic Matrix(11,42,6,23) (-4/1,-7/2) -> (7/4,2/1) Hyperbolic Matrix(13,42,4,13) (-7/2,-3/1) -> (3/1,7/2) Hyperbolic Matrix(31,84,-24,-65) (-3/1,-8/3) -> (-4/3,-9/7) Hyperbolic Matrix(127,336,48,127) (-8/3,-21/8) -> (21/8,8/3) Hyperbolic Matrix(209,546,80,209) (-21/8,-13/5) -> (13/5,21/8) Hyperbolic Matrix(131,336,-108,-277) (-18/7,-5/2) -> (-17/14,-6/5) Hyperbolic Matrix(17,42,2,5) (-5/2,-12/5) -> (6/1,1/0) Hyperbolic Matrix(71,168,30,71) (-12/5,-7/3) -> (7/3,12/5) Hyperbolic Matrix(19,42,14,31) (-7/3,-2/1) -> (4/3,7/5) Hyperbolic Matrix(23,42,6,11) (-2/1,-7/4) -> (7/2,4/1) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(25,42,22,37) (-12/7,-5/3) -> (1/1,6/5) Hyperbolic Matrix(103,168,-84,-137) (-5/3,-13/8) -> (-5/4,-11/9) Hyperbolic Matrix(337,546,208,337) (-13/8,-21/13) -> (21/13,13/8) Hyperbolic Matrix(209,336,130,209) (-21/13,-8/5) -> (8/5,21/13) Hyperbolic Matrix(269,420,-212,-331) (-11/7,-14/9) -> (-14/11,-19/15) Hyperbolic Matrix(109,168,24,37) (-14/9,-3/2) -> (9/2,14/3) Hyperbolic Matrix(29,42,20,29) (-3/2,-7/5) -> (7/5,3/2) Hyperbolic Matrix(31,42,14,19) (-7/5,-4/3) -> (2/1,7/3) Hyperbolic Matrix(131,168,46,59) (-9/7,-14/11) -> (14/5,3/1) Hyperbolic Matrix(233,294,42,53) (-19/15,-5/4) -> (11/2,17/3) Hyperbolic Matrix(241,294,50,61) (-11/9,-17/14) -> (19/4,5/1) Hyperbolic Matrix(37,42,22,25) (-6/5,-1/1) -> (5/3,12/7) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(137,-168,84,-103) (6/5,5/4) -> (13/8,18/11) Hyperbolic Matrix(65,-84,24,-31) (5/4,4/3) -> (8/3,11/4) Hyperbolic Matrix(53,-84,12,-19) (3/2,8/5) -> (4/1,9/2) Hyperbolic Matrix(205,-336,36,-59) (18/11,5/3) -> (17/3,6/1) Hyperbolic Matrix(65,-168,12,-31) (5/2,13/5) -> (5/1,11/2) Hyperbolic Matrix(151,-420,32,-89) (11/4,14/5) -> (14/3,19/4) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,42,2,17) -> Matrix(3,2,4,3) Matrix(31,168,-12,-65) -> Matrix(5,2,-18,-7) Matrix(19,84,-12,-53) -> Matrix(5,2,-28,-11) Matrix(11,42,6,23) -> Matrix(1,0,6,1) Matrix(13,42,4,13) -> Matrix(1,0,4,1) Matrix(31,84,-24,-65) -> Matrix(7,2,-46,-13) Matrix(127,336,48,127) -> Matrix(7,2,10,3) Matrix(209,546,80,209) -> Matrix(7,2,10,3) Matrix(131,336,-108,-277) -> Matrix(1,0,-4,1) Matrix(17,42,2,5) -> Matrix(7,2,-4,-1) Matrix(71,168,30,71) -> Matrix(31,8,58,15) Matrix(19,42,14,31) -> Matrix(9,2,40,9) Matrix(23,42,6,11) -> Matrix(1,0,6,1) Matrix(97,168,56,97) -> Matrix(1,0,8,1) Matrix(25,42,22,37) -> Matrix(1,0,10,1) Matrix(103,168,-84,-137) -> Matrix(9,2,-68,-15) Matrix(337,546,208,337) -> Matrix(39,8,112,23) Matrix(209,336,130,209) -> Matrix(31,6,98,19) Matrix(269,420,-212,-331) -> Matrix(11,2,-72,-13) Matrix(109,168,24,37) -> Matrix(1,0,6,1) Matrix(29,42,20,29) -> Matrix(11,2,38,7) Matrix(31,42,14,19) -> Matrix(13,2,32,5) Matrix(131,168,46,59) -> Matrix(1,0,8,1) Matrix(233,294,42,53) -> Matrix(13,2,6,1) Matrix(241,294,50,61) -> Matrix(1,0,8,1) Matrix(37,42,22,25) -> Matrix(1,0,10,1) Matrix(1,0,2,1) -> Matrix(1,0,14,1) Matrix(137,-168,84,-103) -> Matrix(11,-2,28,-5) Matrix(65,-84,24,-31) -> Matrix(9,-2,14,-3) Matrix(53,-84,12,-19) -> Matrix(7,-2,4,-1) Matrix(205,-336,36,-59) -> Matrix(1,0,-2,1) Matrix(65,-168,12,-31) -> Matrix(3,-2,2,-1) Matrix(151,-420,32,-89) -> Matrix(3,-2,2,-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): 8 Degree of the the map X: 8 Degree of the the map Y: 32 Permutation triple for Y: ((2,6,10,9,14,30,20,19,8,13,4,3,12,5,18,23,28,11,16,15,7)(17,21,29,25,22,24,31)(26,27,32); (1,4,16,21,6,20,31,26,9,25,28,23,24,19,32,29,14,13,17,5,2)(3,10,8,7,18,27,11)(12,22,15); (1,2,8,24,12,11,25,32,18,17,20,30,29,16,27,31,23,7,22,9,3)(4,14,26,19,6,5,15)(10,21,13)) ----------------------------------------------------------------------- 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 7 1 1/1 1/6 1 21 6/5 0/1 1 7 5/4 (1/5,2/9) 0 21 4/3 (1/5,1/4) 0 21 7/5 1/4 7 3 3/2 (1/4,1/3) 0 7 8/5 (3/10,1/3) 0 21 21/13 1/3 7 1 13/8 (1/3,4/11) 0 21 18/11 2/5 1 7 5/3 1/2 1 21 12/7 0/1 1 7 7/4 (0/1,1/3) 0 3 2/1 (1/3,1/2) 0 21 7/3 1/2 7 3 12/5 4/7 1 7 5/2 (2/3,1/1) 0 21 13/5 1/2 1 21 21/8 (1/2,1/1) 0 1 8/3 (1/2,1/1) 0 21 11/4 (0/1,1/1) 0 21 14/5 (1/2,1/1) 0 3 3/1 1/1 1 7 7/2 (0/1,1/1) 0 3 4/1 (1/2,1/1) 0 21 9/2 (1/1,1/0) 0 7 14/3 (1/1,1/0) 0 3 19/4 (1/1,2/1) 0 21 5/1 1/0 1 21 11/2 (0/1,1/1) 0 21 17/3 1/0 1 21 6/1 2/1 1 7 1/0 (-1/1,0/1) 0 21 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(37,-42,22,-25) (1/1,6/5) -> (5/3,12/7) Glide Reflection Matrix(137,-168,84,-103) (6/5,5/4) -> (13/8,18/11) Hyperbolic Matrix(65,-84,24,-31) (5/4,4/3) -> (8/3,11/4) Hyperbolic Matrix(31,-42,14,-19) (4/3,7/5) -> (2/1,7/3) Glide Reflection Matrix(29,-42,20,-29) (7/5,3/2) -> (7/5,3/2) Reflection Matrix(53,-84,12,-19) (3/2,8/5) -> (4/1,9/2) Hyperbolic Matrix(209,-336,130,-209) (8/5,21/13) -> (8/5,21/13) Reflection Matrix(337,-546,208,-337) (21/13,13/8) -> (21/13,13/8) Reflection Matrix(205,-336,36,-59) (18/11,5/3) -> (17/3,6/1) Hyperbolic Matrix(97,-168,56,-97) (12/7,7/4) -> (12/7,7/4) Reflection Matrix(23,-42,6,-11) (7/4,2/1) -> (7/2,4/1) Glide Reflection Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(17,-42,2,-5) (12/5,5/2) -> (6/1,1/0) Glide Reflection Matrix(65,-168,12,-31) (5/2,13/5) -> (5/1,11/2) Hyperbolic Matrix(209,-546,80,-209) (13/5,21/8) -> (13/5,21/8) Reflection Matrix(127,-336,48,-127) (21/8,8/3) -> (21/8,8/3) Reflection Matrix(151,-420,32,-89) (11/4,14/5) -> (14/3,19/4) Hyperbolic Matrix(29,-84,10,-29) (14/5,3/1) -> (14/5,3/1) Reflection Matrix(13,-42,4,-13) (3/1,7/2) -> (3/1,7/2) Reflection Matrix(55,-252,12,-55) (9/2,14/3) -> (9/2,14/3) Reflection Matrix(79,-378,14,-67) (19/4,5/1) -> (11/2,17/3) Glide 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(1,0,2,-1) -> Matrix(1,0,12,-1) (0/1,1/1) -> (0/1,1/6) Matrix(37,-42,22,-25) -> Matrix(1,0,8,-1) *** -> (0/1,1/4) Matrix(137,-168,84,-103) -> Matrix(11,-2,28,-5) Matrix(65,-84,24,-31) -> Matrix(9,-2,14,-3) Matrix(31,-42,14,-19) -> Matrix(9,-2,22,-5) Matrix(29,-42,20,-29) -> Matrix(7,-2,24,-7) (7/5,3/2) -> (1/4,1/3) Matrix(53,-84,12,-19) -> Matrix(7,-2,4,-1) Matrix(209,-336,130,-209) -> Matrix(19,-6,60,-19) (8/5,21/13) -> (3/10,1/3) Matrix(337,-546,208,-337) -> Matrix(23,-8,66,-23) (21/13,13/8) -> (1/3,4/11) Matrix(205,-336,36,-59) -> Matrix(1,0,-2,1) 0/1 Matrix(97,-168,56,-97) -> Matrix(1,0,6,-1) (12/7,7/4) -> (0/1,1/3) Matrix(23,-42,6,-11) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(71,-168,30,-71) -> Matrix(15,-8,28,-15) (7/3,12/5) -> (1/2,4/7) Matrix(17,-42,2,-5) -> Matrix(3,-2,-2,1) Matrix(65,-168,12,-31) -> Matrix(3,-2,2,-1) 1/1 Matrix(209,-546,80,-209) -> Matrix(3,-2,4,-3) (13/5,21/8) -> (1/2,1/1) Matrix(127,-336,48,-127) -> Matrix(3,-2,4,-3) (21/8,8/3) -> (1/2,1/1) Matrix(151,-420,32,-89) -> Matrix(3,-2,2,-1) 1/1 Matrix(29,-84,10,-29) -> Matrix(3,-2,4,-3) (14/5,3/1) -> (1/2,1/1) Matrix(13,-42,4,-13) -> Matrix(1,0,2,-1) (3/1,7/2) -> (0/1,1/1) Matrix(55,-252,12,-55) -> Matrix(-1,2,0,1) (9/2,14/3) -> (1/1,1/0) Matrix(79,-378,14,-67) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.