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): 384 Minimal number of generators: 65 Number of equivalence classes of cusps: 32 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -4/1 -2/1 -4/3 -8/7 0/1 1/1 16/13 4/3 16/11 3/2 8/5 16/9 2/1 16/7 5/2 8/3 48/17 3/1 16/5 7/2 15/4 4/1 9/2 5/1 16/3 11/2 6/1 13/2 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 0/1 -7/1 0/1 1/1 -6/1 1/0 -11/2 -1/1 0/1 -16/3 0/1 -5/1 0/1 1/0 -9/2 0/1 1/0 -13/3 -1/1 0/1 -4/1 0/1 -7/2 0/1 1/0 -10/3 1/0 -13/4 -2/1 -1/1 -16/5 -1/1 -3/1 -1/1 0/1 -17/6 -2/1 -1/1 -14/5 -1/2 -11/4 -1/1 0/1 -8/3 -1/1 -5/2 -1/1 -1/2 -7/3 -1/3 0/1 -16/7 0/1 -9/4 0/1 1/0 -11/5 0/1 1/0 -2/1 -1/2 -9/5 -1/3 0/1 -16/9 0/1 -7/4 -1/2 0/1 -12/7 0/1 -29/17 -1/1 0/1 -17/10 -1/1 -2/3 -5/3 -1/2 -2/5 -18/11 -3/8 -49/30 -6/17 -1/3 -80/49 -1/3 -31/19 -1/3 0/1 -13/8 -2/5 -1/3 -8/5 -1/3 -3/2 -1/3 0/1 -16/11 -1/3 -13/9 -1/3 -2/7 -23/16 -2/7 -1/4 -10/7 -1/4 -17/12 -1/5 0/1 -24/17 0/1 -7/5 -1/3 0/1 -18/13 -1/2 -11/8 -1/3 0/1 -26/19 -1/2 -41/30 -1/2 0/1 -15/11 -2/5 -1/3 -4/3 -1/3 -17/13 -1/3 0/1 -13/10 -1/3 -2/7 -22/17 -3/10 -9/7 -1/3 -2/7 -5/4 -3/11 -1/4 -16/13 -1/4 -11/9 -1/4 -6/25 -17/14 -3/13 -2/9 -6/5 -1/4 -7/6 -1/4 0/1 -8/7 -1/4 -9/8 -1/4 -2/9 -1/1 -1/5 0/1 0/1 0/1 1/1 0/1 1/5 6/5 1/4 11/9 6/25 1/4 16/13 1/4 5/4 1/4 3/11 4/3 1/3 15/11 1/3 2/5 11/8 0/1 1/3 7/5 0/1 1/3 24/17 0/1 17/12 0/1 1/5 10/7 1/4 13/9 2/7 1/3 16/11 1/3 3/2 0/1 1/3 8/5 1/3 13/8 1/3 2/5 31/19 0/1 1/3 18/11 3/8 5/3 2/5 1/2 7/4 0/1 1/2 16/9 0/1 9/5 0/1 1/3 11/6 0/1 1/3 2/1 1/2 9/4 0/1 1/0 16/7 0/1 7/3 0/1 1/3 19/8 0/1 1/3 31/13 1/3 2/5 12/5 1/3 5/2 1/2 1/1 8/3 1/1 11/4 0/1 1/1 14/5 1/2 31/11 4/5 1/1 48/17 1/1 17/6 1/1 2/1 3/1 0/1 1/1 16/5 1/1 13/4 1/1 2/1 10/3 1/0 17/5 0/1 1/1 41/12 2/1 1/0 24/7 1/0 7/2 0/1 1/0 11/3 2/1 1/0 26/7 1/0 15/4 -2/1 -1/1 4/1 0/1 17/4 2/3 1/1 13/3 0/1 1/1 22/5 1/0 9/2 0/1 1/0 14/3 1/2 5/1 0/1 1/0 16/3 0/1 11/2 0/1 1/1 17/3 1/1 2/1 23/4 0/1 1/0 6/1 1/0 13/2 -1/1 0/1 7/1 -1/1 0/1 8/1 0/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,16,0,1) (-8/1,1/0) -> (8/1,1/0) Parabolic Matrix(31,224,22,159) (-8/1,-7/1) -> (7/5,24/17) Hyperbolic Matrix(31,208,-24,-161) (-7/1,-6/1) -> (-22/17,-9/7) Hyperbolic Matrix(63,352,-46,-257) (-6/1,-11/2) -> (-11/8,-26/19) Hyperbolic Matrix(65,352,12,65) (-11/2,-16/3) -> (16/3,11/2) Hyperbolic Matrix(31,160,6,31) (-16/3,-5/1) -> (5/1,16/3) Hyperbolic Matrix(31,144,-14,-65) (-5/1,-9/2) -> (-9/4,-11/5) Hyperbolic Matrix(95,416,-66,-289) (-9/2,-13/3) -> (-13/9,-23/16) Hyperbolic Matrix(65,272,-38,-159) (-13/3,-4/1) -> (-12/7,-29/17) Hyperbolic Matrix(31,112,-18,-65) (-4/1,-7/2) -> (-7/4,-12/7) Hyperbolic Matrix(33,112,-28,-95) (-7/2,-10/3) -> (-6/5,-7/6) Hyperbolic Matrix(127,416,-98,-321) (-10/3,-13/4) -> (-13/10,-22/17) Hyperbolic Matrix(129,416,40,129) (-13/4,-16/5) -> (16/5,13/4) Hyperbolic Matrix(31,96,10,31) (-16/5,-3/1) -> (3/1,16/5) Hyperbolic Matrix(95,272,22,63) (-3/1,-17/6) -> (17/4,13/3) Hyperbolic Matrix(353,992,-216,-607) (-17/6,-14/5) -> (-18/11,-49/30) Hyperbolic Matrix(63,176,34,95) (-14/5,-11/4) -> (11/6,2/1) Hyperbolic Matrix(65,176,24,65) (-11/4,-8/3) -> (8/3,11/4) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(33,80,-26,-63) (-5/2,-7/3) -> (-9/7,-5/4) Hyperbolic Matrix(97,224,42,97) (-7/3,-16/7) -> (16/7,7/3) Hyperbolic Matrix(127,288,56,127) (-16/7,-9/4) -> (9/4,16/7) Hyperbolic Matrix(95,208,58,127) (-11/5,-2/1) -> (18/11,5/3) Hyperbolic Matrix(97,176,-70,-127) (-2/1,-9/5) -> (-7/5,-18/13) Hyperbolic Matrix(161,288,90,161) (-9/5,-16/9) -> (16/9,9/5) Hyperbolic Matrix(127,224,72,127) (-16/9,-7/4) -> (7/4,16/9) Hyperbolic Matrix(319,544,112,191) (-29/17,-17/10) -> (17/6,3/1) Hyperbolic Matrix(161,272,-132,-223) (-17/10,-5/3) -> (-11/9,-17/14) Hyperbolic Matrix(97,160,20,33) (-5/3,-18/11) -> (14/3,5/1) Hyperbolic Matrix(2273,3712,804,1313) (-49/30,-80/49) -> (48/17,17/6) Hyperbolic Matrix(2431,3968,862,1407) (-80/49,-31/19) -> (31/11,48/17) Hyperbolic Matrix(609,992,256,417) (-31/19,-13/8) -> (19/8,31/13) Hyperbolic Matrix(129,208,80,129) (-13/8,-8/5) -> (8/5,13/8) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(65,96,44,65) (-3/2,-16/11) -> (16/11,3/2) Hyperbolic Matrix(287,416,198,287) (-16/11,-13/9) -> (13/9,16/11) Hyperbolic Matrix(703,1008,-514,-737) (-23/16,-10/7) -> (-26/19,-41/30) Hyperbolic Matrix(191,272,-158,-225) (-10/7,-17/12) -> (-17/14,-6/5) Hyperbolic Matrix(577,816,408,577) (-17/12,-24/17) -> (24/17,17/12) Hyperbolic Matrix(159,224,22,31) (-24/17,-7/5) -> (7/1,8/1) Hyperbolic Matrix(255,352,92,127) (-18/13,-11/8) -> (11/4,14/5) Hyperbolic Matrix(961,1312,282,385) (-41/30,-15/11) -> (17/5,41/12) Hyperbolic Matrix(95,128,-72,-97) (-15/11,-4/3) -> (-4/3,-17/13) Parabolic Matrix(479,624,294,383) (-17/13,-13/10) -> (13/8,31/19) Hyperbolic Matrix(129,160,104,129) (-5/4,-16/13) -> (16/13,5/4) Hyperbolic Matrix(287,352,234,287) (-16/13,-11/9) -> (11/9,16/13) Hyperbolic Matrix(193,224,56,65) (-7/6,-8/7) -> (24/7,7/2) Hyperbolic Matrix(479,544,140,159) (-8/7,-9/8) -> (41/12,24/7) Hyperbolic Matrix(159,176,28,31) (-9/8,-1/1) -> (17/3,23/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(95,-112,28,-33) (1/1,6/5) -> (10/3,17/5) Hyperbolic Matrix(289,-352,78,-95) (6/5,11/9) -> (11/3,26/7) Hyperbolic Matrix(63,-80,26,-33) (5/4,4/3) -> (12/5,5/2) Hyperbolic Matrix(225,-304,94,-127) (4/3,15/11) -> (31/13,12/5) Hyperbolic Matrix(257,-352,46,-63) (15/11,11/8) -> (11/2,17/3) Hyperbolic Matrix(127,-176,70,-97) (11/8,7/5) -> (9/5,11/6) Hyperbolic Matrix(417,-592,112,-159) (17/12,10/7) -> (26/7,15/4) Hyperbolic Matrix(289,-416,66,-95) (10/7,13/9) -> (13/3,22/5) Hyperbolic Matrix(607,-992,216,-353) (31/19,18/11) -> (14/5,31/11) Hyperbolic Matrix(65,-112,18,-31) (5/3,7/4) -> (7/2,11/3) Hyperbolic Matrix(65,-144,14,-31) (2/1,9/4) -> (9/2,14/3) Hyperbolic Matrix(95,-224,14,-33) (7/3,19/8) -> (13/2,7/1) Hyperbolic Matrix(63,-208,10,-33) (13/4,10/3) -> (6/1,13/2) Hyperbolic Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic Matrix(127,-560,22,-97) (22/5,9/2) -> (23/4,6/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,16,0,1) -> Matrix(1,0,0,1) Matrix(31,224,22,159) -> Matrix(1,0,2,1) Matrix(31,208,-24,-161) -> Matrix(3,-2,-10,7) Matrix(63,352,-46,-257) -> Matrix(1,0,-2,1) Matrix(65,352,12,65) -> Matrix(1,0,2,1) Matrix(31,160,6,31) -> Matrix(1,0,0,1) Matrix(31,144,-14,-65) -> Matrix(1,0,0,1) Matrix(95,416,-66,-289) -> Matrix(1,2,-4,-7) Matrix(65,272,-38,-159) -> Matrix(1,0,0,1) Matrix(31,112,-18,-65) -> Matrix(1,0,-2,1) Matrix(33,112,-28,-95) -> Matrix(1,0,-4,1) Matrix(127,416,-98,-321) -> Matrix(3,4,-10,-13) Matrix(129,416,40,129) -> Matrix(3,4,2,3) Matrix(31,96,10,31) -> Matrix(1,0,2,1) Matrix(95,272,22,63) -> Matrix(1,0,2,1) Matrix(353,992,-216,-607) -> Matrix(5,4,-14,-11) Matrix(63,176,34,95) -> Matrix(1,0,4,1) Matrix(65,176,24,65) -> Matrix(1,0,2,1) Matrix(31,80,12,31) -> Matrix(3,2,4,3) Matrix(33,80,-26,-63) -> Matrix(5,2,-18,-7) Matrix(97,224,42,97) -> Matrix(1,0,6,1) Matrix(127,288,56,127) -> Matrix(1,0,0,1) Matrix(95,208,58,127) -> Matrix(1,2,2,5) Matrix(97,176,-70,-127) -> Matrix(1,0,0,1) Matrix(161,288,90,161) -> Matrix(1,0,6,1) Matrix(127,224,72,127) -> Matrix(1,0,4,1) Matrix(319,544,112,191) -> Matrix(1,0,2,1) Matrix(161,272,-132,-223) -> Matrix(7,4,-30,-17) Matrix(97,160,20,33) -> Matrix(5,2,2,1) Matrix(2273,3712,804,1313) -> Matrix(23,8,20,7) Matrix(2431,3968,862,1407) -> Matrix(13,4,16,5) Matrix(609,992,256,417) -> Matrix(5,2,12,5) Matrix(129,208,80,129) -> Matrix(11,4,30,11) Matrix(31,48,20,31) -> Matrix(1,0,6,1) Matrix(65,96,44,65) -> Matrix(1,0,6,1) Matrix(287,416,198,287) -> Matrix(13,4,42,13) Matrix(703,1008,-514,-737) -> Matrix(7,2,-18,-5) Matrix(191,272,-158,-225) -> Matrix(7,2,-32,-9) Matrix(577,816,408,577) -> Matrix(1,0,10,1) Matrix(159,224,22,31) -> Matrix(1,0,2,1) Matrix(255,352,92,127) -> Matrix(1,0,4,1) Matrix(961,1312,282,385) -> Matrix(5,2,2,1) Matrix(95,128,-72,-97) -> Matrix(5,2,-18,-7) Matrix(479,624,294,383) -> Matrix(1,0,6,1) Matrix(129,160,104,129) -> Matrix(23,6,88,23) Matrix(287,352,234,287) -> Matrix(49,12,200,49) Matrix(193,224,56,65) -> Matrix(1,0,4,1) Matrix(479,544,140,159) -> Matrix(17,4,4,1) Matrix(159,176,28,31) -> Matrix(9,2,4,1) Matrix(1,0,2,1) -> Matrix(1,0,10,1) Matrix(95,-112,28,-33) -> Matrix(1,0,-4,1) Matrix(289,-352,78,-95) -> Matrix(17,-4,-4,1) Matrix(63,-80,26,-33) -> Matrix(7,-2,18,-5) Matrix(225,-304,94,-127) -> Matrix(1,0,0,1) Matrix(257,-352,46,-63) -> Matrix(1,0,-2,1) Matrix(127,-176,70,-97) -> Matrix(1,0,0,1) Matrix(417,-592,112,-159) -> Matrix(9,-2,-4,1) Matrix(289,-416,66,-95) -> Matrix(7,-2,4,-1) Matrix(607,-992,216,-353) -> Matrix(11,-4,14,-5) Matrix(65,-112,18,-31) -> Matrix(1,0,-2,1) Matrix(65,-144,14,-31) -> Matrix(1,0,0,1) Matrix(95,-224,14,-33) -> Matrix(1,0,-4,1) Matrix(63,-208,10,-33) -> Matrix(1,-2,0,1) Matrix(33,-128,8,-31) -> Matrix(1,0,2,1) Matrix(127,-560,22,-97) -> 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): 15 Degree of the the map X: 15 Degree of the the map Y: 64 Permutation triple for Y: ((2,6,23,36,63,41,24,7)(3,12,31,55,57,39,13,4)(5,18,50,19)(8,22,56,27)(9,32,33,10)(11,25)(14,21)(15,44,37,16)(17,26,49,28,51,53,20,47)(29,35,34,45,43,42,61,30)(46,62)(52,60); (1,4,16,34,62,51,50,63,64,55,44,61,46,17,5,2)(3,10,35,58,26,8,7,25,57,32,42,54,53,56,36,11)(6,21,31,9,30,59,28,27,41,14,13,33,45,48,47,22)(12,37,29,52,20,19,23,40,39,15,43,60,49,18,24,38); (1,2,8,28,60,29,9,3)(4,14,6,5,20,54,42,15)(7,18,17,48,45,16,12,11)(10,13,40,23,22,26,46,34)(19,51,59,30,44,39,25,36)(21,41,50,49,58,35,37,55)(24,27,53,62,61,32,31,38)(33,57,64,63,56,47,52,43)) ----------------------------------------------------------------------- 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 5 1 1/1 (0/1,1/5) 0 16 6/5 1/4 1 8 11/9 (6/25,1/4) 0 16 16/13 1/4 9 1 5/4 (1/4,3/11) 0 16 4/3 1/3 1 4 15/11 (1/3,2/5) 0 16 11/8 (0/1,1/3) 0 16 7/5 (0/1,1/3) 0 16 24/17 0/1 3 2 17/12 (0/1,1/5) 0 16 10/7 1/4 1 8 13/9 (2/7,1/3) 0 16 16/11 1/3 2 1 3/2 (0/1,1/3) 0 16 8/5 1/3 2 2 13/8 (1/3,2/5) 0 16 31/19 (0/1,1/3) 0 16 18/11 3/8 1 8 5/3 (2/5,1/2) 0 16 7/4 (0/1,1/2) 0 16 16/9 0/1 1 1 9/5 (0/1,1/3) 0 16 11/6 (0/1,1/3) 0 16 2/1 1/2 1 8 9/4 (0/1,1/0) 0 16 16/7 0/1 3 1 7/3 (0/1,1/3) 0 16 19/8 (0/1,1/3) 0 16 31/13 (1/3,2/5) 0 16 12/5 1/3 1 4 5/2 (1/2,1/1) 0 16 8/3 1/1 1 2 11/4 (0/1,1/1) 0 16 14/5 1/2 1 8 31/11 (4/5,1/1) 0 16 48/17 1/1 6 1 17/6 (1/1,2/1) 0 16 3/1 (0/1,1/1) 0 16 16/5 1/1 2 1 13/4 (1/1,2/1) 0 16 10/3 1/0 1 8 17/5 (0/1,1/1) 0 16 41/12 (2/1,1/0) 0 16 24/7 1/0 2 2 7/2 (0/1,1/0) 0 16 11/3 (2/1,1/0) 0 16 26/7 1/0 1 8 15/4 (-2/1,-1/1) 0 16 4/1 0/1 1 4 17/4 (2/3,1/1) 0 16 13/3 (0/1,1/1) 0 16 22/5 1/0 1 8 9/2 (0/1,1/0) 0 16 14/3 1/2 1 8 5/1 (0/1,1/0) 0 16 16/3 0/1 1 1 11/2 (0/1,1/1) 0 16 17/3 (1/1,2/1) 0 16 23/4 (0/1,1/0) 0 16 6/1 1/0 1 8 13/2 (-1/1,0/1) 0 16 7/1 (-1/1,0/1) 0 16 8/1 0/1 3 2 1/0 (0/1,1/0) 0 16 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(95,-112,28,-33) (1/1,6/5) -> (10/3,17/5) Hyperbolic Matrix(289,-352,78,-95) (6/5,11/9) -> (11/3,26/7) Hyperbolic Matrix(287,-352,234,-287) (11/9,16/13) -> (11/9,16/13) Reflection Matrix(129,-160,104,-129) (16/13,5/4) -> (16/13,5/4) Reflection Matrix(63,-80,26,-33) (5/4,4/3) -> (12/5,5/2) Hyperbolic Matrix(225,-304,94,-127) (4/3,15/11) -> (31/13,12/5) Hyperbolic Matrix(257,-352,46,-63) (15/11,11/8) -> (11/2,17/3) Hyperbolic Matrix(127,-176,70,-97) (11/8,7/5) -> (9/5,11/6) Hyperbolic Matrix(159,-224,22,-31) (7/5,24/17) -> (7/1,8/1) Glide Reflection Matrix(577,-816,408,-577) (24/17,17/12) -> (24/17,17/12) Reflection Matrix(417,-592,112,-159) (17/12,10/7) -> (26/7,15/4) Hyperbolic Matrix(289,-416,66,-95) (10/7,13/9) -> (13/3,22/5) Hyperbolic Matrix(287,-416,198,-287) (13/9,16/11) -> (13/9,16/11) Reflection Matrix(65,-96,44,-65) (16/11,3/2) -> (16/11,3/2) Reflection Matrix(31,-48,20,-31) (3/2,8/5) -> (3/2,8/5) Reflection Matrix(129,-208,80,-129) (8/5,13/8) -> (8/5,13/8) Reflection Matrix(609,-992,256,-417) (13/8,31/19) -> (19/8,31/13) Glide Reflection Matrix(607,-992,216,-353) (31/19,18/11) -> (14/5,31/11) Hyperbolic Matrix(97,-160,20,-33) (18/11,5/3) -> (14/3,5/1) Glide Reflection Matrix(65,-112,18,-31) (5/3,7/4) -> (7/2,11/3) Hyperbolic Matrix(127,-224,72,-127) (7/4,16/9) -> (7/4,16/9) Reflection Matrix(161,-288,90,-161) (16/9,9/5) -> (16/9,9/5) Reflection Matrix(95,-176,34,-63) (11/6,2/1) -> (11/4,14/5) Glide Reflection Matrix(65,-144,14,-31) (2/1,9/4) -> (9/2,14/3) Hyperbolic Matrix(127,-288,56,-127) (9/4,16/7) -> (9/4,16/7) Reflection Matrix(97,-224,42,-97) (16/7,7/3) -> (16/7,7/3) Reflection Matrix(95,-224,14,-33) (7/3,19/8) -> (13/2,7/1) Hyperbolic Matrix(31,-80,12,-31) (5/2,8/3) -> (5/2,8/3) Reflection Matrix(65,-176,24,-65) (8/3,11/4) -> (8/3,11/4) Reflection Matrix(1055,-2976,374,-1055) (31/11,48/17) -> (31/11,48/17) Reflection Matrix(577,-1632,204,-577) (48/17,17/6) -> (48/17,17/6) Reflection Matrix(95,-272,22,-63) (17/6,3/1) -> (17/4,13/3) Glide Reflection Matrix(31,-96,10,-31) (3/1,16/5) -> (3/1,16/5) Reflection Matrix(129,-416,40,-129) (16/5,13/4) -> (16/5,13/4) Reflection Matrix(63,-208,10,-33) (13/4,10/3) -> (6/1,13/2) Hyperbolic Matrix(319,-1088,56,-191) (17/5,41/12) -> (17/3,23/4) Glide Reflection Matrix(575,-1968,168,-575) (41/12,24/7) -> (41/12,24/7) Reflection Matrix(97,-336,28,-97) (24/7,7/2) -> (24/7,7/2) Reflection Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic Matrix(127,-560,22,-97) (22/5,9/2) -> (23/4,6/1) Hyperbolic Matrix(31,-160,6,-31) (5/1,16/3) -> (5/1,16/3) Reflection Matrix(65,-352,12,-65) (16/3,11/2) -> (16/3,11/2) Reflection Matrix(-1,16,0,1) (8/1,1/0) -> (8/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,2,-1) -> Matrix(1,0,10,-1) (0/1,1/1) -> (0/1,1/5) Matrix(95,-112,28,-33) -> Matrix(1,0,-4,1) 0/1 Matrix(289,-352,78,-95) -> Matrix(17,-4,-4,1) Matrix(287,-352,234,-287) -> Matrix(49,-12,200,-49) (11/9,16/13) -> (6/25,1/4) Matrix(129,-160,104,-129) -> Matrix(23,-6,88,-23) (16/13,5/4) -> (1/4,3/11) Matrix(63,-80,26,-33) -> Matrix(7,-2,18,-5) 1/3 Matrix(225,-304,94,-127) -> Matrix(1,0,0,1) Matrix(257,-352,46,-63) -> Matrix(1,0,-2,1) 0/1 Matrix(127,-176,70,-97) -> Matrix(1,0,0,1) Matrix(159,-224,22,-31) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(577,-816,408,-577) -> Matrix(1,0,10,-1) (24/17,17/12) -> (0/1,1/5) Matrix(417,-592,112,-159) -> Matrix(9,-2,-4,1) Matrix(289,-416,66,-95) -> Matrix(7,-2,4,-1) Matrix(287,-416,198,-287) -> Matrix(13,-4,42,-13) (13/9,16/11) -> (2/7,1/3) Matrix(65,-96,44,-65) -> Matrix(1,0,6,-1) (16/11,3/2) -> (0/1,1/3) Matrix(31,-48,20,-31) -> Matrix(1,0,6,-1) (3/2,8/5) -> (0/1,1/3) Matrix(129,-208,80,-129) -> Matrix(11,-4,30,-11) (8/5,13/8) -> (1/3,2/5) Matrix(609,-992,256,-417) -> Matrix(5,-2,12,-5) *** -> (1/3,1/2) Matrix(607,-992,216,-353) -> Matrix(11,-4,14,-5) Matrix(97,-160,20,-33) -> Matrix(5,-2,2,-1) Matrix(65,-112,18,-31) -> Matrix(1,0,-2,1) 0/1 Matrix(127,-224,72,-127) -> Matrix(1,0,4,-1) (7/4,16/9) -> (0/1,1/2) Matrix(161,-288,90,-161) -> Matrix(1,0,6,-1) (16/9,9/5) -> (0/1,1/3) Matrix(95,-176,34,-63) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(65,-144,14,-31) -> Matrix(1,0,0,1) Matrix(127,-288,56,-127) -> Matrix(1,0,0,-1) (9/4,16/7) -> (0/1,1/0) Matrix(97,-224,42,-97) -> Matrix(1,0,6,-1) (16/7,7/3) -> (0/1,1/3) Matrix(95,-224,14,-33) -> Matrix(1,0,-4,1) 0/1 Matrix(31,-80,12,-31) -> Matrix(3,-2,4,-3) (5/2,8/3) -> (1/2,1/1) Matrix(65,-176,24,-65) -> Matrix(1,0,2,-1) (8/3,11/4) -> (0/1,1/1) Matrix(1055,-2976,374,-1055) -> Matrix(9,-8,10,-9) (31/11,48/17) -> (4/5,1/1) Matrix(577,-1632,204,-577) -> Matrix(3,-4,2,-3) (48/17,17/6) -> (1/1,2/1) Matrix(95,-272,22,-63) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(31,-96,10,-31) -> Matrix(1,0,2,-1) (3/1,16/5) -> (0/1,1/1) Matrix(129,-416,40,-129) -> Matrix(3,-4,2,-3) (16/5,13/4) -> (1/1,2/1) Matrix(63,-208,10,-33) -> Matrix(1,-2,0,1) 1/0 Matrix(319,-1088,56,-191) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(575,-1968,168,-575) -> Matrix(-1,4,0,1) (41/12,24/7) -> (2/1,1/0) Matrix(97,-336,28,-97) -> Matrix(1,0,0,-1) (24/7,7/2) -> (0/1,1/0) Matrix(33,-128,8,-31) -> Matrix(1,0,2,1) 0/1 Matrix(127,-560,22,-97) -> Matrix(1,0,0,1) Matrix(31,-160,6,-31) -> Matrix(1,0,0,-1) (5/1,16/3) -> (0/1,1/0) Matrix(65,-352,12,-65) -> Matrix(1,0,2,-1) (16/3,11/2) -> (0/1,1/1) Matrix(-1,16,0,1) -> Matrix(1,0,0,-1) (8/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.