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): 192 Minimal number of generators: 33 Number of equivalence classes of cusps: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -5/1 -4/1 -3/1 -13/5 -7/3 -2/1 -9/5 -3/2 -1/1 -2/3 -3/5 0/1 1/2 3/5 9/11 1/1 6/5 3/2 2/1 3/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 1/1 -5/1 1/1 1/0 -4/1 1/0 -3/1 1/0 -8/3 1/0 -13/5 -1/1 1/0 -18/7 1/0 -5/2 -3/1 -1/1 -12/5 -1/1 -7/3 -1/1 0/1 -2/1 0/1 -9/5 0/1 -7/4 1/3 1/1 -12/7 1/1 -5/3 1/1 -8/5 1/0 -11/7 1/1 1/0 -3/2 1/0 -1/1 0/1 1/0 -3/4 1/0 -11/15 -1/1 1/0 -8/11 1/0 -5/7 -1/1 -12/17 -1/1 -7/10 -1/1 -1/3 -9/13 0/1 -2/3 0/1 -7/11 0/1 1/1 -12/19 1/1 -5/8 1/1 3/1 -3/5 1/0 -7/12 -1/1 1/1 -11/19 -1/1 1/0 -4/7 1/0 -1/2 -1/1 1/1 0/1 0/1 1/2 1/3 1/1 3/5 0/1 5/8 1/5 1/3 7/11 1/3 2/3 1/2 3/4 1/1 4/5 0/1 9/11 0/1 5/6 1/3 1/1 1/1 1/1 7/6 1/1 3/1 6/5 1/0 11/9 -1/1 1/0 5/4 -1/1 1/1 9/7 0/1 4/3 0/1 7/5 0/1 1/1 10/7 0/1 3/2 1/1 8/5 1/0 5/3 1/1 1/0 12/7 1/1 7/4 1/1 3/1 2/1 1/0 3/1 0/1 4/1 1/2 5/1 1/1 11/2 5/7 1/1 6/1 1/1 1/0 -1/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,54,-10,-77) (-6/1,1/0) -> (-12/17,-7/10) Hyperbolic Matrix(17,90,10,53) (-6/1,-5/1) -> (5/3,12/7) Hyperbolic Matrix(13,60,8,37) (-5/1,-4/1) -> (8/5,5/3) Hyperbolic Matrix(11,36,-4,-13) (-4/1,-3/1) -> (-3/1,-8/3) Parabolic Matrix(73,192,-100,-263) (-8/3,-13/5) -> (-11/15,-8/11) Hyperbolic Matrix(107,276,88,227) (-13/5,-18/7) -> (6/5,11/9) Hyperbolic Matrix(61,156,52,133) (-18/7,-5/2) -> (7/6,6/5) Hyperbolic Matrix(17,42,2,5) (-5/2,-12/5) -> (6/1,1/0) Hyperbolic Matrix(71,168,-112,-265) (-12/5,-7/3) -> (-7/11,-12/19) Hyperbolic Matrix(19,42,14,31) (-7/3,-2/1) -> (4/3,7/5) Hyperbolic Matrix(29,54,22,41) (-2/1,-9/5) -> (9/7,4/3) Hyperbolic Matrix(61,108,48,85) (-9/5,-7/4) -> (5/4,9/7) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(71,120,-100,-169) (-12/7,-5/3) -> (-5/7,-12/17) Hyperbolic Matrix(37,60,8,13) (-5/3,-8/5) -> (4/1,5/1) Hyperbolic Matrix(83,132,-144,-229) (-8/5,-11/7) -> (-11/19,-4/7) Hyperbolic Matrix(43,66,-58,-89) (-11/7,-3/2) -> (-3/4,-11/15) Hyperbolic Matrix(5,6,-6,-7) (-3/2,-1/1) -> (-1/1,-3/4) Parabolic Matrix(91,66,142,103) (-8/11,-5/7) -> (7/11,2/3) Hyperbolic Matrix(155,108,188,131) (-7/10,-9/13) -> (9/11,5/6) Hyperbolic Matrix(79,54,98,67) (-9/13,-2/3) -> (4/5,9/11) Hyperbolic Matrix(131,84,92,59) (-2/3,-7/11) -> (7/5,10/7) Hyperbolic Matrix(257,162,46,29) (-12/19,-5/8) -> (11/2,6/1) Hyperbolic Matrix(59,36,-100,-61) (-5/8,-3/5) -> (-3/5,-7/12) Parabolic Matrix(227,132,184,107) (-7/12,-11/19) -> (11/9,5/4) Hyperbolic Matrix(53,30,30,17) (-4/7,-1/2) -> (7/4,2/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(31,-18,50,-29) (1/2,3/5) -> (3/5,5/8) Parabolic Matrix(161,-102,30,-19) (5/8,7/11) -> (5/1,11/2) Hyperbolic Matrix(41,-30,26,-19) (2/3,3/4) -> (3/2,8/5) Hyperbolic Matrix(55,-42,38,-29) (3/4,4/5) -> (10/7,3/2) Hyperbolic Matrix(13,-12,12,-11) (5/6,1/1) -> (1/1,7/6) Parabolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,54,-10,-77) -> Matrix(1,0,-2,1) Matrix(17,90,10,53) -> Matrix(1,0,0,1) Matrix(13,60,8,37) -> Matrix(1,0,0,1) Matrix(11,36,-4,-13) -> Matrix(1,-2,0,1) Matrix(73,192,-100,-263) -> Matrix(1,0,0,1) Matrix(107,276,88,227) -> Matrix(1,0,0,1) Matrix(61,156,52,133) -> Matrix(1,4,0,1) Matrix(17,42,2,5) -> Matrix(1,2,0,1) Matrix(71,168,-112,-265) -> Matrix(1,0,2,1) Matrix(19,42,14,31) -> Matrix(1,0,2,1) Matrix(29,54,22,41) -> Matrix(1,0,0,1) Matrix(61,108,48,85) -> Matrix(1,0,-2,1) Matrix(97,168,56,97) -> Matrix(3,-2,2,-1) Matrix(71,120,-100,-169) -> Matrix(1,0,-2,1) Matrix(37,60,8,13) -> Matrix(1,-2,2,-3) Matrix(83,132,-144,-229) -> Matrix(1,-2,0,1) Matrix(43,66,-58,-89) -> Matrix(1,-2,0,1) Matrix(5,6,-6,-7) -> Matrix(1,0,0,1) Matrix(91,66,142,103) -> Matrix(1,2,2,5) Matrix(155,108,188,131) -> Matrix(1,0,4,1) Matrix(79,54,98,67) -> Matrix(1,0,2,1) Matrix(131,84,92,59) -> Matrix(1,0,0,1) Matrix(257,162,46,29) -> Matrix(3,-4,4,-5) Matrix(59,36,-100,-61) -> Matrix(1,-2,0,1) Matrix(227,132,184,107) -> Matrix(1,0,0,1) Matrix(53,30,30,17) -> Matrix(1,2,0,1) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(31,-18,50,-29) -> Matrix(1,0,2,1) Matrix(161,-102,30,-19) -> Matrix(5,-2,8,-3) Matrix(41,-30,26,-19) -> Matrix(3,-2,2,-1) Matrix(55,-42,38,-29) -> Matrix(1,0,0,1) Matrix(13,-12,12,-11) -> Matrix(3,-2,2,-1) Matrix(7,-18,2,-5) -> 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): 7 Degree of the the map X: 7 Degree of the the map Y: 32 Permutation triple for Y: ((1,2)(3,9,25,16,15,10)(4,12,23,19,13,5)(6,14,11,27,22,18)(7,20,17,32,21,8)(24,28)(26,29)(30,31); (1,5,16,31,17,6)(2,8,3)(7,19)(9,12,29,22,21,24)(10,11)(13,28,14)(15,20,26)(23,27,30); (1,3,11,28,21,32,31,27,10,26,12,4)(2,6,18,29,20,19,30,16,25,24,13,7)(5,14,17,15)(8,22,23,9)) ----------------------------------------------------------------------- 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 -4/1 1/0 1 6 -3/1 1/0 2 4 -8/3 1/0 1 6 -13/5 (-1/1,1/0) 0 12 -5/2 0 12 -12/5 -1/1 3 2 -7/3 (-1/1,0/1) 0 12 -2/1 0/1 1 6 -9/5 0/1 4 4 -7/4 0 12 -12/7 1/1 1 2 -5/3 1/1 2 12 -8/5 1/0 1 6 -11/7 (1/1,1/0) 0 12 -3/2 1/0 1 4 -1/1 (0/1,1/0) 0 12 0/1 0/1 1 2 1/1 1/1 2 12 6/5 1/0 2 2 11/9 (-1/1,1/0) 0 12 5/4 0 12 9/7 0/1 4 4 4/3 0/1 1 6 7/5 (0/1,1/1) 0 12 3/2 1/1 1 4 5/3 (1/1,1/0) 0 12 12/7 1/1 1 2 7/4 0 12 2/1 1/0 1 6 3/1 0/1 4 4 4/1 1/2 1 6 5/1 1/1 2 12 6/1 1/1 3 2 1/0 0 12 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,30,4,17) (-4/1,1/0) -> (7/4,2/1) Glide Reflection Matrix(11,36,-4,-13) (-4/1,-3/1) -> (-3/1,-8/3) Parabolic Matrix(73,192,-46,-121) (-8/3,-13/5) -> (-8/5,-11/7) Glide Reflection Matrix(47,120,38,97) (-13/5,-5/2) -> (11/9,5/4) Glide Reflection Matrix(17,42,2,5) (-5/2,-12/5) -> (6/1,1/0) Hyperbolic Matrix(71,168,-30,-71) (-12/5,-7/3) -> (-12/5,-7/3) Reflection Matrix(19,42,14,31) (-7/3,-2/1) -> (4/3,7/5) Hyperbolic Matrix(29,54,22,41) (-2/1,-9/5) -> (9/7,4/3) Hyperbolic Matrix(61,108,48,85) (-9/5,-7/4) -> (5/4,9/7) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(71,120,-42,-71) (-12/7,-5/3) -> (-12/7,-5/3) Reflection Matrix(37,60,8,13) (-5/3,-8/5) -> (4/1,5/1) Hyperbolic Matrix(43,66,-28,-43) (-11/7,-3/2) -> (-11/7,-3/2) Reflection Matrix(5,6,-4,-5) (-3/2,-1/1) -> (-3/2,-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(11,-12,10,-11) (1/1,6/5) -> (1/1,6/5) Reflection Matrix(109,-132,90,-109) (6/5,11/9) -> (6/5,11/9) Reflection Matrix(29,-42,20,-29) (7/5,3/2) -> (7/5,3/2) Reflection Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(71,-120,42,-71) (5/3,12/7) -> (5/3,12/7) Reflection Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic Matrix(11,-60,2,-11) (5/1,6/1) -> (5/1,6/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(7,30,4,17) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(11,36,-4,-13) -> Matrix(1,-2,0,1) 1/0 Matrix(73,192,-46,-121) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(47,120,38,97) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(17,42,2,5) -> Matrix(1,2,0,1) 1/0 Matrix(71,168,-30,-71) -> Matrix(-1,0,2,1) (-12/5,-7/3) -> (-1/1,0/1) Matrix(19,42,14,31) -> Matrix(1,0,2,1) 0/1 Matrix(29,54,22,41) -> Matrix(1,0,0,1) Matrix(61,108,48,85) -> Matrix(1,0,-2,1) 0/1 Matrix(97,168,56,97) -> Matrix(3,-2,2,-1) 1/1 Matrix(71,120,-42,-71) -> Matrix(1,0,2,-1) (-12/7,-5/3) -> (0/1,1/1) Matrix(37,60,8,13) -> Matrix(1,-2,2,-3) 1/1 Matrix(43,66,-28,-43) -> Matrix(-1,2,0,1) (-11/7,-3/2) -> (1/1,1/0) Matrix(5,6,-4,-5) -> Matrix(1,0,0,-1) (-3/2,-1/1) -> (0/1,1/0) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(1,0,2,-1) -> Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Matrix(11,-12,10,-11) -> Matrix(-1,2,0,1) (1/1,6/5) -> (1/1,1/0) Matrix(109,-132,90,-109) -> Matrix(1,2,0,-1) (6/5,11/9) -> (-1/1,1/0) Matrix(29,-42,20,-29) -> Matrix(1,0,2,-1) (7/5,3/2) -> (0/1,1/1) Matrix(19,-30,12,-19) -> Matrix(-1,2,0,1) (3/2,5/3) -> (1/1,1/0) Matrix(71,-120,42,-71) -> Matrix(-1,2,0,1) (5/3,12/7) -> (1/1,1/0) Matrix(7,-18,2,-5) -> Matrix(1,0,2,1) 0/1 Matrix(11,-60,2,-11) -> Matrix(3,-2,4,-3) (5/1,6/1) -> (1/2,1/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.