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): 576 Minimal number of generators: 97 Number of equivalence classes of cusps: 48 Genus: 25 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -6/13 -5/12 -4/11 -1/3 -3/10 -2/9 -1/8 0/1 1/6 3/11 1/3 2/5 1/2 5/9 3/4 1/1 4/3 3/2 17/11 5/3 9/5 2/1 11/5 29/13 7/3 5/2 71/27 8/3 3/1 43/13 10/3 17/5 7/2 11/3 4/1 13/3 9/2 23/5 5/1 47/9 11/2 6/1 7/1 8/1 25/3 9/1 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 -6/13 1/0 -5/11 -3/1 -2/1 1/0 -4/9 -2/1 -1/1 1/0 -7/16 1/0 -3/7 -2/1 -1/1 1/0 -8/19 -2/1 -3/2 -1/1 -5/12 -1/1 -12/29 -1/1 0/1 1/0 -7/17 -2/1 -1/1 1/0 -2/5 -2/1 -1/1 1/0 -7/18 -1/2 -5/13 -2/1 -1/1 1/0 -13/34 -1/1 -8/21 -1/1 0/1 1/0 -3/8 1/0 -7/19 -3/2 -4/3 -1/1 -4/11 -1/1 -9/25 -1/1 -3/4 -2/3 -5/14 -1/2 -1/3 -1/1 0/1 1/0 -4/13 -2/1 -1/1 1/0 -3/10 -1/1 -8/27 -1/1 -5/6 -4/5 -5/17 -1/1 -3/4 -2/3 -2/7 -1/1 -1/2 0/1 -5/18 -1/2 -8/29 -1/2 -3/11 -1/2 -1/3 0/1 -4/15 -1/2 -1/3 0/1 -1/4 1/0 -3/13 -2/1 -1/1 1/0 -2/9 -1/1 -5/23 -1/1 -3/4 -2/3 -3/14 -1/2 -1/5 -1/1 -1/2 0/1 -2/11 -1/3 -1/4 0/1 -1/6 1/0 -1/7 -1/1 0/1 1/0 -1/8 -1/1 0/1 -1/1 0/1 1/0 1/6 0/1 1/5 0/1 1/1 1/0 2/9 -1/1 0/1 1/0 3/13 -2/1 -1/1 1/0 1/4 -1/2 3/11 0/1 5/18 1/4 2/7 0/1 1/1 1/0 5/17 1/1 2/1 1/0 8/27 1/0 3/10 1/0 4/13 -1/1 0/1 1/0 1/3 -1/1 -1/2 0/1 5/14 -1/2 4/11 -1/2 -1/3 0/1 3/8 -1/4 2/5 0/1 5/12 1/6 8/19 0/1 1/6 1/5 3/7 0/1 1/4 1/3 7/16 1/2 4/9 1/3 2/5 1/2 5/11 1/2 2/3 1/1 6/13 0/1 1/1 1/0 1/2 1/0 5/9 0/1 9/16 1/4 4/7 0/1 1/2 1/1 15/26 1/2 11/19 0/1 1/2 1/1 7/12 1/0 17/29 -1/1 0/1 1/0 10/17 0/1 1/2 1/1 13/22 1/1 3/5 0/1 1/1 1/0 8/13 1/0 5/8 1/0 12/19 -2/1 -1/1 1/0 7/11 -2/1 -1/1 1/0 2/3 -1/1 -1/2 0/1 9/13 -1/2 -1/3 0/1 7/10 -1/2 19/27 -1/3 -1/4 0/1 12/17 -1/2 -1/3 0/1 5/7 -1/3 -1/4 0/1 3/4 0/1 7/9 0/1 1/5 1/4 11/14 1/4 15/19 1/3 2/5 1/2 4/5 0/1 1/3 1/2 9/11 1/2 2/3 1/1 5/6 1/0 6/7 0/1 1/2 1/1 1/1 0/1 1/1 1/0 6/5 -1/1 0/1 1/0 11/9 -1/1 0/1 1/0 5/4 1/0 4/3 0/1 11/8 1/4 18/13 0/1 1/3 1/2 25/18 1/2 7/5 0/1 1/3 1/2 24/17 3/7 4/9 1/2 41/29 1/2 17/12 1/2 27/19 1/2 2/3 1/1 10/7 0/1 1/2 1/1 23/16 1/2 13/9 0/1 1/2 1/1 3/2 1/2 17/11 1/1 14/9 1/1 3/2 2/1 11/7 0/1 1/1 1/0 19/12 1/0 27/17 2/1 3/1 1/0 8/5 -1/1 0/1 1/0 29/18 1/0 21/13 -1/1 -1/2 0/1 13/8 0/1 18/11 0/1 1/4 1/3 41/25 0/1 1/3 1/2 23/14 1/2 5/3 0/1 1/2 1/1 22/13 1/2 17/10 1/2 12/7 1/2 2/3 1/1 19/11 0/1 1/2 1/1 7/4 1/2 9/5 1/1 11/6 3/2 2/1 0/1 1/1 1/0 15/7 1/1 3/2 2/1 13/6 1/0 11/5 1/1 3/2 2/1 20/9 1/1 2/1 1/0 29/13 2/1 9/4 1/0 16/7 0/1 1/1 1/0 39/17 1/1 23/10 3/2 7/3 1/1 2/1 1/0 5/2 1/0 13/5 -1/1 0/1 1/0 47/18 1/0 81/31 -1/1 1/1 34/13 0/1 1/1 1/0 21/8 1/0 71/27 1/0 50/19 -5/1 -4/1 1/0 29/11 -2/1 -1/1 1/0 8/3 -1/1 0/1 1/0 19/7 0/1 1/1 1/0 30/11 0/1 1/2 1/1 41/15 1/1 11/4 1/0 14/5 -1/1 -1/2 0/1 3/1 0/1 1/1 1/0 13/4 1/0 23/7 1/1 2/1 1/0 33/10 1/0 43/13 1/1 10/3 1/1 2/1 1/0 27/8 1/1 17/5 1/1 2/1 1/0 41/12 1/2 65/19 1/1 24/7 1/1 3/2 2/1 7/2 1/0 11/3 1/0 15/4 1/0 4/1 0/1 1/1 1/0 13/3 1/1 3/2 2/1 35/8 3/2 57/13 2/1 22/5 1/1 2/1 1/0 9/2 1/0 23/5 1/1 3/1 37/8 1/0 14/3 1/1 2/1 1/0 5/1 2/1 3/1 1/0 26/5 5/2 8/3 3/1 47/9 3/1 21/4 7/2 16/3 3/1 7/2 4/1 11/2 1/0 6/1 1/0 7/1 -2/1 -1/1 1/0 8/1 0/1 1/1 1/0 25/3 -1/1 1/1 17/2 1/0 9/1 0/1 1/1 1/0 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(265,124,156,73) (-1/2,-6/13) -> (22/13,17/10) Hyperbolic Matrix(153,70,518,237) (-6/13,-5/11) -> (5/17,8/27) Hyperbolic Matrix(103,46,150,67) (-5/11,-4/9) -> (2/3,9/13) Hyperbolic Matrix(199,88,52,23) (-4/9,-7/16) -> (15/4,4/1) Hyperbolic Matrix(51,22,146,63) (-7/16,-3/7) -> (1/3,5/14) Hyperbolic Matrix(47,20,148,63) (-3/7,-8/19) -> (4/13,1/3) Hyperbolic Matrix(239,100,-576,-241) (-8/19,-5/12) -> (-5/12,-12/29) Parabolic Matrix(953,394,670,277) (-12/29,-7/17) -> (27/19,10/7) Hyperbolic Matrix(239,98,378,155) (-7/17,-2/5) -> (12/19,7/11) Hyperbolic Matrix(91,36,48,19) (-2/5,-7/18) -> (11/6,2/1) Hyperbolic Matrix(227,88,276,107) (-7/18,-5/13) -> (9/11,5/6) Hyperbolic Matrix(271,104,456,175) (-5/13,-13/34) -> (13/22,3/5) Hyperbolic Matrix(907,346,270,103) (-13/34,-8/21) -> (10/3,27/8) Hyperbolic Matrix(179,68,408,155) (-8/21,-3/8) -> (7/16,4/9) Hyperbolic Matrix(43,16,180,67) (-3/8,-7/19) -> (3/13,1/4) Hyperbolic Matrix(175,64,-484,-177) (-7/19,-4/11) -> (-4/11,-9/25) Parabolic Matrix(479,172,220,79) (-9/25,-5/14) -> (13/6,11/5) Hyperbolic Matrix(175,62,302,107) (-5/14,-1/3) -> (11/19,7/12) Hyperbolic Matrix(283,88,164,51) (-1/3,-4/13) -> (12/7,19/11) Hyperbolic Matrix(119,36,-400,-121) (-4/13,-3/10) -> (-3/10,-8/27) Parabolic Matrix(237,70,518,153) (-8/27,-5/17) -> (5/11,6/13) Hyperbolic Matrix(159,46,38,11) (-5/17,-2/7) -> (4/1,13/3) Hyperbolic Matrix(351,98,154,43) (-2/7,-5/18) -> (9/4,16/7) Hyperbolic Matrix(231,64,776,215) (-5/18,-8/29) -> (8/27,3/10) Hyperbolic Matrix(387,106,230,63) (-8/29,-3/11) -> (5/3,22/13) Hyperbolic Matrix(231,62,190,51) (-3/11,-4/15) -> (6/5,11/9) Hyperbolic Matrix(151,40,268,71) (-4/15,-1/4) -> (9/16,4/7) Hyperbolic Matrix(291,68,184,43) (-1/4,-3/13) -> (11/7,19/12) Hyperbolic Matrix(71,16,-324,-73) (-3/13,-2/9) -> (-2/9,-5/23) Parabolic Matrix(427,92,608,131) (-5/23,-3/14) -> (7/10,19/27) Hyperbolic Matrix(107,22,34,7) (-3/14,-1/5) -> (3/1,13/4) Hyperbolic Matrix(103,20,36,7) (-1/5,-2/11) -> (14/5,3/1) Hyperbolic Matrix(67,12,240,43) (-2/11,-1/6) -> (5/18,2/7) Hyperbolic Matrix(99,16,68,11) (-1/6,-1/7) -> (13/9,3/2) Hyperbolic Matrix(325,44,96,13) (-1/7,-1/8) -> (27/8,17/5) Hyperbolic Matrix(93,10,158,17) (-1/8,0/1) -> (10/17,13/22) Hyperbolic Matrix(121,-18,74,-11) (0/1,1/6) -> (13/8,18/11) Hyperbolic Matrix(191,-34,118,-21) (1/6,1/5) -> (21/13,13/8) Hyperbolic Matrix(115,-24,24,-5) (1/5,2/9) -> (14/3,5/1) Hyperbolic Matrix(365,-82,138,-31) (2/9,3/13) -> (29/11,8/3) Hyperbolic Matrix(67,-18,242,-65) (1/4,3/11) -> (3/11,5/18) Parabolic Matrix(173,-50,218,-63) (2/7,5/17) -> (15/19,4/5) Hyperbolic Matrix(235,-72,408,-125) (3/10,4/13) -> (4/7,15/26) Hyperbolic Matrix(431,-156,268,-97) (5/14,4/11) -> (8/5,29/18) Hyperbolic Matrix(103,-38,122,-45) (4/11,3/8) -> (5/6,6/7) Hyperbolic Matrix(41,-16,100,-39) (3/8,2/5) -> (2/5,5/12) Parabolic Matrix(435,-182,98,-41) (5/12,8/19) -> (22/5,9/2) Hyperbolic Matrix(393,-166,670,-283) (8/19,3/7) -> (17/29,10/17) Hyperbolic Matrix(273,-118,118,-51) (3/7,7/16) -> (23/10,7/3) Hyperbolic Matrix(327,-148,232,-105) (4/9,5/11) -> (7/5,24/17) Hyperbolic Matrix(305,-142,58,-27) (6/13,1/2) -> (21/4,16/3) Hyperbolic Matrix(91,-50,162,-89) (1/2,5/9) -> (5/9,9/16) Parabolic Matrix(1225,-708,372,-215) (15/26,11/19) -> (23/7,33/10) Hyperbolic Matrix(1519,-890,582,-341) (7/12,17/29) -> (13/5,47/18) Hyperbolic Matrix(121,-74,18,-11) (3/5,8/13) -> (6/1,7/1) Hyperbolic Matrix(191,-118,34,-21) (8/13,5/8) -> (11/2,6/1) Hyperbolic Matrix(619,-390,446,-281) (5/8,12/19) -> (18/13,25/18) Hyperbolic Matrix(187,-120,120,-77) (7/11,2/3) -> (14/9,11/7) Hyperbolic Matrix(429,-298,298,-207) (9/13,7/10) -> (23/16,13/9) Hyperbolic Matrix(1183,-834,722,-509) (19/27,12/17) -> (18/11,41/25) Hyperbolic Matrix(327,-232,148,-105) (12/17,5/7) -> (11/5,20/9) Hyperbolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(549,-430,346,-271) (7/9,11/14) -> (19/12,27/17) Hyperbolic Matrix(1017,-802,298,-235) (11/14,15/19) -> (17/5,41/12) Hyperbolic Matrix(123,-100,16,-13) (4/5,9/11) -> (7/1,8/1) Hyperbolic Matrix(163,-144,60,-53) (6/7,1/1) -> (19/7,30/11) Hyperbolic Matrix(103,-122,38,-45) (1/1,6/5) -> (8/3,19/7) Hyperbolic Matrix(227,-278,138,-169) (11/9,5/4) -> (23/14,5/3) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(283,-390,82,-113) (11/8,18/13) -> (24/7,7/2) Hyperbolic Matrix(409,-570,94,-131) (25/18,7/5) -> (13/3,35/8) Hyperbolic Matrix(2657,-3754,1010,-1427) (24/17,41/29) -> (71/27,50/19) Hyperbolic Matrix(1461,-2068,556,-787) (41/29,17/12) -> (21/8,71/27) Hyperbolic Matrix(127,-180,12,-17) (17/12,27/19) -> (9/1,1/0) Hyperbolic Matrix(483,-692,104,-149) (10/7,23/16) -> (37/8,14/3) Hyperbolic Matrix(331,-508,144,-221) (3/2,17/11) -> (39/17,23/10) Hyperbolic Matrix(527,-818,230,-357) (17/11,14/9) -> (16/7,39/17) Hyperbolic Matrix(995,-1582,378,-601) (27/17,8/5) -> (50/19,29/11) Hyperbolic Matrix(439,-708,204,-329) (29/18,21/13) -> (15/7,13/6) Hyperbolic Matrix(551,-904,64,-105) (41/25,23/14) -> (17/2,9/1) Hyperbolic Matrix(487,-830,186,-317) (17/10,12/7) -> (34/13,21/8) Hyperbolic Matrix(235,-408,72,-125) (19/11,7/4) -> (13/4,23/7) Hyperbolic Matrix(91,-162,50,-89) (7/4,9/5) -> (9/5,11/6) Parabolic Matrix(187,-400,36,-77) (2/1,15/7) -> (5/1,26/5) Hyperbolic Matrix(799,-1778,182,-405) (20/9,29/13) -> (57/13,22/5) Hyperbolic Matrix(683,-1528,156,-349) (29/13,9/4) -> (35/8,57/13) Hyperbolic Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(977,-2552,116,-303) (47/18,81/31) -> (25/3,17/2) Hyperbolic Matrix(573,-1498,70,-183) (81/31,34/13) -> (8/1,25/3) Hyperbolic Matrix(623,-1700,188,-513) (30/11,41/15) -> (43/13,10/3) Hyperbolic Matrix(667,-1826,202,-553) (41/15,11/4) -> (33/10,43/13) Hyperbolic Matrix(119,-330,22,-61) (11/4,14/5) -> (16/3,11/2) Hyperbolic Matrix(963,-3292,184,-629) (41/12,65/19) -> (47/9,21/4) Hyperbolic Matrix(823,-2818,158,-541) (65/19,24/7) -> (26/5,47/9) Hyperbolic Matrix(67,-242,18,-65) (7/2,11/3) -> (11/3,15/4) Parabolic Matrix(231,-1058,50,-229) (9/2,23/5) -> (23/5,37/8) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,0,1) Matrix(265,124,156,73) -> Matrix(1,0,2,1) Matrix(153,70,518,237) -> Matrix(1,4,0,1) Matrix(103,46,150,67) -> Matrix(1,2,-2,-3) Matrix(199,88,52,23) -> Matrix(1,2,0,1) Matrix(51,22,146,63) -> Matrix(1,2,-2,-3) Matrix(47,20,148,63) -> Matrix(1,2,-2,-3) Matrix(239,100,-576,-241) -> Matrix(1,2,-2,-3) Matrix(953,394,670,277) -> Matrix(1,0,2,1) Matrix(239,98,378,155) -> Matrix(1,0,0,1) Matrix(91,36,48,19) -> Matrix(1,2,0,1) Matrix(227,88,276,107) -> Matrix(1,0,2,1) Matrix(271,104,456,175) -> Matrix(1,2,0,1) Matrix(907,346,270,103) -> Matrix(1,2,0,1) Matrix(179,68,408,155) -> Matrix(1,2,2,5) Matrix(43,16,180,67) -> Matrix(1,2,-2,-3) Matrix(175,64,-484,-177) -> Matrix(5,6,-6,-7) Matrix(479,172,220,79) -> Matrix(1,0,2,1) Matrix(175,62,302,107) -> Matrix(1,0,2,1) Matrix(283,88,164,51) -> Matrix(1,0,2,1) Matrix(119,36,-400,-121) -> Matrix(5,6,-6,-7) Matrix(237,70,518,153) -> Matrix(5,4,6,5) Matrix(159,46,38,11) -> Matrix(1,0,2,1) Matrix(351,98,154,43) -> Matrix(1,0,2,1) Matrix(231,64,776,215) -> Matrix(3,2,-2,-1) Matrix(387,106,230,63) -> Matrix(1,0,4,1) Matrix(231,62,190,51) -> Matrix(1,0,2,1) Matrix(151,40,268,71) -> Matrix(1,0,4,1) Matrix(291,68,184,43) -> Matrix(1,2,0,1) Matrix(71,16,-324,-73) -> Matrix(3,4,-4,-5) Matrix(427,92,608,131) -> Matrix(3,2,-8,-5) Matrix(107,22,34,7) -> Matrix(1,0,2,1) Matrix(103,20,36,7) -> Matrix(1,0,2,1) Matrix(67,12,240,43) -> Matrix(1,0,4,1) Matrix(99,16,68,11) -> Matrix(1,0,2,1) Matrix(325,44,96,13) -> Matrix(1,2,0,1) Matrix(93,10,158,17) -> Matrix(1,0,2,1) Matrix(121,-18,74,-11) -> Matrix(1,0,4,1) Matrix(191,-34,118,-21) -> Matrix(1,0,-2,1) Matrix(115,-24,24,-5) -> Matrix(1,2,0,1) Matrix(365,-82,138,-31) -> Matrix(1,0,0,1) Matrix(67,-18,242,-65) -> Matrix(1,0,6,1) Matrix(173,-50,218,-63) -> Matrix(1,0,2,1) Matrix(235,-72,408,-125) -> Matrix(1,0,2,1) Matrix(431,-156,268,-97) -> Matrix(1,0,2,1) Matrix(103,-38,122,-45) -> Matrix(1,0,4,1) Matrix(41,-16,100,-39) -> Matrix(1,0,10,1) Matrix(435,-182,98,-41) -> Matrix(11,-2,6,-1) Matrix(393,-166,670,-283) -> Matrix(1,0,-4,1) Matrix(273,-118,118,-51) -> Matrix(7,-2,4,-1) Matrix(327,-148,232,-105) -> Matrix(3,-2,8,-5) Matrix(305,-142,58,-27) -> Matrix(7,-4,2,-1) Matrix(91,-50,162,-89) -> Matrix(1,0,4,1) Matrix(1225,-708,372,-215) -> Matrix(3,-2,2,-1) Matrix(1519,-890,582,-341) -> Matrix(1,0,0,1) Matrix(121,-74,18,-11) -> Matrix(1,-2,0,1) Matrix(191,-118,34,-21) -> Matrix(1,6,0,1) Matrix(619,-390,446,-281) -> Matrix(1,2,2,5) Matrix(187,-120,120,-77) -> Matrix(1,2,0,1) Matrix(429,-298,298,-207) -> Matrix(1,0,4,1) Matrix(1183,-834,722,-509) -> Matrix(1,0,6,1) Matrix(327,-232,148,-105) -> Matrix(5,2,2,1) Matrix(49,-36,64,-47) -> Matrix(1,0,8,1) Matrix(549,-430,346,-271) -> Matrix(7,-2,4,-1) Matrix(1017,-802,298,-235) -> Matrix(1,0,-2,1) Matrix(123,-100,16,-13) -> Matrix(1,0,-2,1) Matrix(163,-144,60,-53) -> Matrix(1,0,0,1) Matrix(103,-122,38,-45) -> Matrix(1,0,0,1) Matrix(227,-278,138,-169) -> Matrix(1,0,2,1) Matrix(49,-64,36,-47) -> Matrix(1,0,4,1) Matrix(283,-390,82,-113) -> Matrix(7,-2,4,-1) Matrix(409,-570,94,-131) -> Matrix(7,-2,4,-1) Matrix(2657,-3754,1010,-1427) -> Matrix(17,-8,-2,1) Matrix(1461,-2068,556,-787) -> Matrix(3,-2,2,-1) Matrix(127,-180,12,-17) -> Matrix(3,-2,2,-1) Matrix(483,-692,104,-149) -> Matrix(3,-2,2,-1) Matrix(331,-508,144,-221) -> Matrix(5,-4,4,-3) Matrix(527,-818,230,-357) -> Matrix(1,-2,2,-3) Matrix(995,-1582,378,-601) -> Matrix(1,-4,0,1) Matrix(439,-708,204,-329) -> Matrix(1,2,0,1) Matrix(551,-904,64,-105) -> Matrix(1,0,-2,1) Matrix(487,-830,186,-317) -> Matrix(3,-2,2,-1) Matrix(235,-408,72,-125) -> Matrix(3,-2,2,-1) Matrix(91,-162,50,-89) -> Matrix(5,-4,4,-3) Matrix(187,-400,36,-77) -> Matrix(5,-8,2,-3) Matrix(799,-1778,182,-405) -> Matrix(1,0,0,1) Matrix(683,-1528,156,-349) -> Matrix(3,-8,2,-5) Matrix(41,-100,16,-39) -> Matrix(1,-2,0,1) Matrix(977,-2552,116,-303) -> Matrix(1,0,0,1) Matrix(573,-1498,70,-183) -> Matrix(1,0,0,1) Matrix(623,-1700,188,-513) -> Matrix(3,-2,2,-1) Matrix(667,-1826,202,-553) -> Matrix(1,0,0,1) Matrix(119,-330,22,-61) -> Matrix(1,4,0,1) Matrix(963,-3292,184,-629) -> Matrix(13,-10,4,-3) Matrix(823,-2818,158,-541) -> Matrix(11,-14,4,-5) Matrix(67,-242,18,-65) -> Matrix(1,-2,0,1) Matrix(231,-1058,50,-229) -> 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): 20 Degree of the the map X: 20 Degree of the the map Y: 96 ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- The image of the extended modular group liftables in PGL(2,Z) equals the image of the modular liftables. ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.