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 0/1 1/2 3/4 1/1 3/2 2/1 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 0/1 -2/1 1/0 -3/2 -1/1 1/1 -1/1 1/0 -3/4 -1/1 -2/3 -1/2 -3/5 0/1 -1/2 -1/1 0/1 0/1 0/1 1/2 0/1 1/1 2/3 1/2 3/4 1/1 1/1 1/0 3/2 -1/1 1/1 2/1 1/0 5/2 -1/1 0/1 3/1 0/1 1/0 0/1 1/0 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(5,12,-8,-19) (-3/1,-2/1) -> (-2/3,-3/5) Hyperbolic Matrix(7,12,4,7) (-2/1,-3/2) -> (3/2,2/1) Hyperbolic Matrix(5,6,4,5) (-3/2,-1/1) -> (1/1,3/2) Hyperbolic Matrix(7,6,8,7) (-1/1,-3/4) -> (3/4,1/1) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(31,18,12,7) (-3/5,-1/2) -> (5/2,3/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(19,-12,8,-5) (1/2,2/3) -> (2/1,5/2) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,0,0,1) Matrix(5,12,-8,-19) -> Matrix(1,0,-2,1) Matrix(7,12,4,7) -> Matrix(1,0,0,1) Matrix(5,6,4,5) -> Matrix(1,0,0,1) Matrix(7,6,8,7) -> Matrix(1,2,0,1) Matrix(17,12,24,17) -> Matrix(3,2,4,3) Matrix(31,18,12,7) -> Matrix(1,0,0,1) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(19,-12,8,-5) -> 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): 2 Degree of the the map X: 2 Degree of the the map Y: 8 Permutation triple for Y: ((1,2)(3,5,4,6,8,7); (1,5,3,2,8,6)(4,7); (1,3,4)(2,6,7)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- 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 1/2 3/4 1/1 3/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -1/2 -1/1 0/1 0/1 0/1 1/2 0/1 1/1 2/3 1/2 3/4 1/1 1/1 1/0 3/2 -1/1 1/1 2/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,3,0,1) (-1/1,1/0) -> (2/1,1/0) Parabolic Matrix(5,3,8,5) (-1/1,-1/2) -> (1/2,2/3) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(13,-9,16,-11) (2/3,3/4) -> (3/4,1/1) Parabolic Matrix(7,-9,4,-5) (1/1,3/2) -> (3/2,2/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,3,0,1) -> Matrix(1,0,0,1) Matrix(5,3,8,5) -> Matrix(1,0,2,1) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(13,-9,16,-11) -> Matrix(3,-2,2,-1) Matrix(7,-9,4,-5) -> 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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 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 1 2 1/2 (0/1,1/1) 0 6 3/4 1/1 4 2 1/1 1/0 1 6 3/2 (0/1,1/0) 0 2 1/0 (0/1,1/0) 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(5,-3,8,-5) (1/2,3/4) -> (1/2,3/4) Reflection Matrix(7,-6,8,-7) (3/4,1/1) -> (3/4,1/1) Reflection Matrix(5,-6,4,-5) (1/1,3/2) -> (1/1,3/2) Reflection Matrix(-1,3,0,1) (3/2,1/0) -> (3/2,1/0) 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(1,0,4,-1) -> Matrix(1,0,2,-1) (0/1,1/2) -> (0/1,1/1) Matrix(5,-3,8,-5) -> Matrix(1,0,2,-1) (1/2,3/4) -> (0/1,1/1) Matrix(7,-6,8,-7) -> Matrix(-1,2,0,1) (3/4,1/1) -> (1/1,1/0) Matrix(5,-6,4,-5) -> Matrix(1,0,0,-1) (1/1,3/2) -> (0/1,1/0) Matrix(-1,3,0,1) -> Matrix(1,0,0,-1) (3/2,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.