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 -1/1 -5/11 -7/16 -2/5 -5/14 -1/3 -1/4 0/1 1/5 2/7 1/3 5/13 1/2 7/11 4/5 1/1 5/4 7/5 3/2 11/7 17/10 2/1 19/8 5/2 13/5 11/4 3/1 7/2 15/4 4/1 73/17 9/2 5/1 11/2 6/1 13/2 7/1 8/1 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/2 -6/13 0/1 1/3 -5/11 0/1 -9/20 1/3 1/2 -4/9 0/1 1/2 -7/16 0/1 -10/23 0/1 1/4 -3/7 1/2 -11/26 0/1 1/1 1/0 -8/19 -1/1 0/1 -5/12 0/1 1/4 -7/17 1/2 -2/5 0/1 -7/18 1/3 1/2 -19/49 0/1 -12/31 0/1 1/3 -5/13 1/2 -13/34 0/1 1/3 1/2 -8/21 0/1 1/2 -3/8 0/1 1/3 -4/11 1/3 2/5 -5/14 1/2 -6/17 0/1 1/1 -1/3 1/2 -3/10 1/2 2/3 -5/17 1/1 -2/7 0/1 1/2 -5/18 1/2 3/5 -8/29 2/3 -3/11 1/2 -1/4 0/1 1/2 1/1 -3/13 1/2 -2/9 2/3 1/1 -3/14 1/1 1/0 -4/19 0/1 -1/5 1/2 -1/6 0/1 1/1 0/1 0/1 1/1 1/5 1/1 2/9 1/1 2/1 1/4 0/1 1/1 3/11 1/2 2/7 1/1 5/17 5/4 3/10 1/1 3/2 4/13 1/1 2/1 5/16 1/1 3/2 2/1 1/3 1/0 3/8 1/2 1/1 5/13 1/1 7/18 1/1 5/4 2/5 1/1 2/1 1/2 0/1 1/1 1/0 4/7 1/1 2/1 7/12 1/1 2/1 10/17 2/1 3/5 1/0 5/8 3/1 1/0 7/11 1/0 9/14 -3/1 1/0 2/3 0/1 1/0 7/10 0/1 1/1 12/17 0/1 1/1 5/7 1/1 8/11 1/1 2/1 11/15 3/2 3/4 2/1 1/0 7/9 1/0 4/5 1/0 9/11 1/0 5/6 -1/1 1/0 1/1 1/0 5/4 1/0 9/7 1/0 22/17 -2/1 13/10 -3/1 1/0 17/13 -2/1 4/3 -2/1 -1/1 15/11 1/0 26/19 -2/1 11/8 -2/1 -1/1 18/13 -2/1 -3/2 7/5 -1/1 10/7 -1/1 0/1 3/2 -1/1 0/1 11/7 0/1 19/12 0/1 1/1 8/5 0/1 1/0 29/18 1/1 1/0 21/13 1/0 13/8 -1/1 0/1 1/0 5/3 1/0 17/10 -1/1 29/17 -1/2 12/7 -1/1 0/1 7/4 -1/1 1/0 9/5 -1/2 20/11 0/1 1/0 11/6 -1/1 0/1 2/1 0/1 13/6 0/1 1/1 11/5 1/0 9/4 1/1 1/0 7/3 1/0 19/8 -1/1 31/13 -1/2 12/5 -1/1 0/1 17/7 -1/1 5/2 -1/2 0/1 13/5 0/1 21/8 0/1 1/8 8/3 0/1 1/3 27/10 0/1 1/2 19/7 1/2 11/4 0/1 1/2 1/1 25/9 1/2 39/14 2/3 1/1 53/19 1/1 14/5 0/1 1/1 31/11 0/1 17/6 1/2 1/1 3/1 1/0 7/2 0/1 11/3 1/2 26/7 0/1 41/11 1/2 15/4 0/1 1/1 4/1 0/1 1/0 17/4 0/1 1/1 1/0 30/7 0/1 1/1 73/17 1/1 43/10 1/1 2/1 13/3 1/0 9/2 -1/1 1/0 14/3 -1/1 0/1 5/1 0/1 16/3 0/1 1/1 11/2 1/1 1/0 17/3 1/0 6/1 -1/1 0/1 13/2 0/1 1/1 1/0 7/1 1/0 8/1 0/1 9/1 1/0 1/0 0/1 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(367,170,136,63) (-1/2,-6/13) -> (8/3,27/10) Hyperbolic Matrix(265,122,202,93) (-6/13,-5/11) -> (17/13,4/3) Hyperbolic Matrix(457,206,-1178,-531) (-5/11,-9/20) -> (-7/18,-19/49) Hyperbolic Matrix(421,188,262,117) (-9/20,-4/9) -> (8/5,29/18) Hyperbolic Matrix(223,98,-512,-225) (-4/9,-7/16) -> (-7/16,-10/23) Parabolic Matrix(347,150,192,83) (-10/23,-3/7) -> (9/5,20/11) Hyperbolic Matrix(61,26,190,81) (-3/7,-11/26) -> (5/16,1/3) Hyperbolic Matrix(403,170,64,27) (-11/26,-8/19) -> (6/1,13/2) Hyperbolic Matrix(495,208,188,79) (-8/19,-5/12) -> (21/8,8/3) Hyperbolic Matrix(183,76,248,103) (-5/12,-7/17) -> (11/15,3/4) Hyperbolic Matrix(151,62,-548,-225) (-7/17,-2/5) -> (-8/29,-3/11) Hyperbolic Matrix(87,34,-412,-161) (-2/5,-7/18) -> (-3/14,-4/19) Hyperbolic Matrix(1647,638,586,227) (-19/49,-12/31) -> (14/5,31/11) Hyperbolic Matrix(1117,432,468,181) (-12/31,-5/13) -> (31/13,12/5) Hyperbolic Matrix(407,156,60,23) (-5/13,-13/34) -> (13/2,7/1) Hyperbolic Matrix(493,188,118,45) (-13/34,-8/21) -> (4/1,17/4) Hyperbolic Matrix(463,176,292,111) (-8/21,-3/8) -> (19/12,8/5) Hyperbolic Matrix(27,10,116,43) (-3/8,-4/11) -> (2/9,1/4) Hyperbolic Matrix(139,50,-392,-141) (-4/11,-5/14) -> (-5/14,-6/17) Parabolic Matrix(307,108,54,19) (-6/17,-1/3) -> (17/3,6/1) Hyperbolic Matrix(79,24,102,31) (-1/3,-3/10) -> (3/4,7/9) Hyperbolic Matrix(255,76,104,31) (-3/10,-5/17) -> (17/7,5/2) Hyperbolic Matrix(355,104,256,75) (-5/17,-2/7) -> (18/13,7/5) Hyperbolic Matrix(99,28,152,43) (-2/7,-5/18) -> (9/14,2/3) Hyperbolic Matrix(773,214,596,165) (-5/18,-8/29) -> (22/17,13/10) Hyperbolic Matrix(23,6,-96,-25) (-3/11,-1/4) -> (-1/4,-3/13) Parabolic Matrix(187,42,138,31) (-3/13,-2/9) -> (4/3,15/11) Hyperbolic Matrix(91,20,232,51) (-2/9,-3/14) -> (7/18,2/5) Hyperbolic Matrix(339,70,92,19) (-4/19,-1/5) -> (11/3,26/7) Hyperbolic Matrix(23,4,86,15) (-1/5,-1/6) -> (1/4,3/11) 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(57,-16,196,-55) (3/11,2/7) -> (2/7,5/17) Parabolic Matrix(319,-94,112,-33) (5/17,3/10) -> (17/6,3/1) Hyperbolic Matrix(303,-92,56,-17) (3/10,4/13) -> (16/3,11/2) Hyperbolic Matrix(701,-218,164,-51) (4/13,5/16) -> (17/4,30/7) Hyperbolic Matrix(93,-34,52,-19) (1/3,3/8) -> (7/4,9/5) Hyperbolic Matrix(131,-50,338,-129) (3/8,5/13) -> (5/13,7/18) Parabolic 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(499,-292,364,-213) (7/12,10/17) -> (26/19,11/8) Hyperbolic Matrix(159,-94,22,-13) (10/17,3/5) -> (7/1,8/1) Hyperbolic Matrix(101,-62,44,-27) (3/5,5/8) -> (9/4,7/3) Hyperbolic Matrix(155,-98,242,-153) (5/8,7/11) -> (7/11,9/14) Parabolic Matrix(85,-58,22,-15) (2/3,7/10) -> (15/4,4/1) Hyperbolic Matrix(499,-364,292,-213) (8/11,11/15) -> (29/17,12/7) Hyperbolic Matrix(81,-64,100,-79) (7/9,4/5) -> (4/5,9/11) Parabolic Matrix(445,-366,276,-227) (9/11,5/6) -> (29/18,21/13) Hyperbolic Matrix(87,-74,20,-17) (5/6,1/1) -> (13/3,9/2) Hyperbolic Matrix(41,-50,32,-39) (1/1,5/4) -> (5/4,9/7) Parabolic Matrix(879,-1136,236,-305) (9/7,22/17) -> (26/7,41/11) Hyperbolic Matrix(531,-692,188,-245) (13/10,17/13) -> (31/11,17/6) Hyperbolic Matrix(259,-354,30,-41) (15/11,26/19) -> (8/1,9/1) Hyperbolic Matrix(303,-418,166,-229) (11/8,18/13) -> (20/11,11/6) Hyperbolic Matrix(179,-254,74,-105) (7/5,10/7) -> (12/5,17/7) Hyperbolic Matrix(155,-242,98,-153) (3/2,11/7) -> (11/7,19/12) Parabolic Matrix(343,-556,124,-201) (21/13,13/8) -> (11/4,25/9) Hyperbolic Matrix(185,-302,68,-111) (13/8,5/3) -> (19/7,11/4) Hyperbolic Matrix(341,-578,200,-339) (5/3,17/10) -> (17/10,29/17) Parabolic Matrix(119,-206,26,-45) (12/7,7/4) -> (9/2,14/3) Hyperbolic Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(345,-754,124,-271) (13/6,11/5) -> (25/9,39/14) Hyperbolic Matrix(123,-274,22,-49) (11/5,9/4) -> (11/2,17/3) Hyperbolic Matrix(305,-722,128,-303) (7/3,19/8) -> (19/8,31/13) Parabolic Matrix(131,-338,50,-129) (5/2,13/5) -> (13/5,21/8) Parabolic Matrix(97,-262,10,-27) (27/10,19/7) -> (9/1,1/0) Hyperbolic Matrix(1839,-5126,428,-1193) (39/14,53/19) -> (73/17,43/10) Hyperbolic Matrix(935,-2612,218,-609) (53/19,14/5) -> (30/7,73/17) Hyperbolic Matrix(29,-98,8,-27) (3/1,7/2) -> (7/2,11/3) Parabolic Matrix(525,-1958,122,-455) (41/11,15/4) -> (43/10,13/3) Hyperbolic Matrix(31,-150,6,-29) (14/3,5/1) -> (5/1,16/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,2,1) Matrix(367,170,136,63) -> Matrix(1,0,0,1) Matrix(265,122,202,93) -> Matrix(5,-2,-2,1) Matrix(457,206,-1178,-531) -> Matrix(1,0,0,1) Matrix(421,188,262,117) -> Matrix(1,0,-2,1) Matrix(223,98,-512,-225) -> Matrix(1,0,2,1) Matrix(347,150,192,83) -> Matrix(1,0,-4,1) Matrix(61,26,190,81) -> Matrix(3,-2,2,-1) Matrix(403,170,64,27) -> Matrix(1,0,0,1) Matrix(495,208,188,79) -> Matrix(1,0,4,1) Matrix(183,76,248,103) -> Matrix(7,-2,4,-1) Matrix(151,62,-548,-225) -> Matrix(5,-2,8,-3) Matrix(87,34,-412,-161) -> Matrix(1,0,-2,1) Matrix(1647,638,586,227) -> Matrix(1,0,-2,1) Matrix(1117,432,468,181) -> Matrix(1,0,-4,1) Matrix(407,156,60,23) -> Matrix(1,0,-2,1) Matrix(493,188,118,45) -> Matrix(1,0,-2,1) Matrix(463,176,292,111) -> Matrix(1,0,-2,1) Matrix(27,10,116,43) -> Matrix(1,0,-2,1) Matrix(139,50,-392,-141) -> Matrix(5,-2,8,-3) Matrix(307,108,54,19) -> Matrix(1,0,-2,1) Matrix(79,24,102,31) -> Matrix(7,-4,2,-1) Matrix(255,76,104,31) -> Matrix(3,-2,-4,3) Matrix(355,104,256,75) -> Matrix(1,-2,0,1) Matrix(99,28,152,43) -> Matrix(1,0,-2,1) Matrix(773,214,596,165) -> Matrix(1,0,-2,1) Matrix(23,6,-96,-25) -> Matrix(1,0,0,1) Matrix(187,42,138,31) -> Matrix(1,0,-2,1) Matrix(91,20,232,51) -> Matrix(5,-4,4,-3) Matrix(339,70,92,19) -> Matrix(1,0,0,1) Matrix(23,4,86,15) -> Matrix(1,0,0,1) Matrix(63,10,44,7) -> Matrix(1,0,-2,1) Matrix(45,-8,62,-11) -> Matrix(3,-2,2,-1) Matrix(105,-22,148,-31) -> Matrix(1,-2,2,-3) Matrix(57,-16,196,-55) -> Matrix(7,-6,6,-5) Matrix(319,-94,112,-33) -> Matrix(3,-4,4,-5) Matrix(303,-92,56,-17) -> Matrix(1,-2,2,-3) Matrix(701,-218,164,-51) -> Matrix(1,-2,2,-3) Matrix(93,-34,52,-19) -> Matrix(1,0,-2,1) Matrix(131,-50,338,-129) -> Matrix(7,-6,6,-5) Matrix(13,-6,24,-11) -> Matrix(1,0,0,1) Matrix(193,-112,274,-159) -> Matrix(1,-2,2,-3) Matrix(499,-292,364,-213) -> Matrix(3,-4,-2,3) Matrix(159,-94,22,-13) -> Matrix(1,-2,0,1) Matrix(101,-62,44,-27) -> Matrix(1,-2,0,1) Matrix(155,-98,242,-153) -> Matrix(1,-6,0,1) Matrix(85,-58,22,-15) -> Matrix(1,0,0,1) Matrix(499,-364,292,-213) -> Matrix(1,-2,0,1) Matrix(81,-64,100,-79) -> Matrix(1,-6,0,1) Matrix(445,-366,276,-227) -> Matrix(1,2,0,1) Matrix(87,-74,20,-17) -> Matrix(1,0,0,1) Matrix(41,-50,32,-39) -> Matrix(1,-2,0,1) Matrix(879,-1136,236,-305) -> Matrix(1,2,2,5) Matrix(531,-692,188,-245) -> Matrix(1,2,2,5) Matrix(259,-354,30,-41) -> Matrix(1,2,0,1) Matrix(303,-418,166,-229) -> Matrix(1,2,-2,-3) Matrix(179,-254,74,-105) -> Matrix(1,0,0,1) Matrix(155,-242,98,-153) -> Matrix(1,0,2,1) Matrix(343,-556,124,-201) -> Matrix(1,0,2,1) Matrix(185,-302,68,-111) -> Matrix(1,0,2,1) Matrix(341,-578,200,-339) -> Matrix(1,2,-2,-3) Matrix(119,-206,26,-45) -> Matrix(1,0,0,1) Matrix(25,-48,12,-23) -> Matrix(1,0,2,1) Matrix(345,-754,124,-271) -> Matrix(1,-2,2,-3) Matrix(123,-274,22,-49) -> Matrix(1,0,0,1) Matrix(305,-722,128,-303) -> Matrix(1,2,-2,-3) Matrix(131,-338,50,-129) -> Matrix(1,0,10,1) Matrix(97,-262,10,-27) -> Matrix(1,0,-2,1) Matrix(1839,-5126,428,-1193) -> Matrix(5,-4,4,-3) Matrix(935,-2612,218,-609) -> Matrix(1,0,0,1) Matrix(29,-98,8,-27) -> Matrix(1,0,2,1) Matrix(525,-1958,122,-455) -> Matrix(3,-2,2,-1) Matrix(31,-150,6,-29) -> 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): 14 Degree of the the map X: 14 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 -1/1 0/1 1 1 0/1 (0/1,1/1) 0 18 1/5 1/1 2 3 1/4 (0/1,1/1) 0 18 2/7 1/1 3 2 3/10 (1/1,3/2) 0 18 4/13 (1/1,2/1) 0 18 1/3 1/0 1 9 3/8 (1/2,1/1) 0 18 5/13 1/1 3 1 2/5 (1/1,2/1) 0 18 1/2 0 6 4/7 (1/1,2/1) 0 18 7/12 (1/1,2/1) 0 18 10/17 2/1 1 2 3/5 1/0 1 9 5/8 (3/1,1/0) 0 18 7/11 1/0 3 1 2/3 (0/1,1/0) 0 18 7/10 (0/1,1/1) 0 18 5/7 1/1 2 3 8/11 (1/1,2/1) 0 18 3/4 (2/1,1/0) 0 18 4/5 1/0 3 2 5/6 (-1/1,1/0) 0 18 1/1 1/0 1 9 5/4 1/0 2 2 9/7 1/0 1 9 13/10 (-3/1,1/0) 0 18 4/3 (-2/1,-1/1) 0 18 11/8 (-2/1,-1/1) 0 18 7/5 -1/1 2 3 3/2 (-1/1,0/1) 0 18 11/7 0/1 1 1 8/5 (0/1,1/0) 0 18 21/13 1/0 1 9 13/8 0 6 5/3 1/0 1 9 17/10 -1/1 2 2 12/7 (-1/1,0/1) 0 18 7/4 (-1/1,1/0) 0 18 9/5 -1/2 1 9 11/6 (-1/1,0/1) 0 18 2/1 0/1 1 6 13/6 (0/1,1/1) 0 18 11/5 1/0 1 9 9/4 (1/1,1/0) 0 18 7/3 1/0 1 9 19/8 -1/1 2 2 12/5 (-1/1,0/1) 0 18 5/2 (-1/2,0/1) 0 18 13/5 0/1 5 1 8/3 (0/1,1/3) 0 18 19/7 1/2 1 9 11/4 0 6 25/9 1/2 1 9 39/14 (2/3,1/1) 0 18 53/19 1/1 2 1 14/5 (0/1,1/1) 0 18 3/1 1/0 1 9 7/2 0/1 2 2 11/3 1/2 1 9 15/4 (0/1,1/1) 0 18 4/1 (0/1,1/0) 0 18 13/3 1/0 1 9 9/2 (-1/1,1/0) 0 18 14/3 (-1/1,0/1) 0 18 5/1 0/1 1 3 16/3 (0/1,1/1) 0 18 11/2 (1/1,1/0) 0 18 6/1 (-1/1,0/1) 0 18 7/1 1/0 1 9 8/1 0/1 1 2 1/0 (0/1,1/0) 0 18 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(41,-12,140,-41) (2/7,3/10) -> (2/7,3/10) Reflection Matrix(303,-92,56,-17) (3/10,4/13) -> (16/3,11/2) Hyperbolic Matrix(81,-26,28,-9) (4/13,1/3) -> (14/5,3/1) Glide Reflection Matrix(93,-34,52,-19) (1/3,3/8) -> (7/4,9/5) Hyperbolic Matrix(79,-30,208,-79) (3/8,5/13) -> (3/8,5/13) Reflection Matrix(51,-20,130,-51) (5/13,2/5) -> (5/13,2/5) Reflection Matrix(13,-6,24,-11) (2/5,1/2) -> (1/2,4/7) Parabolic Matrix(125,-72,92,-53) (4/7,7/12) -> (4/3,11/8) Glide Reflection Matrix(239,-140,408,-239) (7/12,10/17) -> (7/12,10/17) Reflection Matrix(159,-94,22,-13) (10/17,3/5) -> (7/1,8/1) Hyperbolic Matrix(101,-62,44,-27) (3/5,5/8) -> (9/4,7/3) Hyperbolic Matrix(111,-70,176,-111) (5/8,7/11) -> (5/8,7/11) Reflection Matrix(43,-28,66,-43) (7/11,2/3) -> (7/11,2/3) Reflection Matrix(85,-58,22,-15) (2/3,7/10) -> (15/4,4/1) Hyperbolic Matrix(147,-104,106,-75) (7/10,5/7) -> (11/8,7/5) 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(49,-40,60,-49) (4/5,5/6) -> (4/5,5/6) Reflection Matrix(87,-74,20,-17) (5/6,1/1) -> (13/3,9/2) Hyperbolic Matrix(41,-50,32,-39) (1/1,5/4) -> (5/4,9/7) Parabolic Matrix(173,-224,78,-101) (9/7,13/10) -> (11/5,9/4) Glide Reflection Matrix(93,-122,16,-21) (13/10,4/3) -> (11/2,6/1) Glide Reflection 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(117,-188,28,-45) (8/5,21/13) -> (4/1,13/3) Glide Reflection Matrix(343,-556,124,-201) (21/13,13/8) -> (11/4,25/9) Hyperbolic Matrix(185,-302,68,-111) (13/8,5/3) -> (19/7,11/4) Hyperbolic Matrix(127,-214,54,-91) (5/3,17/10) -> (7/3,19/8) Glide Reflection Matrix(253,-432,106,-181) (17/10,12/7) -> (19/8,12/5) Glide Reflection Matrix(119,-206,26,-45) (12/7,7/4) -> (9/2,14/3) Hyperbolic Matrix(141,-256,38,-69) (9/5,11/6) -> (11/3,15/4) Glide Reflection Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(345,-754,124,-271) (13/6,11/5) -> (25/9,39/14) Hyperbolic 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(63,-170,10,-27) (8/3,19/7) -> (6/1,7/1) Glide Reflection Matrix(1483,-4134,532,-1483) (39/14,53/19) -> (39/14,53/19) Reflection Matrix(531,-1484,190,-531) (53/19,14/5) -> (53/19,14/5) Reflection Matrix(29,-98,8,-27) (3/1,7/2) -> (7/2,11/3) Parabolic Matrix(31,-150,6,-29) (14/3,5/1) -> (5/1,16/3) Parabolic 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(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(-1,0,2,1) -> Matrix(1,0,2,-1) (-1/1,0/1) -> (0/1,1/1) Matrix(45,-8,62,-11) -> Matrix(3,-2,2,-1) 1/1 Matrix(43,-10,30,-7) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(15,-4,56,-15) -> Matrix(1,0,2,-1) (1/4,2/7) -> (0/1,1/1) Matrix(41,-12,140,-41) -> Matrix(5,-6,4,-5) (2/7,3/10) -> (1/1,3/2) Matrix(303,-92,56,-17) -> Matrix(1,-2,2,-3) 1/1 Matrix(81,-26,28,-9) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(93,-34,52,-19) -> Matrix(1,0,-2,1) 0/1 Matrix(79,-30,208,-79) -> Matrix(3,-2,4,-3) (3/8,5/13) -> (1/2,1/1) Matrix(51,-20,130,-51) -> Matrix(3,-4,2,-3) (5/13,2/5) -> (1/1,2/1) Matrix(13,-6,24,-11) -> Matrix(1,0,0,1) Matrix(125,-72,92,-53) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(239,-140,408,-239) -> Matrix(3,-4,2,-3) (7/12,10/17) -> (1/1,2/1) Matrix(159,-94,22,-13) -> Matrix(1,-2,0,1) 1/0 Matrix(101,-62,44,-27) -> Matrix(1,-2,0,1) 1/0 Matrix(111,-70,176,-111) -> Matrix(-1,6,0,1) (5/8,7/11) -> (3/1,1/0) Matrix(43,-28,66,-43) -> Matrix(1,0,0,-1) (7/11,2/3) -> (0/1,1/0) Matrix(85,-58,22,-15) -> Matrix(1,0,0,1) Matrix(147,-104,106,-75) -> Matrix(3,-2,-2,1) Matrix(103,-76,42,-31) -> Matrix(1,-2,-2,3) Matrix(31,-24,40,-31) -> Matrix(-1,4,0,1) (3/4,4/5) -> (2/1,1/0) Matrix(49,-40,60,-49) -> Matrix(1,2,0,-1) (4/5,5/6) -> (-1/1,1/0) Matrix(87,-74,20,-17) -> Matrix(1,0,0,1) Matrix(41,-50,32,-39) -> Matrix(1,-2,0,1) 1/0 Matrix(173,-224,78,-101) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(93,-122,16,-21) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(43,-66,28,-43) -> Matrix(-1,0,2,1) (3/2,11/7) -> (-1/1,0/1) Matrix(111,-176,70,-111) -> Matrix(1,0,0,-1) (11/7,8/5) -> (0/1,1/0) Matrix(117,-188,28,-45) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(343,-556,124,-201) -> Matrix(1,0,2,1) 0/1 Matrix(185,-302,68,-111) -> Matrix(1,0,2,1) 0/1 Matrix(127,-214,54,-91) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(253,-432,106,-181) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(119,-206,26,-45) -> Matrix(1,0,0,1) Matrix(141,-256,38,-69) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(25,-48,12,-23) -> Matrix(1,0,2,1) 0/1 Matrix(345,-754,124,-271) -> Matrix(1,-2,2,-3) 1/1 Matrix(51,-130,20,-51) -> Matrix(-1,0,4,1) (5/2,13/5) -> (-1/2,0/1) Matrix(79,-208,30,-79) -> Matrix(1,0,6,-1) (13/5,8/3) -> (0/1,1/3) Matrix(63,-170,10,-27) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(1483,-4134,532,-1483) -> Matrix(5,-4,6,-5) (39/14,53/19) -> (2/3,1/1) Matrix(531,-1484,190,-531) -> Matrix(1,0,2,-1) (53/19,14/5) -> (0/1,1/1) Matrix(29,-98,8,-27) -> Matrix(1,0,2,1) 0/1 Matrix(31,-150,6,-29) -> Matrix(1,0,2,1) 0/1 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.