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): 216 Minimal number of generators: 37 Number of equivalence classes of cusps: 24 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -2/5 -1/4 0/1 1/5 2/7 1/2 4/5 1/1 5/4 3/2 11/7 2/1 19/8 5/2 13/5 3/1 7/2 4/1 5/1 6/1 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 1/1 -1/2 1/0 -4/9 0/1 1/0 -3/7 -1/1 1/0 -5/12 1/0 -2/5 -1/1 -7/18 -1/2 -5/13 -1/2 -1/3 -3/8 -1/2 -4/11 -1/6 0/1 -1/3 0/1 1/1 -3/10 1/0 -2/7 0/1 1/0 -5/18 1/0 -3/11 -1/1 1/0 -1/4 0/1 -3/13 1/2 1/1 -2/9 0/1 1/2 -3/14 1/2 -1/5 1/2 1/1 -1/6 3/2 0/1 0/1 1/0 1/5 -3/1 -1/1 2/9 -5/2 -2/1 1/4 -3/2 2/7 -1/1 3/10 -5/6 1/3 -1/1 -1/2 2/5 -1/2 0/1 1/2 0/1 4/7 0/1 1/0 7/12 1/2 10/17 1/1 3/5 1/1 1/0 2/3 0/1 1/0 7/10 1/0 12/17 -1/2 0/1 5/7 -1/1 1/1 8/11 2/1 1/0 11/15 -3/1 1/0 3/4 1/0 4/5 -1/1 5/6 -1/2 1/1 -1/1 0/1 5/4 0/1 9/7 0/1 1/13 4/3 0/1 1/6 7/5 1/5 1/3 10/7 1/4 2/7 3/2 1/2 11/7 1/3 1/1 19/12 1/2 8/5 0/1 1/2 5/3 1/3 1/2 2/1 1/1 7/3 3/1 1/0 19/8 1/0 31/13 -7/1 1/0 12/5 -2/1 1/0 5/2 1/0 13/5 -1/1 1/1 21/8 1/0 8/3 0/1 1/0 3/1 1/1 1/0 7/2 1/0 11/3 -1/1 1/0 4/1 0/1 1/0 9/2 1/0 14/3 -2/1 1/0 5/1 -1/1 1/1 6/1 0/1 1/2 7/1 1/2 1/1 8/1 1/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(67,30,96,43) (-1/2,-4/9) -> (2/3,7/10) Hyperbolic Matrix(127,56,34,15) (-4/9,-3/7) -> (11/3,4/1) Hyperbolic Matrix(33,14,-158,-67) (-3/7,-5/12) -> (-3/14,-1/5) Hyperbolic Matrix(59,24,-150,-61) (-5/12,-2/5) -> (-2/5,-7/18) Parabolic Matrix(237,92,322,125) (-7/18,-5/13) -> (11/15,3/4) Hyperbolic Matrix(89,34,-322,-123) (-5/13,-3/8) -> (-5/18,-3/11) Hyperbolic Matrix(27,10,116,43) (-3/8,-4/11) -> (2/9,1/4) Hyperbolic Matrix(111,40,86,31) (-4/11,-1/3) -> (9/7,4/3) Hyperbolic Matrix(25,8,28,9) (-1/3,-3/10) -> (5/6,1/1) Hyperbolic Matrix(103,30,24,7) (-3/10,-2/7) -> (4/1,9/2) Hyperbolic Matrix(277,78,174,49) (-2/7,-5/18) -> (19/12,8/5) Hyperbolic Matrix(23,6,-96,-25) (-3/11,-1/4) -> (-1/4,-3/13) Parabolic Matrix(141,32,22,5) (-3/13,-2/9) -> (6/1,7/1) Hyperbolic Matrix(301,66,114,25) (-2/9,-3/14) -> (21/8,8/3) Hyperbolic Matrix(21,4,68,13) (-1/5,-1/6) -> (3/10,1/3) Hyperbolic Matrix(63,10,44,7) (-1/6,0/1) -> (10/7,3/2) Hyperbolic Matrix(45,-8,62,-11) (0/1,1/5) -> (5/7,8/11) Hyperbolic Matrix(105,-22,148,-31) (1/5,2/9) -> (12/17,5/7) Hyperbolic Matrix(29,-8,98,-27) (1/4,2/7) -> (2/7,3/10) Parabolic Matrix(39,-14,14,-5) (1/3,2/5) -> (8/3,3/1) Hyperbolic Matrix(13,-6,24,-11) (2/5,1/2) -> (1/2,4/7) Parabolic Matrix(193,-112,274,-159) (4/7,7/12) -> (7/10,12/17) Hyperbolic Matrix(113,-66,12,-7) (7/12,10/17) -> (8/1,1/0) Hyperbolic Matrix(159,-94,22,-13) (10/17,3/5) -> (7/1,8/1) Hyperbolic Matrix(55,-34,34,-21) (3/5,2/3) -> (8/5,5/3) Hyperbolic Matrix(521,-380,218,-159) (8/11,11/15) -> (31/13,12/5) Hyperbolic Matrix(41,-32,50,-39) (3/4,4/5) -> (4/5,5/6) Parabolic Matrix(41,-50,32,-39) (1/1,5/4) -> (5/4,9/7) Parabolic Matrix(45,-62,8,-11) (4/3,7/5) -> (5/1,6/1) Hyperbolic Matrix(105,-148,22,-31) (7/5,10/7) -> (14/3,5/1) Hyperbolic Matrix(155,-242,98,-153) (3/2,11/7) -> (11/7,19/12) Parabolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(305,-722,128,-303) (7/3,19/8) -> (19/8,31/13) Parabolic Matrix(73,-178,16,-39) (12/5,5/2) -> (9/2,14/3) Hyperbolic Matrix(131,-338,50,-129) (5/2,13/5) -> (13/5,21/8) Parabolic Matrix(29,-98,8,-27) (3/1,7/2) -> (7/2,11/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,0,1) Matrix(67,30,96,43) -> Matrix(1,0,0,1) Matrix(127,56,34,15) -> Matrix(1,0,0,1) Matrix(33,14,-158,-67) -> Matrix(1,0,2,1) Matrix(59,24,-150,-61) -> Matrix(1,2,-2,-3) Matrix(237,92,322,125) -> Matrix(3,2,-2,-1) Matrix(89,34,-322,-123) -> Matrix(1,0,2,1) Matrix(27,10,116,43) -> Matrix(7,2,-4,-1) Matrix(111,40,86,31) -> Matrix(1,0,12,1) Matrix(25,8,28,9) -> Matrix(1,0,-2,1) Matrix(103,30,24,7) -> Matrix(1,0,0,1) Matrix(277,78,174,49) -> Matrix(1,0,2,1) Matrix(23,6,-96,-25) -> Matrix(1,0,2,1) Matrix(141,32,22,5) -> Matrix(1,0,0,1) Matrix(301,66,114,25) -> Matrix(1,0,-2,1) Matrix(21,4,68,13) -> Matrix(3,-2,-4,3) Matrix(63,10,44,7) -> Matrix(1,-2,4,-7) Matrix(45,-8,62,-11) -> Matrix(1,2,0,1) Matrix(105,-22,148,-31) -> Matrix(1,2,0,1) Matrix(29,-8,98,-27) -> Matrix(7,8,-8,-9) Matrix(39,-14,14,-5) -> Matrix(1,0,2,1) Matrix(13,-6,24,-11) -> Matrix(1,0,2,1) Matrix(193,-112,274,-159) -> Matrix(1,0,-2,1) Matrix(113,-66,12,-7) -> Matrix(3,-2,2,-1) Matrix(159,-94,22,-13) -> Matrix(1,-2,2,-3) Matrix(55,-34,34,-21) -> Matrix(1,0,2,1) Matrix(521,-380,218,-159) -> Matrix(1,-4,0,1) Matrix(41,-32,50,-39) -> Matrix(1,2,-2,-3) Matrix(41,-50,32,-39) -> Matrix(1,0,14,1) Matrix(45,-62,8,-11) -> Matrix(1,0,-4,1) Matrix(105,-148,22,-31) -> Matrix(1,0,-4,1) Matrix(155,-242,98,-153) -> Matrix(1,0,0,1) Matrix(13,-24,6,-11) -> Matrix(3,-2,2,-1) Matrix(305,-722,128,-303) -> Matrix(1,-10,0,1) Matrix(73,-178,16,-39) -> Matrix(1,0,0,1) Matrix(131,-338,50,-129) -> Matrix(1,0,0,1) Matrix(29,-98,8,-27) -> 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): 9 Degree of the the map X: 9 Degree of the the map Y: 36 Permutation triple for Y: ((1,6,20,34,32,30,21,7,2)(3,12,4)(5,17,28,10,9,24,23,33,13)(8,26,27)(11,18,35,25,15,14,29,22,31); (1,4,15,25,8,7,24,16,5)(3,10,28,27,22,21,19,6,11)(9,14,30)(12,32,34,26,23,29,36,18,13)(17,20,35); (1,2,8,28,35,36,29,9,3)(4,13,33,26,25,20,19,21,14)(5,18,6)(7,22,23)(11,31,27,34,17,16,24,30,12)) ----------------------------------------------------------------------- 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 -1/1 (0/1,1/0) 0 1 0/1 (0/1,1/0) 0 9 1/5 0 3 1/4 -3/2 1 9 2/7 -1/1 4 1 1/3 (-1/1,-1/2) 0 9 2/5 (-1/2,0/1) 0 9 1/2 0/1 1 3 4/7 (0/1,1/0) 0 9 7/12 1/2 1 9 10/17 1/1 2 1 3/5 (1/1,1/0) 0 9 2/3 (0/1,1/0) 0 9 5/7 0 3 8/11 (2/1,1/0) 0 9 3/4 1/0 1 9 4/5 -1/1 1 1 1/1 (-1/1,0/1) 0 9 5/4 0/1 7 1 4/3 (0/1,1/6) 0 9 7/5 0 3 3/2 1/2 1 9 11/7 (0/1,1/2) 0 1 8/5 (0/1,1/2) 0 9 5/3 (1/3,1/2) 0 9 2/1 1/1 1 3 7/3 (3/1,1/0) 0 9 19/8 1/0 5 1 12/5 (-2/1,1/0) 0 9 5/2 1/0 1 9 13/5 (0/1,1/0) 0 1 8/3 (0/1,1/0) 0 9 3/1 (1/1,1/0) 0 9 7/2 1/0 1 1 4/1 (0/1,1/0) 0 9 5/1 0 3 6/1 (0/1,1/2) 0 9 1/0 1/0 1 9 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(45,-8,62,-11) (0/1,1/5) -> (5/7,8/11) Hyperbolic Matrix(43,-10,30,-7) (1/5,1/4) -> (7/5,3/2) Glide Reflection Matrix(15,-4,56,-15) (1/4,2/7) -> (1/4,2/7) Reflection Matrix(13,-4,42,-13) (2/7,1/3) -> (2/7,1/3) Reflection Matrix(39,-14,14,-5) (1/3,2/5) -> (8/3,3/1) Hyperbolic Matrix(13,-6,24,-11) (2/5,1/2) -> (1/2,4/7) Parabolic Matrix(79,-46,12,-7) (4/7,7/12) -> (6/1,1/0) Glide Reflection Matrix(239,-140,408,-239) (7/12,10/17) -> (7/12,10/17) Reflection Matrix(101,-60,170,-101) (10/17,3/5) -> (10/17,3/5) Reflection Matrix(55,-34,34,-21) (3/5,2/3) -> (8/5,5/3) Hyperbolic Matrix(43,-30,10,-7) (2/3,5/7) -> (4/1,5/1) Glide Reflection Matrix(103,-76,42,-31) (8/11,3/4) -> (12/5,5/2) Glide Reflection Matrix(31,-24,40,-31) (3/4,4/5) -> (3/4,4/5) Reflection Matrix(9,-8,10,-9) (4/5,1/1) -> (4/5,1/1) Reflection Matrix(9,-10,8,-9) (1/1,5/4) -> (1/1,5/4) Reflection Matrix(31,-40,24,-31) (5/4,4/3) -> (5/4,4/3) Reflection Matrix(45,-62,8,-11) (4/3,7/5) -> (5/1,6/1) Hyperbolic Matrix(43,-66,28,-43) (3/2,11/7) -> (3/2,11/7) Reflection Matrix(111,-176,70,-111) (11/7,8/5) -> (11/7,8/5) Reflection Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(113,-266,48,-113) (7/3,19/8) -> (7/3,19/8) Reflection Matrix(191,-456,80,-191) (19/8,12/5) -> (19/8,12/5) Reflection Matrix(51,-130,20,-51) (5/2,13/5) -> (5/2,13/5) Reflection Matrix(79,-208,30,-79) (13/5,8/3) -> (13/5,8/3) Reflection Matrix(13,-42,4,-13) (3/1,7/2) -> (3/1,7/2) Reflection Matrix(15,-56,4,-15) (7/2,4/1) -> (7/2,4/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(45,-8,62,-11) -> Matrix(1,2,0,1) 1/0 Matrix(43,-10,30,-7) -> Matrix(1,2,4,7) Matrix(15,-4,56,-15) -> Matrix(5,6,-4,-5) (1/4,2/7) -> (-3/2,-1/1) Matrix(13,-4,42,-13) -> Matrix(3,2,-4,-3) (2/7,1/3) -> (-1/1,-1/2) Matrix(39,-14,14,-5) -> Matrix(1,0,2,1) 0/1 Matrix(13,-6,24,-11) -> Matrix(1,0,2,1) 0/1 Matrix(79,-46,12,-7) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(239,-140,408,-239) -> Matrix(3,-2,4,-3) (7/12,10/17) -> (1/2,1/1) Matrix(101,-60,170,-101) -> Matrix(-1,2,0,1) (10/17,3/5) -> (1/1,1/0) Matrix(55,-34,34,-21) -> Matrix(1,0,2,1) 0/1 Matrix(43,-30,10,-7) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(103,-76,42,-31) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(31,-24,40,-31) -> Matrix(1,2,0,-1) (3/4,4/5) -> (-1/1,1/0) Matrix(9,-8,10,-9) -> Matrix(-1,0,2,1) (4/5,1/1) -> (-1/1,0/1) Matrix(9,-10,8,-9) -> Matrix(-1,0,2,1) (1/1,5/4) -> (-1/1,0/1) Matrix(31,-40,24,-31) -> Matrix(1,0,12,-1) (5/4,4/3) -> (0/1,1/6) Matrix(45,-62,8,-11) -> Matrix(1,0,-4,1) 0/1 Matrix(43,-66,28,-43) -> Matrix(1,0,4,-1) (3/2,11/7) -> (0/1,1/2) Matrix(111,-176,70,-111) -> Matrix(1,0,4,-1) (11/7,8/5) -> (0/1,1/2) Matrix(13,-24,6,-11) -> Matrix(3,-2,2,-1) 1/1 Matrix(113,-266,48,-113) -> Matrix(-1,6,0,1) (7/3,19/8) -> (3/1,1/0) Matrix(191,-456,80,-191) -> Matrix(1,4,0,-1) (19/8,12/5) -> (-2/1,1/0) Matrix(51,-130,20,-51) -> Matrix(1,0,0,-1) (5/2,13/5) -> (0/1,1/0) Matrix(79,-208,30,-79) -> Matrix(1,0,0,-1) (13/5,8/3) -> (0/1,1/0) Matrix(13,-42,4,-13) -> Matrix(-1,2,0,1) (3/1,7/2) -> (1/1,1/0) Matrix(15,-56,4,-15) -> Matrix(1,0,0,-1) (7/2,4/1) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.