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: 32 Genus: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/1 -4/1 -11/3 -3/1 -5/2 -2/1 -5/3 -5/4 -1/1 -5/7 -5/9 0/1 1/2 5/9 5/7 1/1 5/4 10/7 3/2 5/3 2/1 20/9 7/3 5/2 3/1 10/3 7/2 11/3 4/1 5/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -1/1 -9/2 -1/2 -4/1 -1/1 0/1 1/0 -11/3 -2/1 1/0 -7/2 1/0 -10/3 -1/1 -3/1 -1/1 1/0 -5/2 1/0 -7/3 -2/1 1/0 -9/4 1/0 -20/9 -3/1 -1/1 -11/5 -2/1 1/0 -2/1 -2/1 -1/1 1/0 -5/3 -1/1 -8/5 -1/1 -1/2 0/1 -19/12 1/0 -30/19 -1/1 1/1 -11/7 0/1 1/0 -3/2 1/0 -10/7 -3/1 -1/1 -7/5 -2/1 1/0 -11/8 -3/2 -4/3 -2/1 -1/1 1/0 -9/7 -2/1 1/0 -5/4 -2/1 -1/1 -2/1 -1/1 -5/6 -2/1 -9/11 -2/1 -3/2 -4/5 -2/1 -3/2 -1/1 -11/14 1/0 -7/9 -2/1 -3/2 -10/13 -5/3 -1/1 -3/4 -3/2 -5/7 -1/1 -7/10 1/0 -9/13 -2/1 1/0 -2/3 -2/1 -3/2 -1/1 -9/14 -3/2 -7/11 -2/1 -3/2 -5/8 -3/2 -3/5 -3/2 -1/1 -10/17 -1/1 -7/12 -3/2 -11/19 -2/1 -3/2 -4/7 -3/2 -4/3 -1/1 -5/9 -1/1 -6/11 -2/1 -1/1 1/0 -1/2 -3/2 0/1 -1/1 1/2 -3/4 5/9 -1/1 -5/7 4/7 -3/4 -5/7 -2/3 7/12 -3/4 3/5 -5/7 -2/3 2/3 -2/3 -3/5 -1/2 5/7 -1/1 -3/5 8/11 -2/3 -3/5 -1/2 3/4 -1/2 7/9 -2/3 -1/2 4/5 -2/3 -3/5 -1/2 1/1 -1/1 -1/2 6/5 -2/3 -3/5 -1/2 5/4 -1/2 14/11 -1/2 -4/9 -3/7 9/7 -1/2 -2/5 4/3 -1/2 -1/3 0/1 15/11 -1/1 -1/3 11/8 -1/2 7/5 -1/2 0/1 17/12 -1/2 10/7 -1/1 -1/3 3/2 -1/2 5/3 -1/1 -1/3 7/4 -1/2 16/9 -1/2 -1/3 0/1 9/5 -1/2 0/1 20/11 -1/1 -1/3 11/6 -1/2 2/1 -1/2 -1/3 0/1 11/5 -1/4 0/1 20/9 -1/3 -1/5 9/4 -1/4 7/3 -1/6 0/1 5/2 0/1 13/5 0/1 1/8 21/8 1/6 8/3 0/1 1/5 1/4 3/1 0/1 1/1 13/4 1/2 10/3 1/1 7/2 1/0 18/5 0/1 1/1 1/0 11/3 0/1 1/0 4/1 0/1 1/1 1/0 5/1 -1/1 1/1 6/1 0/1 1/1 1/0 7/1 0/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(9,50,-2,-11) (-5/1,1/0) -> (-5/1,-9/2) Parabolic Matrix(19,80,-24,-101) (-9/2,-4/1) -> (-4/5,-11/14) Hyperbolic Matrix(21,80,16,61) (-4/1,-11/3) -> (9/7,4/3) Hyperbolic Matrix(39,140,-56,-201) (-11/3,-7/2) -> (-7/10,-9/13) Hyperbolic Matrix(41,140,12,41) (-7/2,-10/3) -> (10/3,7/2) Hyperbolic Matrix(19,60,-32,-101) (-10/3,-3/1) -> (-3/5,-10/17) Hyperbolic Matrix(11,30,-18,-49) (-3/1,-5/2) -> (-5/8,-3/5) Hyperbolic Matrix(29,70,-46,-111) (-5/2,-7/3) -> (-7/11,-5/8) Hyperbolic Matrix(61,140,44,101) (-7/3,-9/4) -> (11/8,7/5) Hyperbolic Matrix(161,360,72,161) (-9/4,-20/9) -> (20/9,9/4) Hyperbolic Matrix(181,400,100,221) (-20/9,-11/5) -> (9/5,20/11) Hyperbolic Matrix(101,220,28,61) (-11/5,-2/1) -> (18/5,11/3) Hyperbolic Matrix(29,50,-18,-31) (-2/1,-5/3) -> (-5/3,-8/5) Parabolic Matrix(201,320,76,121) (-8/5,-19/12) -> (21/8,8/3) Hyperbolic Matrix(449,710,246,389) (-19/12,-30/19) -> (20/11,11/6) Hyperbolic Matrix(349,550,158,249) (-30/19,-11/7) -> (11/5,20/9) Hyperbolic Matrix(71,110,-122,-189) (-11/7,-3/2) -> (-7/12,-11/19) Hyperbolic Matrix(41,60,28,41) (-3/2,-10/7) -> (10/7,3/2) Hyperbolic Matrix(99,140,-128,-181) (-10/7,-7/5) -> (-7/9,-10/13) Hyperbolic Matrix(101,140,44,61) (-7/5,-11/8) -> (9/4,7/3) Hyperbolic Matrix(51,70,-94,-129) (-11/8,-4/3) -> (-6/11,-1/2) Hyperbolic Matrix(61,80,16,21) (-4/3,-9/7) -> (11/3,4/1) Hyperbolic Matrix(71,90,-86,-109) (-9/7,-5/4) -> (-5/6,-9/11) Hyperbolic Matrix(9,10,-10,-11) (-5/4,-1/1) -> (-1/1,-5/6) Parabolic Matrix(221,180,124,101) (-9/11,-4/5) -> (16/9,9/5) Hyperbolic Matrix(371,290,142,111) (-11/14,-7/9) -> (13/5,21/8) Hyperbolic Matrix(261,200,184,141) (-10/13,-3/4) -> (17/12,10/7) Hyperbolic Matrix(69,50,-98,-71) (-3/4,-5/7) -> (-5/7,-7/10) Parabolic Matrix(59,40,28,19) (-9/13,-2/3) -> (2/1,11/5) Hyperbolic Matrix(61,40,32,21) (-2/3,-9/14) -> (11/6,2/1) Hyperbolic Matrix(109,70,14,9) (-9/14,-7/11) -> (7/1,1/0) Hyperbolic Matrix(341,200,104,61) (-10/17,-7/12) -> (13/4,10/3) Hyperbolic Matrix(329,190,258,149) (-11/19,-4/7) -> (14/11,9/7) Hyperbolic Matrix(89,50,-162,-91) (-4/7,-5/9) -> (-5/9,-6/11) Parabolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(129,-70,94,-51) (1/2,5/9) -> (15/11,11/8) Hyperbolic Matrix(141,-80,104,-59) (5/9,4/7) -> (4/3,15/11) Hyperbolic Matrix(241,-140,136,-79) (4/7,7/12) -> (7/4,16/9) Hyperbolic Matrix(101,-60,32,-19) (7/12,3/5) -> (3/1,13/4) Hyperbolic Matrix(49,-30,18,-11) (3/5,2/3) -> (8/3,3/1) Hyperbolic Matrix(71,-50,98,-69) (2/3,5/7) -> (5/7,8/11) Parabolic Matrix(149,-110,42,-31) (8/11,3/4) -> (7/2,18/5) Hyperbolic Matrix(181,-140,128,-99) (3/4,7/9) -> (7/5,17/12) Hyperbolic Matrix(89,-70,14,-11) (7/9,4/5) -> (6/1,7/1) Hyperbolic Matrix(11,-10,10,-9) (4/5,1/1) -> (1/1,6/5) Parabolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(9,50,-2,-11) -> Matrix(1,2,-2,-3) Matrix(19,80,-24,-101) -> Matrix(3,2,-2,-1) Matrix(21,80,16,61) -> Matrix(1,0,-2,1) Matrix(39,140,-56,-201) -> Matrix(1,0,0,1) Matrix(41,140,12,41) -> Matrix(1,2,0,1) Matrix(19,60,-32,-101) -> Matrix(3,2,-2,-1) Matrix(11,30,-18,-49) -> Matrix(3,2,-2,-1) Matrix(29,70,-46,-111) -> Matrix(3,8,-2,-5) Matrix(61,140,44,101) -> Matrix(1,2,-2,-3) Matrix(161,360,72,161) -> Matrix(1,2,-4,-7) Matrix(181,400,100,221) -> Matrix(1,2,-2,-3) Matrix(101,220,28,61) -> Matrix(1,2,0,1) Matrix(29,50,-18,-31) -> Matrix(1,2,-2,-3) Matrix(201,320,76,121) -> Matrix(1,0,6,1) Matrix(449,710,246,389) -> Matrix(1,0,-2,1) Matrix(349,550,158,249) -> Matrix(1,0,-4,1) Matrix(71,110,-122,-189) -> Matrix(3,2,-2,-1) Matrix(41,60,28,41) -> Matrix(1,2,-2,-3) Matrix(99,140,-128,-181) -> Matrix(3,8,-2,-5) Matrix(101,140,44,61) -> Matrix(1,2,-6,-11) Matrix(51,70,-94,-129) -> Matrix(1,0,0,1) Matrix(61,80,16,21) -> Matrix(1,2,0,1) Matrix(71,90,-86,-109) -> Matrix(3,8,-2,-5) Matrix(9,10,-10,-11) -> Matrix(1,0,0,1) Matrix(221,180,124,101) -> Matrix(1,2,-4,-7) Matrix(371,290,142,111) -> Matrix(1,2,6,13) Matrix(261,200,184,141) -> Matrix(1,2,-4,-7) Matrix(69,50,-98,-71) -> Matrix(1,2,-2,-3) Matrix(59,40,28,19) -> Matrix(1,2,-4,-7) Matrix(61,40,32,21) -> Matrix(1,2,-4,-7) Matrix(109,70,14,9) -> Matrix(1,2,-2,-3) Matrix(341,200,104,61) -> Matrix(1,2,0,1) Matrix(329,190,258,149) -> Matrix(5,8,-12,-19) Matrix(89,50,-162,-91) -> Matrix(1,2,-2,-3) Matrix(1,0,4,1) -> Matrix(5,6,-6,-7) Matrix(129,-70,94,-51) -> Matrix(3,2,-2,-1) Matrix(141,-80,104,-59) -> Matrix(3,2,-2,-1) Matrix(241,-140,136,-79) -> Matrix(3,2,-2,-1) Matrix(101,-60,32,-19) -> Matrix(3,2,10,7) Matrix(49,-30,18,-11) -> Matrix(3,2,10,7) Matrix(71,-50,98,-69) -> Matrix(1,0,0,1) Matrix(149,-110,42,-31) -> Matrix(3,2,-2,-1) Matrix(181,-140,128,-99) -> Matrix(3,2,-8,-5) Matrix(89,-70,14,-11) -> Matrix(3,2,-2,-1) Matrix(11,-10,10,-9) -> Matrix(1,0,0,1) Matrix(81,-100,64,-79) -> Matrix(11,6,-24,-13) Matrix(31,-50,18,-29) -> Matrix(1,0,0,1) Matrix(41,-100,16,-39) -> Matrix(1,0,14,1) Matrix(11,-50,2,-9) -> 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): 8 Degree of the the map X: 8 Degree of the the map Y: 48 Permutation triple for Y: ((1,2)(3,10,32,40,15,6,21,38,33,11)(4,16,41,26,8,7,25,42,17,5)(9,29,14,13,37,20,46,24,23,30)(12,35,19,18,45,22,43,28,27,36)(31,44)(34,39)(47,48); (1,5,19,20,6)(2,8,28,9,3)(4,14,39,22,15)(7,24,34,12,11)(16,40,27,31,30)(18,44,37,25,33)(29,43,42,48,38)(32,46,35,41,47); (1,3,12,36,40,47,42,37,13,4)(2,6,22,45,33,48,41,30,23,7)(5,17,43,39,24,32,10,9,31,18)(8,26,35,34,14,38,21,20,44,27)(11,25)(15,16)(19,46)(28,29)) ----------------------------------------------------------------------- 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 -5/1 -1/1 2 2 -9/2 -1/2 1 10 -4/1 0 10 -11/3 (-2/1,1/0) 0 10 -7/2 1/0 1 10 -10/3 -1/1 1 2 -3/1 (-1/1,1/0) 0 10 -5/2 1/0 1 2 -7/3 (-2/1,1/0) 0 10 -9/4 1/0 1 10 -2/1 0 10 -5/3 -1/1 2 2 -8/5 0 10 -11/7 (0/1,1/0) 0 10 -3/2 1/0 1 10 -10/7 (-2/1,1/0) 0 2 -7/5 (-2/1,1/0) 0 10 -11/8 -3/2 1 10 -4/3 0 10 -9/7 (-2/1,1/0) 0 10 -5/4 -2/1 1 2 -1/1 (-2/1,-1/1) 0 10 0/1 -1/1 3 2 1/1 (-1/1,-1/2) 0 10 5/4 -1/2 3 2 9/7 (-1/2,-2/5) 0 10 4/3 0 10 11/8 -1/2 1 10 7/5 (-1/2,0/1) 0 10 10/7 (-1/2,0/1) 0 2 3/2 -1/2 1 10 5/3 0 2 7/4 -1/2 1 10 9/5 (-1/2,0/1) 0 10 20/11 (-1/2,0/1) 0 2 11/6 -1/2 1 10 2/1 0 10 11/5 (-1/4,0/1) 0 10 20/9 (-1/4,0/1) 0 2 9/4 -1/4 1 10 7/3 (-1/6,0/1) 0 10 5/2 0/1 7 2 3/1 (0/1,1/1) 0 10 10/3 1/1 1 2 7/2 1/0 1 10 11/3 (0/1,1/0) 0 10 4/1 0 10 5/1 0 2 6/1 0 10 1/0 1/0 1 10 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(9,50,-2,-11) (-5/1,1/0) -> (-5/1,-9/2) Parabolic Matrix(19,80,-14,-59) (-9/2,-4/1) -> (-11/8,-4/3) Glide Reflection Matrix(21,80,16,61) (-4/1,-11/3) -> (9/7,4/3) Hyperbolic Matrix(31,110,-20,-71) (-11/3,-7/2) -> (-11/7,-3/2) Glide Reflection Matrix(41,140,12,41) (-7/2,-10/3) -> (10/3,7/2) Hyperbolic Matrix(19,60,-6,-19) (-10/3,-3/1) -> (-10/3,-3/1) Reflection Matrix(11,30,-4,-11) (-3/1,-5/2) -> (-3/1,-5/2) Reflection Matrix(29,70,-12,-29) (-5/2,-7/3) -> (-5/2,-7/3) Reflection Matrix(61,140,44,101) (-7/3,-9/4) -> (11/8,7/5) Hyperbolic Matrix(19,40,10,21) (-9/4,-2/1) -> (11/6,2/1) Glide Reflection Matrix(29,50,-18,-31) (-2/1,-5/3) -> (-5/3,-8/5) Parabolic Matrix(69,110,32,51) (-8/5,-11/7) -> (2/1,11/5) Glide Reflection Matrix(41,60,28,41) (-3/2,-10/7) -> (10/7,3/2) Hyperbolic Matrix(99,140,-70,-99) (-10/7,-7/5) -> (-10/7,-7/5) Reflection Matrix(101,140,44,61) (-7/5,-11/8) -> (9/4,7/3) Hyperbolic Matrix(61,80,16,21) (-4/3,-9/7) -> (11/3,4/1) Hyperbolic Matrix(71,90,-56,-71) (-9/7,-5/4) -> (-9/7,-5/4) Reflection Matrix(9,10,-8,-9) (-5/4,-1/1) -> (-5/4,-1/1) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(9,-10,8,-9) (1/1,5/4) -> (1/1,5/4) Reflection Matrix(71,-90,56,-71) (5/4,9/7) -> (5/4,9/7) Reflection Matrix(51,-70,8,-11) (4/3,11/8) -> (6/1,1/0) Glide Reflection Matrix(99,-140,70,-99) (7/5,10/7) -> (7/5,10/7) Reflection Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(79,-140,22,-39) (7/4,9/5) -> (7/2,11/3) Glide Reflection Matrix(199,-360,110,-199) (9/5,20/11) -> (9/5,20/11) Reflection Matrix(219,-400,98,-179) (20/11,11/6) -> (20/9,9/4) Glide Reflection Matrix(199,-440,90,-199) (11/5,20/9) -> (11/5,20/9) Reflection Matrix(29,-70,12,-29) (7/3,5/2) -> (7/3,5/2) Reflection Matrix(11,-30,4,-11) (5/2,3/1) -> (5/2,3/1) Reflection Matrix(19,-60,6,-19) (3/1,10/3) -> (3/1,10/3) Reflection Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(9,50,-2,-11) -> Matrix(1,2,-2,-3) -1/1 Matrix(19,80,-14,-59) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(21,80,16,61) -> Matrix(1,0,-2,1) 0/1 Matrix(31,110,-20,-71) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(41,140,12,41) -> Matrix(1,2,0,1) 1/0 Matrix(19,60,-6,-19) -> Matrix(1,2,0,-1) (-10/3,-3/1) -> (-1/1,1/0) Matrix(11,30,-4,-11) -> Matrix(1,2,0,-1) (-3/1,-5/2) -> (-1/1,1/0) Matrix(29,70,-12,-29) -> Matrix(1,4,0,-1) (-5/2,-7/3) -> (-2/1,1/0) Matrix(61,140,44,101) -> Matrix(1,2,-2,-3) -1/1 Matrix(19,40,10,21) -> Matrix(1,2,-2,-5) Matrix(29,50,-18,-31) -> Matrix(1,2,-2,-3) -1/1 Matrix(69,110,32,51) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(41,60,28,41) -> Matrix(1,2,-2,-3) -1/1 Matrix(99,140,-70,-99) -> Matrix(1,4,0,-1) (-10/7,-7/5) -> (-2/1,1/0) Matrix(101,140,44,61) -> Matrix(1,2,-6,-11) Matrix(61,80,16,21) -> Matrix(1,2,0,1) 1/0 Matrix(71,90,-56,-71) -> Matrix(1,4,0,-1) (-9/7,-5/4) -> (-2/1,1/0) Matrix(9,10,-8,-9) -> Matrix(3,4,-2,-3) (-5/4,-1/1) -> (-2/1,-1/1) Matrix(-1,0,2,1) -> Matrix(3,4,-2,-3) (-1/1,0/1) -> (-2/1,-1/1) Matrix(1,0,2,-1) -> Matrix(3,2,-4,-3) (0/1,1/1) -> (-1/1,-1/2) Matrix(9,-10,8,-9) -> Matrix(3,2,-4,-3) (1/1,5/4) -> (-1/1,-1/2) Matrix(71,-90,56,-71) -> Matrix(9,4,-20,-9) (5/4,9/7) -> (-1/2,-2/5) Matrix(51,-70,8,-11) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(99,-140,70,-99) -> Matrix(-1,0,4,1) (7/5,10/7) -> (-1/2,0/1) Matrix(31,-50,18,-29) -> Matrix(1,0,0,1) Matrix(79,-140,22,-39) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(199,-360,110,-199) -> Matrix(-1,0,4,1) (9/5,20/11) -> (-1/2,0/1) Matrix(219,-400,98,-179) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(199,-440,90,-199) -> Matrix(-1,0,8,1) (11/5,20/9) -> (-1/4,0/1) Matrix(29,-70,12,-29) -> Matrix(-1,0,12,1) (7/3,5/2) -> (-1/6,0/1) Matrix(11,-30,4,-11) -> Matrix(1,0,2,-1) (5/2,3/1) -> (0/1,1/1) Matrix(19,-60,6,-19) -> Matrix(1,0,2,-1) (3/1,10/3) -> (0/1,1/1) Matrix(11,-50,2,-9) -> Matrix(1,0,0,1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.