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): 48 Minimal number of generators: 9 Number of equivalence classes of cusps: 10 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -2/1 -3/2 -6/5 0/1 1/1 3/2 2/1 5/2 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 0/1 -5/2 0/1 1/0 -2/1 -1/1 0/1 1/0 -3/2 -1/1 1/1 -4/3 -1/1 0/1 1/0 -5/4 0/1 1/0 -6/5 -1/1 1/1 -1/1 1/0 0/1 -1/1 1/1 -1/2 3/2 -1/1 -1/3 5/3 -1/2 2/1 -1/1 -1/2 0/1 7/3 -1/2 12/5 -1/1 -1/3 5/2 -1/2 0/1 3/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,6,0,1) (-3/1,1/0) -> (3/1,1/0) Parabolic Matrix(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(13,30,-10,-23) (-5/2,-2/1) -> (-4/3,-5/4) Hyperbolic Matrix(11,18,-8,-13) (-2/1,-3/2) -> (-3/2,-4/3) Parabolic Matrix(73,90,30,37) (-5/4,-6/5) -> (12/5,5/2) Hyperbolic Matrix(47,54,20,23) (-6/5,-1/1) -> (7/3,12/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,0,0,1) Matrix(11,30,4,11) -> Matrix(1,0,-2,1) Matrix(13,30,-10,-23) -> Matrix(1,0,0,1) Matrix(11,18,-8,-13) -> Matrix(1,0,0,1) Matrix(73,90,30,37) -> Matrix(1,0,-2,1) Matrix(47,54,20,23) -> Matrix(1,0,-2,1) Matrix(1,0,2,1) -> Matrix(1,2,-2,-3) Matrix(13,-18,8,-11) -> Matrix(1,0,0,1) Matrix(13,-24,6,-11) -> 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): 1 Degree of the the map X: 1 Degree of the the map Y: 8 Permutation triple for Y: ((2,5,6)(3,7,4); (1,4,7,8,5,2)(3,6); (1,2,3)(6,8,7)) ----------------------------------------------------------------------- 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 1 1 1/1 -1/2 1 6 3/2 0 2 5/3 -1/2 1 6 2/1 0 3 7/3 -1/2 1 6 12/5 (-1/2,0/1) 0 1 5/2 (-1/2,0/1) 0 6 3/1 0/1 1 2 1/0 (-1/1,0/1) 0 6 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(11,-30,4,-11) (5/2,3/1) -> (5/2,3/1) Reflection Matrix(-1,6,0,1) (3/1,1/0) -> (3/1,1/0) 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(1,0,2,-1) -> Matrix(3,2,-4,-3) (0/1,1/1) -> (-1/1,-1/2) Matrix(13,-18,8,-11) -> Matrix(1,0,0,1) Matrix(13,-24,6,-11) -> Matrix(1,0,0,1) Matrix(71,-168,30,-71) -> Matrix(-1,0,4,1) (7/3,12/5) -> (-1/2,0/1) Matrix(49,-120,20,-49) -> Matrix(-1,0,4,1) (12/5,5/2) -> (-1/2,0/1) Matrix(11,-30,4,-11) -> Matrix(-1,0,4,1) (5/2,3/1) -> (-1/2,0/1) Matrix(-1,6,0,1) -> Matrix(-1,0,2,1) (3/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.