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): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 20 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -3/8 -1/3 0/1 1/4 3/7 1/2 2/3 1/1 3/2 19/11 2/1 7/3 5/2 11/4 3/1 7/2 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/2 -3/7 1/2 -5/12 0/1 1/0 -2/5 0/1 1/3 -3/8 1/3 1/1 -4/11 0/1 1/3 -1/3 1/2 -1/4 1/2 1/1 -2/9 0/1 1/1 -1/5 1/2 -1/6 1/1 0/1 0/1 1/1 1/4 1/1 2/7 1/1 2/1 1/3 1/0 3/8 0/1 1/1 5/13 1/0 2/5 0/1 1/1 3/7 1/1 4/9 1/1 2/1 1/2 1/1 1/0 3/5 1/0 2/3 1/0 5/7 1/0 3/4 0/1 1/0 7/9 1/2 4/5 0/1 1/1 1/1 1/0 3/2 -1/1 1/1 5/3 1/0 12/7 -2/1 -1/1 19/11 -1/1 7/4 -1/1 0/1 2/1 -1/1 0/1 7/3 0/1 12/5 0/1 1/5 5/2 0/1 1/2 13/5 1/2 21/8 2/3 1/1 29/11 1/1 8/3 0/1 1/1 11/4 1/1 3/1 1/0 10/3 -1/1 0/1 7/2 -1/1 0/1 4/1 0/1 9/2 0/1 1/1 5/1 1/0 1/0 0/1 1/0 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(17,8,2,1) (-1/2,-3/7) -> (5/1,1/0) Hyperbolic Matrix(105,44,136,57) (-3/7,-5/12) -> (3/4,7/9) Hyperbolic Matrix(169,70,70,29) (-5/12,-2/5) -> (12/5,5/2) Hyperbolic Matrix(47,18,-128,-49) (-2/5,-3/8) -> (-3/8,-4/11) Parabolic Matrix(61,22,158,57) (-4/11,-1/3) -> (5/13,2/5) Hyperbolic Matrix(15,4,26,7) (-1/3,-1/4) -> (1/2,3/5) Hyperbolic Matrix(25,6,54,13) (-1/4,-2/9) -> (4/9,1/2) Hyperbolic Matrix(77,16,24,5) (-2/9,-1/5) -> (3/1,10/3) Hyperbolic Matrix(73,14,26,5) (-1/5,-1/6) -> (11/4,3/1) Hyperbolic Matrix(59,8,22,3) (-1/6,0/1) -> (8/3,11/4) Hyperbolic Matrix(9,-2,32,-7) (0/1,1/4) -> (1/4,2/7) Parabolic Matrix(61,-18,78,-23) (2/7,1/3) -> (7/9,4/5) Hyperbolic Matrix(67,-24,14,-5) (1/3,3/8) -> (9/2,5/1) Hyperbolic Matrix(377,-144,144,-55) (3/8,5/13) -> (13/5,21/8) Hyperbolic Matrix(43,-18,98,-41) (2/5,3/7) -> (3/7,4/9) Parabolic Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(87,-64,34,-25) (5/7,3/4) -> (5/2,13/5) Hyperbolic Matrix(37,-32,22,-19) (4/5,1/1) -> (5/3,12/7) Hyperbolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(291,-500,110,-189) (12/7,19/11) -> (29/11,8/3) Hyperbolic Matrix(347,-602,132,-229) (19/11,7/4) -> (21/8,29/11) Hyperbolic Matrix(47,-84,14,-25) (7/4,2/1) -> (10/3,7/2) Hyperbolic Matrix(43,-98,18,-41) (2/1,7/3) -> (7/3,12/5) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,2,1) Matrix(17,8,2,1) -> Matrix(1,0,-2,1) Matrix(105,44,136,57) -> Matrix(1,0,0,1) Matrix(169,70,70,29) -> Matrix(1,0,2,1) Matrix(47,18,-128,-49) -> Matrix(1,0,0,1) Matrix(61,22,158,57) -> Matrix(1,0,-2,1) Matrix(15,4,26,7) -> Matrix(3,-2,2,-1) Matrix(25,6,54,13) -> Matrix(3,-2,2,-1) Matrix(77,16,24,5) -> Matrix(1,0,-2,1) Matrix(73,14,26,5) -> Matrix(3,-2,2,-1) Matrix(59,8,22,3) -> Matrix(1,0,0,1) Matrix(9,-2,32,-7) -> Matrix(3,-2,2,-1) Matrix(61,-18,78,-23) -> Matrix(1,-2,2,-3) Matrix(67,-24,14,-5) -> Matrix(1,0,0,1) Matrix(377,-144,144,-55) -> Matrix(1,-2,2,-3) Matrix(43,-18,98,-41) -> Matrix(3,-2,2,-1) Matrix(25,-16,36,-23) -> Matrix(1,-2,0,1) Matrix(87,-64,34,-25) -> Matrix(1,0,2,1) Matrix(37,-32,22,-19) -> Matrix(1,-2,0,1) Matrix(13,-18,8,-11) -> Matrix(1,0,0,1) Matrix(291,-500,110,-189) -> Matrix(1,2,0,1) Matrix(347,-602,132,-229) -> Matrix(3,2,4,3) Matrix(47,-84,14,-25) -> Matrix(1,0,0,1) Matrix(43,-98,18,-41) -> Matrix(1,0,6,1) Matrix(17,-64,4,-15) -> 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): 5 Degree of the the map X: 5 Degree of the the map Y: 24 Permutation triple for Y: ((1,6,18,14,24,23,11,19,7,2)(3,12,22,8,21,17,16,20,13,4)(5,15)(9,10); (1,4,13,14,5)(3,10,20,19,11)(6,17,9,22,18)(7,15,23,21,8); (1,2,8,9,3)(5,7,20,16,6)(10,17,23,24,13)(11,15,14,22,12)) ----------------------------------------------------------------------- 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 2 2 0/1 (0/1,1/1) 0 20 1/4 1/1 4 4 2/7 (1/1,2/1) 0 20 1/3 1/0 2 10 3/8 (0/1,1/1) 0 20 2/5 (0/1,1/1) 0 20 3/7 1/1 2 2 1/2 (1/1,1/0) 0 20 2/3 1/0 2 4 3/4 (0/1,1/0) 0 20 4/5 (0/1,1/1) 0 20 1/1 1/0 2 10 3/2 0 4 5/3 1/0 2 10 12/7 (-2/1,-1/1) 0 20 19/11 -1/1 4 2 7/4 (-1/1,0/1) 0 20 2/1 (-1/1,0/1) 0 20 7/3 0/1 6 2 5/2 (0/1,1/2) 0 20 8/3 (0/1,1/1) 0 20 3/1 1/0 2 10 7/2 (-1/1,0/1) 0 20 4/1 0/1 2 4 1/0 (0/1,1/0) 0 20 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(9,-2,32,-7) (0/1,1/4) -> (1/4,2/7) Parabolic Matrix(45,-14,16,-5) (2/7,1/3) -> (8/3,3/1) Glide Reflection Matrix(45,-16,14,-5) (1/3,3/8) -> (3/1,7/2) Glide Reflection Matrix(51,-20,28,-11) (3/8,2/5) -> (7/4,2/1) Glide Reflection Matrix(29,-12,70,-29) (2/5,3/7) -> (2/5,3/7) Reflection Matrix(13,-6,28,-13) (3/7,1/2) -> (3/7,1/2) Reflection Matrix(7,-4,12,-7) (1/2,2/3) -> (1/2,2/3) Reflection Matrix(17,-12,24,-17) (2/3,3/4) -> (2/3,3/4) Reflection Matrix(57,-44,22,-17) (3/4,4/5) -> (5/2,8/3) Glide Reflection Matrix(37,-32,22,-19) (4/5,1/1) -> (5/3,12/7) Hyperbolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(265,-456,154,-265) (12/7,19/11) -> (12/7,19/11) Reflection Matrix(153,-266,88,-153) (19/11,7/4) -> (19/11,7/4) Reflection Matrix(13,-28,6,-13) (2/1,7/3) -> (2/1,7/3) Reflection Matrix(29,-70,12,-29) (7/3,5/2) -> (7/3,5/2) Reflection Matrix(15,-56,4,-15) (7/2,4/1) -> (7/2,4/1) Reflection Matrix(-1,8,0,1) (4/1,1/0) -> (4/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(-1,0,2,1) -> Matrix(1,0,2,-1) (-1/1,0/1) -> (0/1,1/1) Matrix(9,-2,32,-7) -> Matrix(3,-2,2,-1) 1/1 Matrix(45,-14,16,-5) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(45,-16,14,-5) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(51,-20,28,-11) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(29,-12,70,-29) -> Matrix(1,0,2,-1) (2/5,3/7) -> (0/1,1/1) Matrix(13,-6,28,-13) -> Matrix(-1,2,0,1) (3/7,1/2) -> (1/1,1/0) Matrix(7,-4,12,-7) -> Matrix(-1,2,0,1) (1/2,2/3) -> (1/1,1/0) Matrix(17,-12,24,-17) -> Matrix(1,0,0,-1) (2/3,3/4) -> (0/1,1/0) Matrix(57,-44,22,-17) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(37,-32,22,-19) -> Matrix(1,-2,0,1) 1/0 Matrix(13,-18,8,-11) -> Matrix(1,0,0,1) Matrix(265,-456,154,-265) -> Matrix(3,4,-2,-3) (12/7,19/11) -> (-2/1,-1/1) Matrix(153,-266,88,-153) -> Matrix(-1,0,2,1) (19/11,7/4) -> (-1/1,0/1) Matrix(13,-28,6,-13) -> Matrix(-1,0,2,1) (2/1,7/3) -> (-1/1,0/1) Matrix(29,-70,12,-29) -> Matrix(1,0,4,-1) (7/3,5/2) -> (0/1,1/2) Matrix(15,-56,4,-15) -> Matrix(-1,0,2,1) (7/2,4/1) -> (-1/1,0/1) Matrix(-1,8,0,1) -> Matrix(1,0,0,-1) (4/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.