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): 432 Minimal number of generators: 73 Number of equivalence classes of cusps: 40 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/6 -7/9 -11/15 -2/3 -17/27 -5/9 -1/2 -3/8 -1/3 -5/17 -19/72 -1/4 -1/6 -1/7 0/1 1/6 1/5 2/9 3/13 1/4 3/11 5/18 2/7 1/3 3/8 7/18 2/5 4/9 1/2 5/9 11/18 17/27 2/3 13/18 11/15 7/9 5/6 8/9 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/0 -7/8 -1/1 1/1 -6/7 1/1 -11/13 0/1 1/0 -5/6 0/1 -9/11 0/1 1/0 -4/5 -1/1 -7/9 0/1 -10/13 1/1 -3/4 -1/1 1/1 -14/19 -1/1 -11/15 -1/1 -1/2 0/1 -8/11 -1/3 -13/18 0/1 -5/7 0/1 1/2 -12/17 1/1 -19/27 1/1 -7/10 -1/1 1/1 -2/3 -1/1 1/1 -7/11 0/1 1/0 -19/30 0/1 -12/19 1/1 -17/27 1/1 -5/8 1/1 3/1 -18/29 1/1 -13/21 2/1 3/1 1/0 -34/55 3/1 -21/34 3/1 11/3 -8/13 5/1 -11/18 1/0 -3/5 -1/1 1/0 -7/12 1/0 -11/19 -1/1 0/1 -4/7 -1/1 -5/9 1/0 -6/11 -3/1 -7/13 -2/1 1/0 -1/2 -1/1 1/1 -6/13 -1/3 -5/11 -1/2 0/1 -4/9 0/1 -3/7 0/1 1/0 -5/12 1/0 -7/17 -2/1 -1/1 -9/22 -1/1 -1/3 -2/5 -1/1 -7/18 0/1 -5/13 0/1 1/2 -13/34 3/5 1/1 -8/21 -1/1 1/1 -3/8 -1/1 1/1 -1/3 -1/1 0/1 1/0 -3/10 -1/1 1/1 -11/37 0/1 1/0 -8/27 0/1 -5/17 0/1 1/1 -2/7 1/1 -5/18 1/0 -3/11 -2/1 1/0 -4/15 -3/1 -1/1 -9/34 -7/5 -1/1 -14/53 -1/1 -19/72 -1/1 -5/19 -1/1 -2/3 -1/4 -1/1 1/1 -4/17 1/1 -7/30 2/1 -3/13 2/1 1/0 -2/9 1/0 -1/5 -1/1 1/0 -1/6 1/0 -1/7 -2/1 1/0 0/1 -1/1 1/6 -1/2 1/5 -1/1 -1/2 2/9 -1/2 3/13 -1/2 -2/5 4/17 -1/3 1/4 -1/1 -1/3 3/11 -2/3 -1/2 5/18 -1/2 2/7 -1/3 1/3 -1/1 -1/2 0/1 4/11 -1/3 7/19 -1/3 0/1 10/27 0/1 3/8 -1/1 -1/3 5/13 -1/4 0/1 7/18 0/1 2/5 -1/1 7/17 -1/1 -2/3 5/12 -1/2 8/19 -1/3 3/7 -1/2 0/1 4/9 0/1 5/11 0/1 1/0 1/2 -1/1 -1/3 5/9 -1/2 9/16 -3/7 -1/3 13/23 -1/2 0/1 4/7 -1/1 11/19 -1/1 0/1 7/12 -1/2 3/5 -1/1 -1/2 11/18 -1/2 8/13 -5/11 21/34 -11/25 -3/7 13/21 -1/2 -3/7 -2/5 5/8 -3/7 -1/3 17/27 -1/3 12/19 -1/3 7/11 -1/2 0/1 9/14 -3/7 -1/3 11/17 -4/11 -1/3 2/3 -1/1 -1/3 13/19 -4/11 -1/3 11/16 -1/3 -3/11 9/13 -1/4 0/1 7/10 -1/1 -1/3 19/27 -1/3 12/17 -1/3 5/7 -1/4 0/1 13/18 0/1 8/11 1/1 19/26 -1/1 1/1 11/15 -1/1 0/1 1/0 25/34 -5/3 -1/1 39/53 -8/7 -1/1 53/72 -1/1 14/19 -1/1 3/4 -1/1 -1/3 7/9 0/1 11/14 -1/1 1/1 15/19 -1/1 -2/3 4/5 -1/1 13/16 -1/3 -1/5 9/11 -1/2 0/1 14/17 -1/3 5/6 0/1 16/19 -1/1 11/13 -1/2 0/1 6/7 -1/3 7/8 -1/1 -1/3 8/9 0/1 1/1 -1/2 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(91,80,-306,-269) (-1/1,-7/8) -> (-3/10,-11/37) Hyperbolic Matrix(197,170,270,233) (-7/8,-6/7) -> (8/11,19/26) Hyperbolic Matrix(143,122,252,215) (-6/7,-11/13) -> (13/23,4/7) Hyperbolic Matrix(109,92,-468,-395) (-11/13,-5/6) -> (-7/30,-3/13) Hyperbolic Matrix(251,206,-396,-325) (-5/6,-9/11) -> (-7/11,-19/30) Hyperbolic Matrix(91,74,-198,-161) (-9/11,-4/5) -> (-6/13,-5/11) Hyperbolic Matrix(125,98,-162,-127) (-4/5,-7/9) -> (-7/9,-10/13) Parabolic Matrix(125,96,-306,-235) (-10/13,-3/4) -> (-9/22,-2/5) Hyperbolic Matrix(35,26,144,107) (-3/4,-14/19) -> (4/17,1/4) Hyperbolic Matrix(757,556,-1224,-899) (-14/19,-11/15) -> (-13/21,-34/55) Hyperbolic Matrix(413,302,-666,-487) (-11/15,-8/11) -> (-18/29,-13/21) Hyperbolic Matrix(287,208,396,287) (-8/11,-13/18) -> (13/18,8/11) Hyperbolic Matrix(181,130,252,181) (-13/18,-5/7) -> (5/7,13/18) Hyperbolic Matrix(107,76,252,179) (-5/7,-12/17) -> (8/19,3/7) Hyperbolic Matrix(613,432,972,685) (-12/17,-19/27) -> (17/27,12/19) Hyperbolic Matrix(305,214,486,341) (-19/27,-7/10) -> (5/8,17/27) Hyperbolic Matrix(89,62,-234,-163) (-7/10,-2/3) -> (-8/21,-3/8) Hyperbolic Matrix(53,34,-198,-127) (-2/3,-7/11) -> (-3/11,-4/15) Hyperbolic Matrix(253,160,-1080,-683) (-19/30,-12/19) -> (-4/17,-7/30) Hyperbolic Matrix(685,432,972,613) (-12/19,-17/27) -> (19/27,12/17) Hyperbolic Matrix(341,214,486,305) (-17/27,-5/8) -> (7/10,19/27) Hyperbolic Matrix(251,156,288,179) (-5/8,-18/29) -> (6/7,7/8) Hyperbolic Matrix(971,600,-3672,-2269) (-34/55,-21/34) -> (-9/34,-14/53) Hyperbolic Matrix(305,188,378,233) (-21/34,-8/13) -> (4/5,13/16) Hyperbolic Matrix(287,176,468,287) (-8/13,-11/18) -> (11/18,8/13) Hyperbolic Matrix(109,66,180,109) (-11/18,-3/5) -> (3/5,11/18) Hyperbolic Matrix(17,10,90,53) (-3/5,-7/12) -> (1/6,1/5) Hyperbolic Matrix(179,104,432,251) (-7/12,-11/19) -> (7/17,5/12) Hyperbolic Matrix(125,72,342,197) (-11/19,-4/7) -> (4/11,7/19) Hyperbolic Matrix(89,50,-162,-91) (-4/7,-5/9) -> (-5/9,-6/11) Parabolic Matrix(199,108,234,127) (-6/11,-7/13) -> (11/13,6/7) Hyperbolic Matrix(161,86,234,125) (-7/13,-1/2) -> (11/16,9/13) Hyperbolic Matrix(289,134,468,217) (-1/2,-6/13) -> (8/13,21/34) Hyperbolic Matrix(89,40,198,89) (-5/11,-4/9) -> (4/9,5/11) Hyperbolic Matrix(55,24,126,55) (-4/9,-3/7) -> (3/7,4/9) Hyperbolic Matrix(19,8,-126,-53) (-3/7,-5/12) -> (-1/6,-1/7) Hyperbolic Matrix(251,104,432,179) (-5/12,-7/17) -> (11/19,7/12) Hyperbolic Matrix(395,162,612,251) (-7/17,-9/22) -> (9/14,11/17) Hyperbolic Matrix(71,28,180,71) (-2/5,-7/18) -> (7/18,2/5) Hyperbolic Matrix(181,70,468,181) (-7/18,-5/13) -> (5/13,7/18) Hyperbolic Matrix(559,214,990,379) (-5/13,-13/34) -> (9/16,13/23) Hyperbolic Matrix(325,124,-1224,-467) (-13/34,-8/21) -> (-4/15,-9/34) Hyperbolic Matrix(17,6,-54,-19) (-3/8,-1/3) -> (-1/3,-3/10) Parabolic Matrix(323,96,360,107) (-11/37,-8/27) -> (8/9,1/1) Hyperbolic Matrix(359,106,972,287) (-8/27,-5/17) -> (7/19,10/27) Hyperbolic Matrix(145,42,252,73) (-5/17,-2/7) -> (4/7,11/19) Hyperbolic Matrix(71,20,252,71) (-2/7,-5/18) -> (5/18,2/7) Hyperbolic Matrix(109,30,396,109) (-5/18,-3/11) -> (3/11,5/18) Hyperbolic Matrix(3817,1008,5184,1369) (-14/53,-19/72) -> (53/72,14/19) Hyperbolic Matrix(3815,1006,5184,1367) (-19/72,-5/19) -> (39/53,53/72) Hyperbolic Matrix(269,70,342,89) (-5/19,-1/4) -> (11/14,15/19) Hyperbolic Matrix(107,26,144,35) (-1/4,-4/17) -> (14/19,3/4) Hyperbolic Matrix(53,12,234,53) (-3/13,-2/9) -> (2/9,3/13) Hyperbolic Matrix(19,4,90,19) (-2/9,-1/5) -> (1/5,2/9) Hyperbolic Matrix(53,10,90,17) (-1/5,-1/6) -> (7/12,3/5) Hyperbolic Matrix(89,12,126,17) (-1/7,0/1) -> (12/17,5/7) Hyperbolic Matrix(53,-8,126,-19) (0/1,1/6) -> (5/12,8/19) Hyperbolic Matrix(395,-92,468,-109) (3/13,4/17) -> (16/19,11/13) Hyperbolic Matrix(127,-34,198,-53) (1/4,3/11) -> (7/11,9/14) Hyperbolic Matrix(19,-6,54,-17) (2/7,1/3) -> (1/3,4/11) Parabolic Matrix(253,-94,288,-107) (10/27,3/8) -> (7/8,8/9) Hyperbolic Matrix(163,-62,234,-89) (3/8,5/13) -> (9/13,7/10) Hyperbolic Matrix(143,-58,180,-73) (2/5,7/17) -> (15/19,4/5) Hyperbolic Matrix(161,-74,198,-91) (5/11,1/2) -> (13/16,9/11) Hyperbolic Matrix(91,-50,162,-89) (1/2,5/9) -> (5/9,9/16) Parabolic Matrix(899,-556,1224,-757) (21/34,13/21) -> (11/15,25/34) Hyperbolic Matrix(487,-302,666,-413) (13/21,5/8) -> (19/26,11/15) Hyperbolic Matrix(325,-206,396,-251) (12/19,7/11) -> (9/11,14/17) Hyperbolic Matrix(73,-48,108,-71) (11/17,2/3) -> (2/3,13/19) Parabolic Matrix(1099,-754,1494,-1025) (13/19,11/16) -> (25/34,39/53) Hyperbolic Matrix(127,-98,162,-125) (3/4,7/9) -> (7/9,11/14) Parabolic Matrix(181,-150,216,-179) (14/17,5/6) -> (5/6,16/19) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(91,80,-306,-269) -> Matrix(1,0,0,1) Matrix(197,170,270,233) -> Matrix(1,0,0,1) Matrix(143,122,252,215) -> Matrix(1,0,-2,1) Matrix(109,92,-468,-395) -> Matrix(1,2,0,1) Matrix(251,206,-396,-325) -> Matrix(1,0,0,1) Matrix(91,74,-198,-161) -> Matrix(1,0,-2,1) Matrix(125,98,-162,-127) -> Matrix(1,0,2,1) Matrix(125,96,-306,-235) -> Matrix(1,0,-2,1) Matrix(35,26,144,107) -> Matrix(1,0,-2,1) Matrix(757,556,-1224,-899) -> Matrix(5,2,2,1) Matrix(413,302,-666,-487) -> Matrix(5,2,2,1) Matrix(287,208,396,287) -> Matrix(1,0,4,1) Matrix(181,130,252,181) -> Matrix(1,0,-6,1) Matrix(107,76,252,179) -> Matrix(1,0,-4,1) Matrix(613,432,972,685) -> Matrix(1,0,-4,1) Matrix(305,214,486,341) -> Matrix(1,-2,-2,5) Matrix(89,62,-234,-163) -> Matrix(1,0,0,1) Matrix(53,34,-198,-127) -> Matrix(1,-2,0,1) Matrix(253,160,-1080,-683) -> Matrix(3,-2,2,-1) Matrix(685,432,972,613) -> Matrix(1,0,-4,1) Matrix(341,214,486,305) -> Matrix(1,-2,-2,5) Matrix(251,156,288,179) -> Matrix(1,-2,-2,5) Matrix(971,600,-3672,-2269) -> Matrix(5,-16,-4,13) Matrix(305,188,378,233) -> Matrix(1,-4,-2,9) Matrix(287,176,468,287) -> Matrix(1,-10,-2,21) Matrix(109,66,180,109) -> Matrix(1,2,-2,-3) Matrix(17,10,90,53) -> Matrix(1,2,-2,-3) Matrix(179,104,432,251) -> Matrix(1,2,-2,-3) Matrix(125,72,342,197) -> Matrix(1,0,-2,1) Matrix(89,50,-162,-91) -> Matrix(1,-2,0,1) Matrix(199,108,234,127) -> Matrix(1,2,-2,-3) Matrix(161,86,234,125) -> Matrix(1,2,-4,-7) Matrix(289,134,468,217) -> Matrix(7,4,-16,-9) Matrix(89,40,198,89) -> Matrix(1,0,2,1) Matrix(55,24,126,55) -> Matrix(1,0,-2,1) Matrix(19,8,-126,-53) -> Matrix(1,-2,0,1) Matrix(251,104,432,179) -> Matrix(1,2,-2,-3) Matrix(395,162,612,251) -> Matrix(3,2,-8,-5) Matrix(71,28,180,71) -> Matrix(1,0,0,1) Matrix(181,70,468,181) -> Matrix(1,0,-6,1) Matrix(559,214,990,379) -> Matrix(1,0,-4,1) Matrix(325,124,-1224,-467) -> Matrix(1,-2,0,1) Matrix(17,6,-54,-19) -> Matrix(1,0,0,1) Matrix(323,96,360,107) -> Matrix(1,0,-2,1) Matrix(359,106,972,287) -> Matrix(1,0,-4,1) Matrix(145,42,252,73) -> Matrix(1,0,-2,1) Matrix(71,20,252,71) -> Matrix(1,-2,-2,5) Matrix(109,30,396,109) -> Matrix(1,4,-2,-7) Matrix(3817,1008,5184,1369) -> Matrix(11,12,-12,-13) Matrix(3815,1006,5184,1367) -> Matrix(11,10,-10,-9) Matrix(269,70,342,89) -> Matrix(1,0,0,1) Matrix(107,26,144,35) -> Matrix(1,0,-2,1) Matrix(53,12,234,53) -> Matrix(1,-4,-2,9) Matrix(19,4,90,19) -> Matrix(1,2,-2,-3) Matrix(53,10,90,17) -> Matrix(1,2,-2,-3) Matrix(89,12,126,17) -> Matrix(1,2,-4,-7) Matrix(53,-8,126,-19) -> Matrix(3,2,-8,-5) Matrix(395,-92,468,-109) -> Matrix(5,2,-8,-3) Matrix(127,-34,198,-53) -> Matrix(3,2,-8,-5) Matrix(19,-6,54,-17) -> Matrix(1,0,0,1) Matrix(253,-94,288,-107) -> Matrix(1,0,0,1) Matrix(163,-62,234,-89) -> Matrix(1,0,0,1) Matrix(143,-58,180,-73) -> Matrix(1,0,0,1) Matrix(161,-74,198,-91) -> Matrix(1,0,-2,1) Matrix(91,-50,162,-89) -> Matrix(3,2,-8,-5) Matrix(899,-556,1224,-757) -> Matrix(5,2,2,1) Matrix(487,-302,666,-413) -> Matrix(5,2,2,1) Matrix(325,-206,396,-251) -> Matrix(1,0,0,1) Matrix(73,-48,108,-71) -> Matrix(1,0,0,1) Matrix(1099,-754,1494,-1025) -> Matrix(13,4,-10,-3) Matrix(127,-98,162,-125) -> Matrix(1,0,2,1) Matrix(181,-150,216,-179) -> Matrix(1,0,2,1) 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): 16 Degree of the the map X: 16 Degree of the the map Y: 72 ----------------------------------------------------------------------- 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 -1/1 1 18 1/6 -1/2 1 3 1/5 (-1/1,-1/2) 0 18 2/9 -1/2 3 2 3/13 (-1/2,-2/5) 0 18 4/17 -1/3 1 18 1/4 0 9 3/11 (-2/3,-1/2) 0 18 5/18 -1/2 3 1 2/7 -1/3 1 18 1/3 0 6 4/11 -1/3 1 18 7/19 (-1/3,0/1) 0 18 10/27 0/1 1 2 3/8 0 9 5/13 (-1/4,0/1) 0 18 7/18 0/1 3 1 2/5 -1/1 1 18 7/17 (-1/1,-2/3) 0 18 5/12 -1/2 1 3 8/19 -1/3 1 18 3/7 (-1/2,0/1) 0 18 4/9 0/1 2 2 5/11 (0/1,1/0) 0 18 1/2 0 9 5/9 -1/2 2 2 9/16 0 9 13/23 (-1/2,0/1) 0 18 4/7 -1/1 1 18 11/19 (-1/1,0/1) 0 18 7/12 -1/2 1 3 3/5 (-1/1,-1/2) 0 18 11/18 -1/2 6 1 8/13 -5/11 1 18 21/34 0 9 13/21 0 6 5/8 0 9 17/27 -1/3 2 2 12/19 -1/3 1 18 7/11 (-1/2,0/1) 0 18 9/14 0 9 11/17 (-4/11,-1/3) 0 18 2/3 0 6 13/19 (-4/11,-1/3) 0 18 11/16 0 9 9/13 (-1/4,0/1) 0 18 7/10 0 9 19/27 -1/3 2 2 12/17 -1/3 1 18 5/7 (-1/4,0/1) 0 18 13/18 0/1 5 1 8/11 1/1 1 18 19/26 0 9 11/15 0 6 25/34 0 9 39/53 (-8/7,-1/1) 0 18 53/72 -1/1 11 1 14/19 -1/1 1 18 3/4 0 9 7/9 0/1 2 2 11/14 0 9 15/19 (-1/1,-2/3) 0 18 4/5 -1/1 1 18 13/16 0 9 9/11 (-1/2,0/1) 0 18 14/17 -1/3 1 18 5/6 0/1 1 3 16/19 -1/1 1 18 11/13 (-1/2,0/1) 0 18 6/7 -1/3 1 18 7/8 0 9 8/9 0/1 1 2 1/1 (-1/2,0/1) 0 18 1/0 0/1 1 1 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(53,-8,126,-19) (0/1,1/6) -> (5/12,8/19) Hyperbolic Matrix(53,-10,90,-17) (1/6,1/5) -> (7/12,3/5) Glide Reflection Matrix(19,-4,90,-19) (1/5,2/9) -> (1/5,2/9) Reflection Matrix(53,-12,234,-53) (2/9,3/13) -> (2/9,3/13) Reflection Matrix(395,-92,468,-109) (3/13,4/17) -> (16/19,11/13) Hyperbolic Matrix(107,-26,144,-35) (4/17,1/4) -> (14/19,3/4) Glide Reflection Matrix(127,-34,198,-53) (1/4,3/11) -> (7/11,9/14) Hyperbolic Matrix(109,-30,396,-109) (3/11,5/18) -> (3/11,5/18) Reflection Matrix(71,-20,252,-71) (5/18,2/7) -> (5/18,2/7) Reflection Matrix(19,-6,54,-17) (2/7,1/3) -> (1/3,4/11) Parabolic Matrix(197,-72,342,-125) (4/11,7/19) -> (4/7,11/19) Glide Reflection Matrix(379,-140,1026,-379) (7/19,10/27) -> (7/19,10/27) Reflection Matrix(253,-94,288,-107) (10/27,3/8) -> (7/8,8/9) Hyperbolic Matrix(163,-62,234,-89) (3/8,5/13) -> (9/13,7/10) Hyperbolic Matrix(181,-70,468,-181) (5/13,7/18) -> (5/13,7/18) Reflection Matrix(71,-28,180,-71) (7/18,2/5) -> (7/18,2/5) Reflection Matrix(143,-58,180,-73) (2/5,7/17) -> (15/19,4/5) Hyperbolic Matrix(251,-104,432,-179) (7/17,5/12) -> (11/19,7/12) Glide Reflection Matrix(179,-76,252,-107) (8/19,3/7) -> (12/17,5/7) Glide Reflection Matrix(55,-24,126,-55) (3/7,4/9) -> (3/7,4/9) Reflection Matrix(89,-40,198,-89) (4/9,5/11) -> (4/9,5/11) Reflection Matrix(161,-74,198,-91) (5/11,1/2) -> (13/16,9/11) Hyperbolic Matrix(91,-50,162,-89) (1/2,5/9) -> (5/9,9/16) Parabolic Matrix(397,-224,576,-325) (9/16,13/23) -> (11/16,9/13) Glide Reflection Matrix(215,-122,252,-143) (13/23,4/7) -> (11/13,6/7) Glide Reflection Matrix(109,-66,180,-109) (3/5,11/18) -> (3/5,11/18) Reflection Matrix(287,-176,468,-287) (11/18,8/13) -> (11/18,8/13) Reflection Matrix(305,-188,378,-233) (8/13,21/34) -> (4/5,13/16) Glide Reflection Matrix(899,-556,1224,-757) (21/34,13/21) -> (11/15,25/34) Hyperbolic Matrix(487,-302,666,-413) (13/21,5/8) -> (19/26,11/15) Hyperbolic Matrix(341,-214,486,-305) (5/8,17/27) -> (7/10,19/27) Glide Reflection Matrix(685,-432,972,-613) (17/27,12/19) -> (19/27,12/17) Glide Reflection Matrix(325,-206,396,-251) (12/19,7/11) -> (9/11,14/17) Hyperbolic Matrix(397,-256,504,-325) (9/14,11/17) -> (11/14,15/19) Glide Reflection Matrix(73,-48,108,-71) (11/17,2/3) -> (2/3,13/19) Parabolic Matrix(1099,-754,1494,-1025) (13/19,11/16) -> (25/34,39/53) Hyperbolic Matrix(181,-130,252,-181) (5/7,13/18) -> (5/7,13/18) Reflection Matrix(287,-208,396,-287) (13/18,8/11) -> (13/18,8/11) Reflection Matrix(233,-170,270,-197) (8/11,19/26) -> (6/7,7/8) Glide Reflection Matrix(5617,-4134,7632,-5617) (39/53,53/72) -> (39/53,53/72) Reflection Matrix(2015,-1484,2736,-2015) (53/72,14/19) -> (53/72,14/19) Reflection Matrix(127,-98,162,-125) (3/4,7/9) -> (7/9,11/14) Parabolic Matrix(181,-150,216,-179) (14/17,5/6) -> (5/6,16/19) Parabolic Matrix(17,-16,18,-17) (8/9,1/1) -> (8/9,1/1) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) 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(53,-8,126,-19) -> Matrix(3,2,-8,-5) -1/2 Matrix(53,-10,90,-17) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(19,-4,90,-19) -> Matrix(3,2,-4,-3) (1/5,2/9) -> (-1/1,-1/2) Matrix(53,-12,234,-53) -> Matrix(9,4,-20,-9) (2/9,3/13) -> (-1/2,-2/5) Matrix(395,-92,468,-109) -> Matrix(5,2,-8,-3) -1/2 Matrix(107,-26,144,-35) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(127,-34,198,-53) -> Matrix(3,2,-8,-5) -1/2 Matrix(109,-30,396,-109) -> Matrix(7,4,-12,-7) (3/11,5/18) -> (-2/3,-1/2) Matrix(71,-20,252,-71) -> Matrix(5,2,-12,-5) (5/18,2/7) -> (-1/2,-1/3) Matrix(19,-6,54,-17) -> Matrix(1,0,0,1) Matrix(197,-72,342,-125) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(379,-140,1026,-379) -> Matrix(-1,0,6,1) (7/19,10/27) -> (-1/3,0/1) Matrix(253,-94,288,-107) -> Matrix(1,0,0,1) Matrix(163,-62,234,-89) -> Matrix(1,0,0,1) Matrix(181,-70,468,-181) -> Matrix(-1,0,8,1) (5/13,7/18) -> (-1/4,0/1) Matrix(71,-28,180,-71) -> Matrix(-1,0,2,1) (7/18,2/5) -> (-1/1,0/1) Matrix(143,-58,180,-73) -> Matrix(1,0,0,1) Matrix(251,-104,432,-179) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(179,-76,252,-107) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(55,-24,126,-55) -> Matrix(-1,0,4,1) (3/7,4/9) -> (-1/2,0/1) Matrix(89,-40,198,-89) -> Matrix(1,0,0,-1) (4/9,5/11) -> (0/1,1/0) Matrix(161,-74,198,-91) -> Matrix(1,0,-2,1) 0/1 Matrix(91,-50,162,-89) -> Matrix(3,2,-8,-5) -1/2 Matrix(397,-224,576,-325) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(215,-122,252,-143) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(109,-66,180,-109) -> Matrix(3,2,-4,-3) (3/5,11/18) -> (-1/1,-1/2) Matrix(287,-176,468,-287) -> Matrix(21,10,-44,-21) (11/18,8/13) -> (-1/2,-5/11) Matrix(305,-188,378,-233) -> Matrix(9,4,-20,-9) *** -> (-1/2,-2/5) Matrix(899,-556,1224,-757) -> Matrix(5,2,2,1) Matrix(487,-302,666,-413) -> Matrix(5,2,2,1) Matrix(341,-214,486,-305) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(685,-432,972,-613) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(325,-206,396,-251) -> Matrix(1,0,0,1) Matrix(397,-256,504,-325) -> Matrix(5,2,-2,-1) Matrix(73,-48,108,-71) -> Matrix(1,0,0,1) Matrix(1099,-754,1494,-1025) -> Matrix(13,4,-10,-3) Matrix(181,-130,252,-181) -> Matrix(-1,0,8,1) (5/7,13/18) -> (-1/4,0/1) Matrix(287,-208,396,-287) -> Matrix(1,0,2,-1) (13/18,8/11) -> (0/1,1/1) Matrix(233,-170,270,-197) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(5617,-4134,7632,-5617) -> Matrix(15,16,-14,-15) (39/53,53/72) -> (-8/7,-1/1) Matrix(2015,-1484,2736,-2015) -> Matrix(7,6,-8,-7) (53/72,14/19) -> (-1/1,-3/4) Matrix(127,-98,162,-125) -> Matrix(1,0,2,1) 0/1 Matrix(181,-150,216,-179) -> Matrix(1,0,2,1) 0/1 Matrix(17,-16,18,-17) -> Matrix(-1,0,4,1) (8/9,1/1) -> (-1/2,0/1) Matrix(-1,2,0,1) -> Matrix(-1,0,4,1) (1/1,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.