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: 8 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/1 -2/1 0/1 1/1 2/1 8/3 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -1/1 0/1 1/0 -3/1 1/0 -2/1 -1/1 0/1 -5/3 1/0 -8/5 -1/1 -3/2 -1/2 -4/3 -1/1 -1/2 0/1 -1/1 -1/2 0/1 0/1 1/1 1/2 4/3 0/1 1/2 1/1 3/2 1/2 2/1 0/1 1/1 5/2 1/2 8/3 1/1 3/1 1/0 4/1 0/1 1/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,16,2,11) (-4/1,1/0) -> (4/3,3/2) Hyperbolic Matrix(5,16,4,13) (-4/1,-3/1) -> (1/1,4/3) Hyperbolic Matrix(7,16,-4,-9) (-3/1,-2/1) -> (-2/1,-5/3) Parabolic Matrix(39,64,14,23) (-5/3,-8/5) -> (8/3,3/1) Hyperbolic Matrix(41,64,16,25) (-8/5,-3/2) -> (5/2,8/3) Hyperbolic Matrix(11,16,2,3) (-3/2,-4/3) -> (4/1,1/0) Hyperbolic Matrix(13,16,4,5) (-4/3,-1/1) -> (3/1,4/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(3,16,2,11) -> Matrix(1,0,2,1) Matrix(5,16,4,13) -> Matrix(1,0,2,1) Matrix(7,16,-4,-9) -> Matrix(1,0,0,1) Matrix(39,64,14,23) -> Matrix(1,2,0,1) Matrix(41,64,16,25) -> Matrix(3,2,4,3) Matrix(11,16,2,3) -> Matrix(1,0,2,1) Matrix(13,16,4,5) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(9,-16,4,-7) -> 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): 2 Degree of the the map X: 2 Degree of the the map Y: 8 Permutation triple for Y: ((2,6)(3,4); (1,4,7,6,8,3,5,2); (1,2,7,4,8,6,5,3)) ----------------------------------------------------------------------- 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): 12 Minimal number of generators: 3 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: 4 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/2 2/1 0/1 1/1 3/1 1/0 4/1 0/1 1/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(5,-8,2,-3) (1/1,2/1) -> (2/1,3/1) Parabolic Matrix(5,-16,1,-3) (3/1,4/1) -> (4/1,1/0) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,2,1) Matrix(5,-8,2,-3) -> Matrix(1,-1,2,-1) Matrix(5,-16,1,-3) -> Matrix(1,0,0,1) 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 0/1 2 1 2/1 (0/1,1/1).(1/2,1/0) 0 4 4/1 (1/2,1/0) 0 2 1/0 1/0 1 8 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(3,-8,1,-3) (2/1,4/1) -> (2/1,4/1) 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(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,1,-1) -> Matrix(1,0,2,-1) (0/1,2/1) -> (0/1,1/1) Matrix(3,-8,1,-3) -> Matrix(-1,1,0,1) (2/1,4/1) -> (1/2,1/0) Matrix(-1,8,0,1) -> Matrix(-1,1,0,1) (4/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.