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): 288 Minimal number of generators: 49 Number of equivalence classes of cusps: 30 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -4/11 -1/3 -2/9 0/1 1/6 3/11 1/3 2/5 1/2 5/9 3/4 1/1 4/3 3/2 5/3 22/13 9/5 2/1 5/2 8/3 3/1 43/13 10/3 11/3 4/1 5/1 6/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 1/1 -4/9 2/1 1/0 -7/16 -3/1 -3/7 -1/1 1/1 -2/5 0/1 1/0 -7/18 1/1 -5/13 -1/1 1/1 -8/21 0/1 1/1 -3/8 -1/1 -4/11 0/1 -5/14 1/1 -1/3 -1/1 1/1 -2/7 0/1 1/0 -5/18 -1/1 -8/29 -2/1 0/1 -3/11 -1/1 1/1 -4/15 1/1 1/0 -1/4 -1/1 -2/9 0/1 -3/14 1/1 -1/5 -1/1 1/1 -2/11 -1/2 0/1 -1/6 1/1 -1/7 -1/1 1/1 0/1 0/1 1/0 1/6 0/1 1/5 -1/1 1/1 1/4 -1/1 3/11 0/1 5/18 1/5 2/7 0/1 1/2 3/10 1/1 1/3 -1/1 1/1 2/5 0/1 3/7 1/3 1/1 7/16 3/5 4/9 2/3 1/1 1/2 1/1 5/9 1/0 9/16 -3/1 4/7 -1/1 1/0 11/19 -1/1 -1/3 7/12 -1/1 3/5 -1/1 1/1 2/3 0/1 1/0 5/7 -1/1 1/1 3/4 1/0 7/9 -5/1 -3/1 4/5 -2/1 1/0 9/11 -3/1 -1/1 5/6 -1/1 6/7 -4/3 -1/1 1/1 -1/1 1/1 4/3 0/1 7/5 1/3 1/1 10/7 1/1 1/0 13/9 -1/1 1/1 3/2 1/1 8/5 -1/1 0/1 21/13 -1/1 -1/3 13/8 0/1 18/11 0/1 1/0 5/3 -1/1 1/1 22/13 -2/1 0/1 17/10 -1/1 12/7 0/1 1/0 19/11 -3/1 -1/1 7/4 -1/1 9/5 0/1 11/6 1/1 2/1 0/1 1/0 7/3 -1/1 1/1 5/2 1/0 13/5 -3/1 -1/1 21/8 -1/1 29/11 -1/1 -1/3 8/3 -1/1 1/0 19/7 -5/3 -1/1 30/11 -6/5 -1/1 41/15 -1/1 11/4 -1/1 14/5 -1/2 0/1 3/1 -1/1 1/1 13/4 1/1 23/7 3/5 1/1 33/10 1/1 43/13 1/1 10/3 1/1 2/1 7/2 3/1 11/3 1/0 15/4 -7/1 4/1 -2/1 1/0 5/1 -3/1 -1/1 6/1 -2/1 0/1 7/1 -3/1 -1/1 8/1 -2/1 1/0 1/0 -1/1 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(49,22,-176,-79) (-1/2,-4/9) -> (-2/7,-5/18) Hyperbolic Matrix(199,88,52,23) (-4/9,-7/16) -> (15/4,4/1) Hyperbolic Matrix(97,42,224,97) (-7/16,-3/7) -> (3/7,7/16) Hyperbolic Matrix(49,20,22,9) (-3/7,-2/5) -> (2/1,7/3) 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(545,208,338,129) (-5/13,-8/21) -> (8/5,21/13) Hyperbolic Matrix(179,68,408,155) (-8/21,-3/8) -> (7/16,4/9) Hyperbolic Matrix(87,32,-242,-89) (-3/8,-4/11) -> (-4/11,-5/14) Parabolic Matrix(175,62,302,107) (-5/14,-1/3) -> (11/19,7/12) Hyperbolic Matrix(41,12,58,17) (-1/3,-2/7) -> (2/3,5/7) Hyperbolic Matrix(889,246,524,145) (-5/18,-8/29) -> (22/13,17/10) Hyperbolic Matrix(387,106,230,63) (-8/29,-3/11) -> (5/3,22/13) Hyperbolic Matrix(305,82,212,57) (-3/11,-4/15) -> (10/7,13/9) Hyperbolic Matrix(151,40,268,71) (-4/15,-1/4) -> (9/16,4/7) Hyperbolic Matrix(35,8,-162,-37) (-1/4,-2/9) -> (-2/9,-3/14) Parabolic 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(131,18,80,11) (-1/7,0/1) -> (18/11,5/3) 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(47,-10,80,-17) (1/5,1/4) -> (7/12,3/5) Hyperbolic Matrix(67,-18,242,-65) (1/4,3/11) -> (3/11,5/18) Parabolic Matrix(239,-70,140,-41) (2/7,3/10) -> (17/10,12/7) Hyperbolic Matrix(193,-60,74,-23) (3/10,1/3) -> (13/5,21/8) Hyperbolic Matrix(21,-8,50,-19) (1/3,2/5) -> (2/5,3/7) Parabolic Matrix(57,-26,68,-31) (4/9,1/2) -> (5/6,6/7) Hyperbolic Matrix(91,-50,162,-89) (1/2,5/9) -> (5/9,9/16) Parabolic Matrix(355,-204,134,-77) (4/7,11/19) -> (29/11,8/3) Hyperbolic Matrix(35,-22,8,-5) (3/5,2/3) -> (4/1,5/1) Hyperbolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(203,-160,118,-93) (7/9,4/5) -> (12/7,19/11) 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(25,-32,18,-23) (1/1,4/3) -> (4/3,7/5) Parabolic Matrix(173,-246,64,-91) (7/5,10/7) -> (8/3,19/7) Hyperbolic Matrix(55,-86,16,-25) (3/2,8/5) -> (10/3,7/2) 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(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(547,-1440,166,-437) (21/8,29/11) -> (23/7,33/10) 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(37,-102,4,-11) (11/4,14/5) -> (8/1,1/0) Hyperbolic Matrix(67,-242,18,-65) (7/2,11/3) -> (11/3,15/4) Parabolic Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,2,1) Matrix(49,22,-176,-79) -> Matrix(1,-2,0,1) Matrix(199,88,52,23) -> Matrix(1,-4,0,1) Matrix(97,42,224,97) -> Matrix(1,0,2,1) Matrix(49,20,22,9) -> Matrix(1,0,0,1) Matrix(91,36,48,19) -> Matrix(1,0,0,1) Matrix(227,88,276,107) -> Matrix(1,-2,0,1) Matrix(545,208,338,129) -> Matrix(1,0,-2,1) Matrix(179,68,408,155) -> Matrix(1,-2,2,-3) Matrix(87,32,-242,-89) -> Matrix(1,0,2,1) Matrix(175,62,302,107) -> Matrix(1,0,-2,1) Matrix(41,12,58,17) -> Matrix(1,0,0,1) Matrix(889,246,524,145) -> Matrix(1,0,0,1) Matrix(387,106,230,63) -> Matrix(1,0,0,1) Matrix(305,82,212,57) -> Matrix(1,0,0,1) Matrix(151,40,268,71) -> Matrix(1,-2,0,1) Matrix(35,8,-162,-37) -> Matrix(1,0,2,1) Matrix(107,22,34,7) -> Matrix(1,0,0,1) Matrix(103,20,36,7) -> Matrix(1,0,0,1) Matrix(67,12,240,43) -> Matrix(1,0,4,1) Matrix(99,16,68,11) -> Matrix(1,0,0,1) Matrix(131,18,80,11) -> Matrix(1,0,0,1) Matrix(121,-18,74,-11) -> Matrix(1,0,0,1) Matrix(191,-34,118,-21) -> Matrix(1,0,-2,1) Matrix(47,-10,80,-17) -> Matrix(1,0,0,1) Matrix(67,-18,242,-65) -> Matrix(1,0,6,1) Matrix(239,-70,140,-41) -> Matrix(1,0,-2,1) Matrix(193,-60,74,-23) -> Matrix(1,-2,0,1) Matrix(21,-8,50,-19) -> Matrix(1,0,2,1) Matrix(57,-26,68,-31) -> Matrix(1,-2,0,1) Matrix(91,-50,162,-89) -> Matrix(1,-4,0,1) Matrix(355,-204,134,-77) -> Matrix(1,0,0,1) Matrix(35,-22,8,-5) -> Matrix(1,-2,0,1) Matrix(49,-36,64,-47) -> Matrix(1,-4,0,1) Matrix(203,-160,118,-93) -> Matrix(1,2,0,1) Matrix(123,-100,16,-13) -> Matrix(1,0,0,1) Matrix(163,-144,60,-53) -> Matrix(3,2,-2,-1) Matrix(25,-32,18,-23) -> Matrix(1,0,2,1) Matrix(173,-246,64,-91) -> Matrix(1,-2,0,1) Matrix(55,-86,16,-25) -> Matrix(1,2,0,1) Matrix(235,-408,72,-125) -> Matrix(1,0,2,1) Matrix(91,-162,50,-89) -> Matrix(1,0,2,1) Matrix(41,-100,16,-39) -> Matrix(1,-2,0,1) Matrix(547,-1440,166,-437) -> Matrix(3,2,4,3) Matrix(623,-1700,188,-513) -> Matrix(7,8,6,7) Matrix(667,-1826,202,-553) -> Matrix(7,6,8,7) Matrix(37,-102,4,-11) -> Matrix(3,2,-2,-1) Matrix(67,-242,18,-65) -> Matrix(1,-10,0,1) Matrix(13,-72,2,-11) -> 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): 10 Degree of the the map X: 10 Degree of the the map Y: 48 Permutation triple for Y: ((1,6,21,37,22,7,2)(3,12,8,26,34,13,4)(5,17,24,27,29,10,9)(11,32,39,43,18,42,33)(14,38,44,40,48,47,30)(16,41,31,45,20,36,35); (1,4,15,32,40,16,5)(3,10,30,23,22,31,11)(6,19,44,27,12,33,20)(7,17,35,47,42,25,8)(9,28,41,26,38,21,18)(13,36,46,24,39,37,14); (1,2,8,27,46,36,33,47,48,32,31,28,9,3)(4,14,10,29,44,26,25,42,21,20,45,22,39,15)(5,18,43,24,7,23,30,35,13,34,41,40,19,6)(11,12)(16,17)(37,38)) ----------------------------------------------------------------------- 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 (0/1,1/0) 0 14 1/6 0/1 1 2 1/5 0 7 1/4 -1/1 1 14 3/11 0/1 3 1 2/7 (0/1,1/2) 0 14 3/10 1/1 1 14 1/3 0 7 2/5 0/1 2 2 3/7 0 7 4/9 (2/3,1/1) 0 14 1/2 1/1 1 14 5/9 1/0 2 1 4/7 (-1/1,1/0) 0 14 7/12 -1/1 1 14 3/5 0 7 2/3 (0/1,1/0) 0 14 3/4 1/0 2 2 4/5 (-2/1,1/0) 0 14 5/6 -1/1 1 14 6/7 (-4/3,-1/1) 0 14 1/1 0 7 4/3 0/1 2 2 7/5 0 7 10/7 (1/1,1/0) 0 14 3/2 1/1 1 14 8/5 (-1/1,0/1) 0 14 13/8 0/1 1 2 5/3 0 7 17/10 -1/1 1 14 12/7 (0/1,1/0) 0 14 7/4 -1/1 1 14 9/5 0/1 1 1 2/1 (0/1,1/0) 0 14 5/2 1/0 1 2 8/3 (-1/1,1/0) 0 14 19/7 0 7 30/11 (-6/5,-1/1) 0 14 41/15 -1/1 7 1 11/4 -1/1 1 14 3/1 0 7 10/3 (1/1,2/1) 0 14 7/2 3/1 1 14 11/3 1/0 5 1 4/1 (-2/1,1/0) 0 14 5/1 0 7 6/1 0 2 7/1 0 7 1/0 -1/1 1 14 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(1,0,12,-1) (0/1,1/6) -> (0/1,1/6) Reflection Matrix(95,-18,58,-11) (1/6,1/5) -> (13/8,5/3) Glide Reflection Matrix(47,-10,80,-17) (1/5,1/4) -> (7/12,3/5) Hyperbolic Matrix(23,-6,88,-23) (1/4,3/11) -> (1/4,3/11) Reflection Matrix(43,-12,154,-43) (3/11,2/7) -> (3/11,2/7) Reflection Matrix(239,-70,140,-41) (2/7,3/10) -> (17/10,12/7) Hyperbolic Matrix(63,-20,22,-7) (3/10,1/3) -> (11/4,3/1) Glide Reflection Matrix(21,-8,50,-19) (1/3,2/5) -> (2/5,3/7) Parabolic Matrix(97,-42,30,-13) (3/7,4/9) -> (3/1,10/3) Glide Reflection Matrix(57,-26,68,-31) (4/9,1/2) -> (5/6,6/7) Hyperbolic Matrix(19,-10,36,-19) (1/2,5/9) -> (1/2,5/9) Reflection Matrix(71,-40,126,-71) (5/9,4/7) -> (5/9,4/7) Reflection Matrix(141,-82,98,-57) (4/7,7/12) -> (10/7,3/2) Glide Reflection Matrix(35,-22,8,-5) (3/5,2/3) -> (4/1,5/1) Hyperbolic Matrix(17,-12,24,-17) (2/3,3/4) -> (2/3,3/4) Reflection Matrix(31,-24,40,-31) (3/4,4/5) -> (3/4,4/5) Reflection Matrix(107,-88,62,-51) (4/5,5/6) -> (12/7,7/4) Glide Reflection Matrix(163,-144,60,-53) (6/7,1/1) -> (19/7,30/11) Hyperbolic Matrix(25,-32,18,-23) (1/1,4/3) -> (4/3,7/5) Parabolic Matrix(173,-246,64,-91) (7/5,10/7) -> (8/3,19/7) Hyperbolic Matrix(55,-86,16,-25) (3/2,8/5) -> (10/3,7/2) Hyperbolic Matrix(129,-208,80,-129) (8/5,13/8) -> (8/5,13/8) Reflection Matrix(73,-124,10,-17) (5/3,17/10) -> (7/1,1/0) Glide Reflection Matrix(71,-126,40,-71) (7/4,9/5) -> (7/4,9/5) Reflection Matrix(19,-36,10,-19) (9/5,2/1) -> (9/5,2/1) Reflection Matrix(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) Reflection Matrix(31,-80,12,-31) (5/2,8/3) -> (5/2,8/3) Reflection Matrix(901,-2460,330,-901) (30/11,41/15) -> (30/11,41/15) Reflection Matrix(329,-902,120,-329) (41/15,11/4) -> (41/15,11/4) Reflection Matrix(43,-154,12,-43) (7/2,11/3) -> (7/2,11/3) Reflection Matrix(23,-88,6,-23) (11/3,4/1) -> (11/3,4/1) Reflection Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(1,0,12,-1) -> Matrix(1,0,0,-1) (0/1,1/6) -> (0/1,1/0) Matrix(95,-18,58,-11) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(47,-10,80,-17) -> Matrix(1,0,0,1) Matrix(23,-6,88,-23) -> Matrix(-1,0,2,1) (1/4,3/11) -> (-1/1,0/1) Matrix(43,-12,154,-43) -> Matrix(1,0,4,-1) (3/11,2/7) -> (0/1,1/2) Matrix(239,-70,140,-41) -> Matrix(1,0,-2,1) 0/1 Matrix(63,-20,22,-7) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(21,-8,50,-19) -> Matrix(1,0,2,1) 0/1 Matrix(97,-42,30,-13) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(57,-26,68,-31) -> Matrix(1,-2,0,1) 1/0 Matrix(19,-10,36,-19) -> Matrix(-1,2,0,1) (1/2,5/9) -> (1/1,1/0) Matrix(71,-40,126,-71) -> Matrix(1,2,0,-1) (5/9,4/7) -> (-1/1,1/0) Matrix(141,-82,98,-57) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(35,-22,8,-5) -> Matrix(1,-2,0,1) 1/0 Matrix(17,-12,24,-17) -> Matrix(1,0,0,-1) (2/3,3/4) -> (0/1,1/0) Matrix(31,-24,40,-31) -> Matrix(1,4,0,-1) (3/4,4/5) -> (-2/1,1/0) Matrix(107,-88,62,-51) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(163,-144,60,-53) -> Matrix(3,2,-2,-1) -1/1 Matrix(25,-32,18,-23) -> Matrix(1,0,2,1) 0/1 Matrix(173,-246,64,-91) -> Matrix(1,-2,0,1) 1/0 Matrix(55,-86,16,-25) -> Matrix(1,2,0,1) 1/0 Matrix(129,-208,80,-129) -> Matrix(-1,0,2,1) (8/5,13/8) -> (-1/1,0/1) Matrix(73,-124,10,-17) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(71,-126,40,-71) -> Matrix(-1,0,2,1) (7/4,9/5) -> (-1/1,0/1) Matrix(19,-36,10,-19) -> Matrix(1,0,0,-1) (9/5,2/1) -> (0/1,1/0) Matrix(9,-20,4,-9) -> Matrix(1,0,0,-1) (2/1,5/2) -> (0/1,1/0) Matrix(31,-80,12,-31) -> Matrix(1,2,0,-1) (5/2,8/3) -> (-1/1,1/0) Matrix(901,-2460,330,-901) -> Matrix(11,12,-10,-11) (30/11,41/15) -> (-6/5,-1/1) Matrix(329,-902,120,-329) -> Matrix(3,2,-4,-3) (41/15,11/4) -> (-1/1,-1/2) Matrix(43,-154,12,-43) -> Matrix(-1,6,0,1) (7/2,11/3) -> (3/1,1/0) Matrix(23,-88,6,-23) -> Matrix(1,4,0,-1) (11/3,4/1) -> (-2/1,1/0) Matrix(13,-72,2,-11) -> Matrix(1,0,0,1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.