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 -4/5 -3/5 -1/2 -9/20 -2/5 -3/8 -7/20 -1/4 -1/5 -1/6 -3/20 0/1 1/7 1/6 1/5 2/9 1/4 2/7 3/10 1/3 11/30 3/8 2/5 1/2 5/9 3/5 19/30 2/3 7/10 4/5 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -6/7 -3/1 -2/1 -5/6 -2/1 -1/1 1/0 -4/5 -2/1 0/1 -11/14 -2/1 -1/1 1/0 -7/9 1/0 -3/4 -2/1 -1/1 1/0 -8/11 -2/1 1/0 -5/7 -3/2 -7/10 -1/1 -2/3 -1/1 0/1 -9/14 -1/1 0/1 1/0 -7/11 1/0 -19/30 -1/1 -12/19 -1/1 0/1 -5/8 -1/1 0/1 1/0 -3/5 -1/1 -7/12 -1/1 -2/3 -1/2 -11/19 -1/2 -4/7 -1/1 0/1 -5/9 -1/2 -1/2 -1/1 0/1 1/0 -5/11 -1/2 -9/20 0/1 -4/9 0/1 1/0 -3/7 -1/2 -2/5 0/1 -5/13 1/0 -8/21 0/1 1/1 -3/8 0/1 1/1 1/0 -7/19 1/0 -11/30 1/0 -4/11 -2/1 1/0 -5/14 -1/1 0/1 1/0 -6/17 -1/1 0/1 -7/20 0/1 -1/3 1/0 -3/10 -1/1 -2/7 -1/1 0/1 -3/11 1/0 -1/4 -1/1 0/1 1/0 -2/9 0/1 1/0 -1/5 -1/1 -2/11 -2/3 -1/2 -1/6 -1/1 -1/2 0/1 -2/13 -1/3 0/1 -3/20 0/1 -1/7 1/0 0/1 -1/1 0/1 1/7 -1/2 1/6 -1/1 0/1 1/0 1/5 -1/1 3/14 -1/1 -2/3 -1/2 2/9 -1/2 0/1 1/4 -1/1 -1/2 0/1 3/11 -1/2 2/7 -1/1 0/1 3/10 -1/1 1/3 -1/2 5/14 -1/1 -1/2 0/1 4/11 -2/3 -1/2 11/30 -1/2 7/19 -1/2 3/8 -1/2 -1/3 0/1 2/5 0/1 5/12 -1/1 0/1 1/0 8/19 -1/1 0/1 3/7 1/0 4/9 -1/2 0/1 1/2 -1/1 -1/2 0/1 6/11 -1/2 0/1 11/20 0/1 5/9 1/0 4/7 -1/1 0/1 3/5 -1/1 8/13 -1/1 -2/3 13/21 -1/2 5/8 -1/1 -1/2 0/1 12/19 -1/1 0/1 19/30 -1/1 7/11 -1/2 9/14 -1/1 -1/2 0/1 11/17 -1/2 13/20 0/1 2/3 -1/1 0/1 7/10 -1/1 5/7 -3/4 8/11 -2/3 -1/2 3/4 -1/1 -2/3 -1/2 7/9 -1/2 4/5 -2/3 0/1 9/11 -1/2 5/6 -1/1 -2/3 -1/2 11/13 -3/4 17/20 -2/3 6/7 -2/3 -3/5 1/1 -1/2 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(99,86,160,139) (-1/1,-6/7) -> (8/13,13/21) Hyperbolic Matrix(19,16,-120,-101) (-6/7,-5/6) -> (-1/6,-2/13) Hyperbolic Matrix(79,64,-100,-81) (-5/6,-4/5) -> (-4/5,-11/14) Parabolic Matrix(179,140,280,219) (-11/14,-7/9) -> (7/11,9/14) Hyperbolic Matrix(21,16,80,61) (-7/9,-3/4) -> (1/4,3/11) Hyperbolic Matrix(19,14,80,59) (-3/4,-8/11) -> (2/9,1/4) Hyperbolic Matrix(61,44,140,101) (-8/11,-5/7) -> (3/7,4/9) Hyperbolic Matrix(99,70,140,99) (-5/7,-7/10) -> (7/10,5/7) Hyperbolic Matrix(41,28,60,41) (-7/10,-2/3) -> (2/3,7/10) Hyperbolic Matrix(99,64,-280,-181) (-2/3,-9/14) -> (-5/14,-6/17) Hyperbolic Matrix(181,116,220,141) (-9/14,-7/11) -> (9/11,5/6) Hyperbolic Matrix(419,266,660,419) (-7/11,-19/30) -> (19/30,7/11) Hyperbolic Matrix(721,456,1140,721) (-19/30,-12/19) -> (12/19,19/30) Hyperbolic Matrix(159,100,380,239) (-12/19,-5/8) -> (5/12,8/19) Hyperbolic Matrix(59,36,-100,-61) (-5/8,-3/5) -> (-3/5,-7/12) Parabolic Matrix(141,82,380,221) (-7/12,-11/19) -> (7/19,3/8) Hyperbolic Matrix(121,70,140,81) (-11/19,-4/7) -> (6/7,1/1) Hyperbolic Matrix(39,22,140,79) (-4/7,-5/9) -> (3/11,2/7) Hyperbolic Matrix(19,10,-40,-21) (-5/9,-1/2) -> (-1/2,-5/11) Parabolic Matrix(221,100,400,181) (-5/11,-9/20) -> (11/20,5/9) Hyperbolic Matrix(219,98,400,179) (-9/20,-4/9) -> (6/11,11/20) Hyperbolic Matrix(101,44,140,61) (-4/9,-3/7) -> (5/7,8/11) Hyperbolic Matrix(39,16,-100,-41) (-3/7,-2/5) -> (-2/5,-5/13) Parabolic Matrix(21,8,160,61) (-5/13,-8/21) -> (0/1,1/7) Hyperbolic Matrix(201,76,320,121) (-8/21,-3/8) -> (5/8,12/19) Hyperbolic Matrix(199,74,320,119) (-3/8,-7/19) -> (13/21,5/8) Hyperbolic Matrix(419,154,1140,419) (-7/19,-11/30) -> (11/30,7/19) Hyperbolic Matrix(241,88,660,241) (-11/30,-4/11) -> (4/11,11/30) Hyperbolic Matrix(61,22,280,101) (-4/11,-5/14) -> (3/14,2/9) Hyperbolic Matrix(261,92,400,141) (-6/17,-7/20) -> (13/20,2/3) Hyperbolic Matrix(259,90,400,139) (-7/20,-1/3) -> (11/17,13/20) Hyperbolic Matrix(19,6,60,19) (-1/3,-3/10) -> (3/10,1/3) Hyperbolic Matrix(41,12,140,41) (-3/10,-2/7) -> (2/7,3/10) Hyperbolic Matrix(79,22,140,39) (-2/7,-3/11) -> (5/9,4/7) Hyperbolic Matrix(61,16,80,21) (-3/11,-1/4) -> (3/4,7/9) Hyperbolic Matrix(59,14,80,19) (-1/4,-2/9) -> (8/11,3/4) Hyperbolic Matrix(19,4,-100,-21) (-2/9,-1/5) -> (-1/5,-2/11) Parabolic Matrix(79,14,220,39) (-2/11,-1/6) -> (5/14,4/11) Hyperbolic Matrix(341,52,400,61) (-2/13,-3/20) -> (17/20,6/7) Hyperbolic Matrix(339,50,400,59) (-3/20,-1/7) -> (11/13,17/20) Hyperbolic Matrix(59,8,140,19) (-1/7,0/1) -> (8/19,3/7) Hyperbolic Matrix(101,-16,120,-19) (1/7,1/6) -> (5/6,11/13) Hyperbolic Matrix(21,-4,100,-19) (1/6,1/5) -> (1/5,3/14) Parabolic Matrix(181,-64,280,-99) (1/3,5/14) -> (9/14,11/17) Hyperbolic Matrix(41,-16,100,-39) (3/8,2/5) -> (2/5,5/12) Parabolic Matrix(21,-10,40,-19) (4/9,1/2) -> (1/2,6/11) Parabolic Matrix(61,-36,100,-59) (4/7,3/5) -> (3/5,8/13) Parabolic Matrix(81,-64,100,-79) (7/9,4/5) -> (4/5,9/11) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(99,86,160,139) -> Matrix(1,4,-2,-7) Matrix(19,16,-120,-101) -> Matrix(1,2,-2,-3) Matrix(79,64,-100,-81) -> Matrix(1,0,0,1) Matrix(179,140,280,219) -> Matrix(1,2,-2,-3) Matrix(21,16,80,61) -> Matrix(1,2,-2,-3) Matrix(19,14,80,59) -> Matrix(1,2,-2,-3) Matrix(61,44,140,101) -> Matrix(1,2,-2,-3) Matrix(99,70,140,99) -> Matrix(5,6,-6,-7) Matrix(41,28,60,41) -> Matrix(1,0,0,1) Matrix(99,64,-280,-181) -> Matrix(1,0,0,1) Matrix(181,116,220,141) -> Matrix(1,2,-2,-3) Matrix(419,266,660,419) -> Matrix(1,2,-2,-3) Matrix(721,456,1140,721) -> Matrix(1,0,0,1) Matrix(159,100,380,239) -> Matrix(1,0,0,1) Matrix(59,36,-100,-61) -> Matrix(1,2,-2,-3) Matrix(141,82,380,221) -> Matrix(3,2,-8,-5) Matrix(121,70,140,81) -> Matrix(5,2,-8,-3) Matrix(39,22,140,79) -> Matrix(1,0,0,1) Matrix(19,10,-40,-21) -> Matrix(1,0,0,1) Matrix(221,100,400,181) -> Matrix(1,0,2,1) Matrix(219,98,400,179) -> Matrix(1,0,-2,1) Matrix(101,44,140,61) -> Matrix(1,2,-2,-3) Matrix(39,16,-100,-41) -> Matrix(1,0,2,1) Matrix(21,8,160,61) -> Matrix(1,0,-2,1) Matrix(201,76,320,121) -> Matrix(1,0,-2,1) Matrix(199,74,320,119) -> Matrix(1,0,-2,1) Matrix(419,154,1140,419) -> Matrix(1,-6,-2,13) Matrix(241,88,660,241) -> Matrix(1,4,-2,-7) Matrix(61,22,280,101) -> Matrix(1,2,-2,-3) Matrix(261,92,400,141) -> Matrix(1,0,0,1) Matrix(259,90,400,139) -> Matrix(1,0,-2,1) Matrix(19,6,60,19) -> Matrix(1,2,-2,-3) Matrix(41,12,140,41) -> Matrix(1,0,0,1) Matrix(79,22,140,39) -> Matrix(1,0,0,1) Matrix(61,16,80,21) -> Matrix(1,2,-2,-3) Matrix(59,14,80,19) -> Matrix(1,2,-2,-3) Matrix(19,4,-100,-21) -> Matrix(1,2,-2,-3) Matrix(79,14,220,39) -> Matrix(1,0,0,1) Matrix(341,52,400,61) -> Matrix(9,2,-14,-3) Matrix(339,50,400,59) -> Matrix(3,-2,-4,3) Matrix(59,8,140,19) -> Matrix(1,0,0,1) Matrix(101,-16,120,-19) -> Matrix(1,2,-2,-3) Matrix(21,-4,100,-19) -> Matrix(1,2,-2,-3) Matrix(181,-64,280,-99) -> Matrix(1,0,0,1) Matrix(41,-16,100,-39) -> Matrix(1,0,2,1) Matrix(21,-10,40,-19) -> Matrix(1,0,0,1) Matrix(61,-36,100,-59) -> Matrix(1,2,-2,-3) Matrix(81,-64,100,-79) -> 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): 9 Degree of the the map X: 9 Degree of the the map Y: 48 Permutation triple for Y: ((1,4,16,41,34,46,37,17,5,2)(3,10,27,8,7,26,43,48,33,11)(6,22)(9,32)(12,35,42,47,31,25,24,15,20,19)(13,14)(18,39,38,44,21,40,28,36,45,23)(29,30); (1,2,8,30,35,44,38,31,9,3)(4,14,39,48,43,21,6,5,20,15)(7,25)(10,23,22,41,47,42,37,13,28,27)(11,12)(16,40)(17,18)(19,32,26,46,34,33,29,24,45,36); (2,6,23,24,7)(3,12,36,13,4)(5,18,10,9,19)(8,28,16,15,29)(11,34,22,21,35)(14,37,26,25,38)(17,42,30,33,39)(31,41,40,43,32)) ----------------------------------------------------------------------- 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 (-1/1,0/1) 0 10 1/7 -1/2 1 10 1/6 0 5 1/5 -1/1 1 2 3/14 0 5 2/9 (-1/2,0/1) 0 10 1/4 0 5 3/11 -1/2 1 10 2/7 (-1/1,0/1) 0 10 3/10 -1/1 1 1 1/3 -1/2 1 10 5/14 0 5 4/11 (-2/3,-1/2) 0 10 11/30 -1/2 5 1 7/19 -1/2 1 10 3/8 0 5 2/5 0/1 1 2 5/12 0 5 8/19 (-1/1,0/1) 0 10 3/7 1/0 1 10 4/9 (-1/2,0/1) 0 10 1/2 0 5 6/11 (-1/2,0/1) 0 10 11/20 0/1 2 1 5/9 1/0 1 10 4/7 (-1/1,0/1) 0 10 3/5 -1/1 1 2 8/13 (-1/1,-2/3) 0 10 13/21 -1/2 1 10 5/8 0 5 12/19 (-1/1,0/1) 0 10 19/30 -1/1 1 1 7/11 -1/2 1 10 9/14 0 5 11/17 -1/2 1 10 13/20 0/1 1 1 2/3 (-1/1,0/1) 0 10 7/10 -1/1 3 1 5/7 -3/4 1 10 8/11 (-2/3,-1/2) 0 10 3/4 0 5 7/9 -1/2 1 10 4/5 0 2 9/11 -1/2 1 10 5/6 0 5 11/13 -3/4 1 10 17/20 -2/3 3 1 6/7 (-2/3,-3/5) 0 10 1/1 -1/2 1 10 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(59,-8,140,-19) (0/1,1/7) -> (8/19,3/7) Glide Reflection Matrix(101,-16,120,-19) (1/7,1/6) -> (5/6,11/13) Hyperbolic Matrix(21,-4,100,-19) (1/6,1/5) -> (1/5,3/14) Parabolic Matrix(101,-22,280,-61) (3/14,2/9) -> (5/14,4/11) Glide Reflection Matrix(59,-14,80,-19) (2/9,1/4) -> (8/11,3/4) Glide Reflection Matrix(61,-16,80,-21) (1/4,3/11) -> (3/4,7/9) Glide Reflection Matrix(79,-22,140,-39) (3/11,2/7) -> (5/9,4/7) Glide Reflection Matrix(41,-12,140,-41) (2/7,3/10) -> (2/7,3/10) Reflection Matrix(19,-6,60,-19) (3/10,1/3) -> (3/10,1/3) Reflection Matrix(181,-64,280,-99) (1/3,5/14) -> (9/14,11/17) Hyperbolic Matrix(241,-88,660,-241) (4/11,11/30) -> (4/11,11/30) Reflection Matrix(419,-154,1140,-419) (11/30,7/19) -> (11/30,7/19) Reflection Matrix(199,-74,320,-119) (7/19,3/8) -> (13/21,5/8) Glide Reflection Matrix(41,-16,100,-39) (3/8,2/5) -> (2/5,5/12) Parabolic Matrix(239,-100,380,-159) (5/12,8/19) -> (5/8,12/19) Glide Reflection Matrix(101,-44,140,-61) (3/7,4/9) -> (5/7,8/11) Glide Reflection Matrix(21,-10,40,-19) (4/9,1/2) -> (1/2,6/11) Parabolic Matrix(241,-132,440,-241) (6/11,11/20) -> (6/11,11/20) Reflection Matrix(199,-110,360,-199) (11/20,5/9) -> (11/20,5/9) Reflection Matrix(61,-36,100,-59) (4/7,3/5) -> (3/5,8/13) Parabolic Matrix(139,-86,160,-99) (8/13,13/21) -> (6/7,1/1) Glide Reflection Matrix(721,-456,1140,-721) (12/19,19/30) -> (12/19,19/30) Reflection Matrix(419,-266,660,-419) (19/30,7/11) -> (19/30,7/11) Reflection Matrix(181,-116,220,-141) (7/11,9/14) -> (9/11,5/6) Glide Reflection Matrix(441,-286,680,-441) (11/17,13/20) -> (11/17,13/20) Reflection Matrix(79,-52,120,-79) (13/20,2/3) -> (13/20,2/3) Reflection Matrix(41,-28,60,-41) (2/3,7/10) -> (2/3,7/10) Reflection Matrix(99,-70,140,-99) (7/10,5/7) -> (7/10,5/7) Reflection Matrix(81,-64,100,-79) (7/9,4/5) -> (4/5,9/11) Parabolic Matrix(441,-374,520,-441) (11/13,17/20) -> (11/13,17/20) Reflection Matrix(239,-204,280,-239) (17/20,6/7) -> (17/20,6/7) 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(59,-8,140,-19) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(101,-16,120,-19) -> Matrix(1,2,-2,-3) -1/1 Matrix(21,-4,100,-19) -> Matrix(1,2,-2,-3) -1/1 Matrix(101,-22,280,-61) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(59,-14,80,-19) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(61,-16,80,-21) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(79,-22,140,-39) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(41,-12,140,-41) -> Matrix(-1,0,2,1) (2/7,3/10) -> (-1/1,0/1) Matrix(19,-6,60,-19) -> Matrix(3,2,-4,-3) (3/10,1/3) -> (-1/1,-1/2) Matrix(181,-64,280,-99) -> Matrix(1,0,0,1) Matrix(241,-88,660,-241) -> Matrix(7,4,-12,-7) (4/11,11/30) -> (-2/3,-1/2) Matrix(419,-154,1140,-419) -> Matrix(13,6,-28,-13) (11/30,7/19) -> (-1/2,-3/7) Matrix(199,-74,320,-119) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(41,-16,100,-39) -> Matrix(1,0,2,1) 0/1 Matrix(239,-100,380,-159) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(101,-44,140,-61) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(21,-10,40,-19) -> Matrix(1,0,0,1) Matrix(241,-132,440,-241) -> Matrix(-1,0,4,1) (6/11,11/20) -> (-1/2,0/1) Matrix(199,-110,360,-199) -> Matrix(1,0,0,-1) (11/20,5/9) -> (0/1,1/0) Matrix(61,-36,100,-59) -> Matrix(1,2,-2,-3) -1/1 Matrix(139,-86,160,-99) -> Matrix(7,4,-12,-7) *** -> (-2/3,-1/2) Matrix(721,-456,1140,-721) -> Matrix(-1,0,2,1) (12/19,19/30) -> (-1/1,0/1) Matrix(419,-266,660,-419) -> Matrix(3,2,-4,-3) (19/30,7/11) -> (-1/1,-1/2) Matrix(181,-116,220,-141) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(441,-286,680,-441) -> Matrix(-1,0,4,1) (11/17,13/20) -> (-1/2,0/1) Matrix(79,-52,120,-79) -> Matrix(-1,0,2,1) (13/20,2/3) -> (-1/1,0/1) Matrix(41,-28,60,-41) -> Matrix(-1,0,2,1) (2/3,7/10) -> (-1/1,0/1) Matrix(99,-70,140,-99) -> Matrix(7,6,-8,-7) (7/10,5/7) -> (-1/1,-3/4) Matrix(81,-64,100,-79) -> Matrix(1,0,0,1) Matrix(441,-374,520,-441) -> Matrix(17,12,-24,-17) (11/13,17/20) -> (-3/4,-2/3) Matrix(239,-204,280,-239) -> Matrix(19,12,-30,-19) (17/20,6/7) -> (-2/3,-3/5) 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.