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 -5/3 -4/3 -1/1 -2/3 -1/2 -1/3 -1/4 -1/5 0/1 1/6 1/5 1/4 1/3 2/5 5/12 1/2 2/3 4/5 5/6 1/1 4/3 5/3 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 0/1 2/1 -9/5 -1/1 1/1 -7/4 -1/2 1/0 -5/3 0/1 -8/5 0/1 -3/2 1/2 -7/5 1/1 -4/3 1/1 -9/7 1/1 -5/4 3/2 1/0 -1/1 -1/1 1/1 -5/6 1/0 -4/5 0/1 -7/9 0/1 -3/4 1/2 1/0 -2/3 1/1 -5/8 3/2 1/0 -3/5 1/1 -7/12 3/2 1/0 -4/7 2/1 -1/2 1/0 -3/7 -1/1 1/1 -5/12 1/2 1/0 -2/5 0/1 2/1 -1/3 1/0 -2/7 -2/1 0/1 -5/18 1/0 -3/11 -3/1 -1/1 -1/4 -1/2 1/0 -1/5 -1/1 1/1 -2/11 0/1 2/1 -1/6 1/0 0/1 0/1 1/6 1/2 1/5 1/1 1/4 1/2 1/0 1/3 1/1 3/8 3/2 1/0 2/5 0/1 2/1 5/12 1/2 1/0 3/7 1/1 1/2 1/0 4/7 0/1 7/12 1/2 1/0 3/5 -1/1 1/1 2/3 0/1 5/7 1/3 1/1 3/4 1/2 1/0 4/5 0/1 5/6 1/2 1/1 1/1 7/6 1/0 6/5 0/1 2/1 5/4 3/2 1/0 4/3 1/0 11/8 -1/2 1/0 7/5 -1/1 1/1 3/2 1/0 8/5 0/1 5/3 0/1 12/7 0/1 7/4 1/2 1/0 9/5 1/1 11/6 1/0 2/1 0/1 2/1 1/0 1/2 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(11,20,-60,-109) (-2/1,-9/5) -> (-1/5,-2/11) Hyperbolic Matrix(83,148,60,107) (-9/5,-7/4) -> (11/8,7/5) Hyperbolic Matrix(37,64,-48,-83) (-7/4,-5/3) -> (-7/9,-3/4) Hyperbolic Matrix(47,76,-60,-97) (-5/3,-8/5) -> (-4/5,-7/9) Hyperbolic Matrix(13,20,24,37) (-8/5,-3/2) -> (1/2,4/7) Hyperbolic Matrix(11,16,24,35) (-3/2,-7/5) -> (3/7,1/2) Hyperbolic Matrix(47,64,-36,-49) (-7/5,-4/3) -> (-4/3,-9/7) Parabolic Matrix(85,108,48,61) (-9/7,-5/4) -> (7/4,9/5) Hyperbolic Matrix(13,16,-48,-59) (-5/4,-1/1) -> (-3/11,-1/4) Hyperbolic Matrix(23,20,-84,-73) (-1/1,-5/6) -> (-5/18,-3/11) Hyperbolic Matrix(49,40,60,49) (-5/6,-4/5) -> (4/5,5/6) Hyperbolic Matrix(23,16,-36,-25) (-3/4,-2/3) -> (-2/3,-5/8) Parabolic Matrix(13,8,60,37) (-5/8,-3/5) -> (1/5,1/4) Hyperbolic Matrix(61,36,144,85) (-3/5,-7/12) -> (5/12,3/7) Hyperbolic Matrix(97,56,168,97) (-7/12,-4/7) -> (4/7,7/12) Hyperbolic Matrix(37,20,24,13) (-4/7,-1/2) -> (3/2,8/5) Hyperbolic Matrix(35,16,24,11) (-1/2,-3/7) -> (7/5,3/2) Hyperbolic Matrix(85,36,144,61) (-3/7,-5/12) -> (7/12,3/5) Hyperbolic Matrix(49,20,120,49) (-5/12,-2/5) -> (2/5,5/12) Hyperbolic Matrix(11,4,-36,-13) (-2/5,-1/3) -> (-1/3,-2/7) Parabolic Matrix(157,44,132,37) (-2/7,-5/18) -> (7/6,6/5) Hyperbolic Matrix(35,8,48,11) (-1/4,-1/5) -> (5/7,3/4) Hyperbolic Matrix(133,24,72,13) (-2/11,-1/6) -> (11/6,2/1) Hyperbolic Matrix(1,0,12,1) (-1/6,0/1) -> (0/1,1/6) Parabolic Matrix(109,-20,60,-11) (1/6,1/5) -> (9/5,11/6) Hyperbolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(73,-28,60,-23) (3/8,2/5) -> (6/5,5/4) Hyperbolic Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(83,-64,48,-37) (3/4,4/5) -> (12/7,7/4) Hyperbolic Matrix(13,-12,12,-11) (5/6,1/1) -> (1/1,7/6) Parabolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,0,1) Matrix(11,20,-60,-109) -> Matrix(1,0,0,1) Matrix(83,148,60,107) -> Matrix(1,0,0,1) Matrix(37,64,-48,-83) -> Matrix(1,0,2,1) Matrix(47,76,-60,-97) -> Matrix(1,0,-2,1) Matrix(13,20,24,37) -> Matrix(1,0,-2,1) Matrix(11,16,24,35) -> Matrix(3,-2,2,-1) Matrix(47,64,-36,-49) -> Matrix(5,-4,4,-3) Matrix(85,108,48,61) -> Matrix(1,-2,2,-3) Matrix(13,16,-48,-59) -> Matrix(1,-2,0,1) Matrix(23,20,-84,-73) -> Matrix(1,-2,0,1) Matrix(49,40,60,49) -> Matrix(1,0,2,1) Matrix(23,16,-36,-25) -> Matrix(3,-2,2,-1) Matrix(13,8,60,37) -> Matrix(1,-2,2,-3) Matrix(61,36,144,85) -> Matrix(1,-2,2,-3) Matrix(97,56,168,97) -> Matrix(1,-2,2,-3) Matrix(37,20,24,13) -> Matrix(1,-2,0,1) Matrix(35,16,24,11) -> Matrix(1,0,0,1) Matrix(85,36,144,61) -> Matrix(1,0,0,1) Matrix(49,20,120,49) -> Matrix(1,0,0,1) Matrix(11,4,-36,-13) -> Matrix(1,-2,0,1) Matrix(157,44,132,37) -> Matrix(1,2,0,1) Matrix(35,8,48,11) -> Matrix(1,0,2,1) Matrix(133,24,72,13) -> Matrix(1,0,0,1) Matrix(1,0,12,1) -> Matrix(1,0,2,1) Matrix(109,-20,60,-11) -> Matrix(3,-2,2,-1) Matrix(13,-4,36,-11) -> Matrix(3,-2,2,-1) Matrix(73,-28,60,-23) -> Matrix(1,0,0,1) Matrix(25,-16,36,-23) -> Matrix(1,0,2,1) Matrix(83,-64,48,-37) -> Matrix(1,0,0,1) Matrix(13,-12,12,-11) -> Matrix(3,-2,2,-1) Matrix(49,-64,36,-47) -> Matrix(1,-2,0,1) Matrix(61,-100,36,-59) -> 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): 6 Degree of the the map X: 6 Degree of the the map Y: 32 Permutation triple for Y: ((1,6,22,23,7,2)(3,12,30,31,13,4)(5,18)(8,24,32,21,20,25)(9,10)(11,29)(14,17,28,27,19,26)(15,16); (1,4,16,32,17,5)(2,10,28,24,11,3)(6,21)(7,8)(9,26,20,29,31,22)(12,27)(13,14)(15,25,19,18,23,30); (1,3)(2,8,15,4,14,9)(5,19,12,11,20,6)(7,18,17,13,29,24)(10,22,21,16,30,27)(23,31)(25,26)(28,32)) ----------------------------------------------------------------------- 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 -3/2 1/2 1 6 -7/5 1/1 1 6 -4/3 1/1 2 2 -5/4 0 6 -1/1 0 6 -5/6 1/0 1 2 -4/5 0/1 1 6 -7/9 0/1 2 2 -3/4 0 6 -2/3 1/1 1 2 -5/8 0 6 -3/5 1/1 1 6 -7/12 (3/2,1/0) 0 2 -1/2 1/0 1 6 -5/12 (1/2,1/0) 0 2 -2/5 0 6 -1/3 1/0 1 2 -2/7 0 6 -3/11 0 6 -1/4 0 6 -1/5 0 6 -2/11 0 6 -1/6 1/0 1 2 0/1 0/1 1 6 1/6 1/2 1 2 1/5 1/1 1 6 1/4 0 6 1/3 1/1 1 2 3/8 0 6 2/5 0 6 5/12 (1/2,1/0) 0 2 3/7 1/1 1 6 1/2 1/0 1 6 1/0 (1/2,1/0) 0 2 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,3,0,-1) (-3/2,1/0) -> (-3/2,1/0) Reflection Matrix(11,16,24,35) (-3/2,-7/5) -> (3/7,1/2) Hyperbolic Matrix(47,65,-60,-83) (-7/5,-4/3) -> (-4/5,-7/9) Glide Reflection Matrix(37,47,-48,-61) (-4/3,-5/4) -> (-7/9,-3/4) Glide Reflection Matrix(13,16,-48,-59) (-5/4,-1/1) -> (-3/11,-1/4) Hyperbolic Matrix(13,11,-72,-61) (-1/1,-5/6) -> (-2/11,-1/6) Glide Reflection Matrix(11,9,60,49) (-5/6,-4/5) -> (1/6,1/5) Glide Reflection Matrix(23,16,-36,-25) (-3/4,-2/3) -> (-2/3,-5/8) Parabolic Matrix(13,8,60,37) (-5/8,-3/5) -> (1/5,1/4) Hyperbolic Matrix(61,36,144,85) (-3/5,-7/12) -> (5/12,3/7) Hyperbolic Matrix(13,7,-24,-13) (-7/12,-1/2) -> (-7/12,-1/2) Reflection Matrix(11,5,-24,-11) (-1/2,-5/12) -> (-1/2,-5/12) Reflection Matrix(49,20,120,49) (-5/12,-2/5) -> (2/5,5/12) Hyperbolic Matrix(11,4,-36,-13) (-2/5,-1/3) -> (-1/3,-2/7) Parabolic Matrix(25,7,-132,-37) (-2/7,-3/11) -> (-1/5,-2/11) Glide Reflection Matrix(23,5,60,13) (-1/4,-1/5) -> (3/8,2/5) Glide Reflection Matrix(1,0,12,1) (-1/6,0/1) -> (0/1,1/6) Parabolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,3,0,-1) -> Matrix(-1,1,0,1) (-3/2,1/0) -> (1/2,1/0) Matrix(11,16,24,35) -> Matrix(3,-2,2,-1) 1/1 Matrix(47,65,-60,-83) -> Matrix(1,-1,-2,1) Matrix(37,47,-48,-61) -> Matrix(1,-1,2,-3) Matrix(13,16,-48,-59) -> Matrix(1,-2,0,1) 1/0 Matrix(13,11,-72,-61) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(11,9,60,49) -> Matrix(1,1,2,1) Matrix(23,16,-36,-25) -> Matrix(3,-2,2,-1) 1/1 Matrix(13,8,60,37) -> Matrix(1,-2,2,-3) 1/1 Matrix(61,36,144,85) -> Matrix(1,-2,2,-3) 1/1 Matrix(13,7,-24,-13) -> Matrix(-1,3,0,1) (-7/12,-1/2) -> (3/2,1/0) Matrix(11,5,-24,-11) -> Matrix(-1,1,0,1) (-1/2,-5/12) -> (1/2,1/0) Matrix(49,20,120,49) -> Matrix(1,0,0,1) Matrix(11,4,-36,-13) -> Matrix(1,-2,0,1) 1/0 Matrix(25,7,-132,-37) -> Matrix(1,1,0,-1) *** -> (-1/2,1/0) Matrix(23,5,60,13) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(1,0,12,1) -> Matrix(1,0,2,1) 0/1 Matrix(13,-4,36,-11) -> Matrix(3,-2,2,-1) 1/1 Matrix(-1,1,0,1) -> Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.