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 -1/2 -1/3 -1/5 -1/6 0/1 1/3 1/2 2/3 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/0 -2/3 -1/1 1/1 -3/5 0/1 1/0 -1/2 -1/1 1/1 -1/3 -1/1 0/1 1/0 -1/4 -1/1 1/1 -1/5 0/1 1/0 -1/6 1/0 0/1 -1/1 1/3 -1/1 -1/2 0/1 2/5 -1/1 1/2 -1/1 -1/3 4/7 -1/1 7/12 -1/2 3/5 -1/2 0/1 2/3 -1/1 -1/3 1/1 -1/2 0/1 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(5,4,6,5) (-1/1,-2/3) -> (2/3,1/1) Hyperbolic Matrix(19,12,30,19) (-2/3,-3/5) -> (3/5,2/3) Hyperbolic Matrix(7,4,-30,-17) (-3/5,-1/2) -> (-1/4,-1/5) Hyperbolic Matrix(5,2,-18,-7) (-1/2,-1/3) -> (-1/3,-1/4) Parabolic Matrix(53,10,90,17) (-1/5,-1/6) -> (7/12,3/5) Hyperbolic Matrix(31,4,54,7) (-1/6,0/1) -> (4/7,7/12) Hyperbolic Matrix(7,-2,18,-5) (0/1,1/3) -> (1/3,2/5) Parabolic Matrix(13,-6,24,-11) (2/5,1/2) -> (1/2,4/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(5,4,6,5) -> Matrix(1,0,-2,1) Matrix(19,12,30,19) -> Matrix(1,0,-2,1) Matrix(7,4,-30,-17) -> Matrix(1,0,0,1) Matrix(5,2,-18,-7) -> Matrix(1,0,0,1) Matrix(53,10,90,17) -> Matrix(1,0,-2,1) Matrix(31,4,54,7) -> Matrix(1,2,-2,-3) Matrix(7,-2,18,-5) -> Matrix(1,0,0,1) Matrix(13,-6,24,-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: ((1,4,7,8,5,2)(3,6); (1,2,3)(6,8,7); (2,5,6)(3,7,4)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 24 Minimal number of generators: 5 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 6 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/3 1/2 2/3 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/0 -2/3 -1/1 1/1 -1/2 -1/1 1/1 -1/3 -1/1 0/1 1/0 0/1 -1/1 1/3 -1/1 -1/2 0/1 1/2 -1/1 -1/3 2/3 -1/1 -1/3 1/1 -1/2 0/1 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(5,4,6,5) (-1/1,-2/3) -> (2/3,1/1) Hyperbolic Matrix(7,4,12,7) (-2/3,-1/2) -> (1/2,2/3) Hyperbolic Matrix(5,2,12,5) (-1/2,-1/3) -> (1/3,1/2) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(5,4,6,5) -> Matrix(1,0,-2,1) Matrix(7,4,12,7) -> Matrix(0,-1,1,2) Matrix(5,2,12,5) -> Matrix(0,-1,1,2) Matrix(1,0,6,1) -> Matrix(0,-1,1,2) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 1 ----------------------------------------------------------------------- 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 This is a reflection group. CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 -1/1 1 6 1/3 (-1/1,-1/3) 0 2 1/2 (-1/1,-1/3) 0 3 2/3 (-1/1,-1/3).(-1/2,0/1) 0 2 1/1 (-1/2,0/1) 0 6 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(1,0,6,-1) (0/1,1/3) -> (0/1,1/3) Reflection Matrix(5,-2,12,-5) (1/3,1/2) -> (1/3,1/2) Reflection Matrix(7,-4,12,-7) (1/2,2/3) -> (1/2,2/3) Reflection Matrix(5,-4,6,-5) (2/3,1/1) -> (2/3,1/1) 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(1,0,6,-1) -> Matrix(2,1,-3,-2) (0/1,1/3) -> (-1/1,-1/3) Matrix(5,-2,12,-5) -> Matrix(2,1,-3,-2) (1/3,1/2) -> (-1/1,-1/3) Matrix(7,-4,12,-7) -> Matrix(2,1,-3,-2) (1/2,2/3) -> (-1/1,-1/3) Matrix(5,-4,6,-5) -> Matrix(-1,0,4,1) (2/3,1/1) -> (-1/2,0/1) Matrix(-1,2,0,1) -> Matrix(-1,0,4,1) (1/1,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.