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 -3/8 0/1 1/4 1/3 1/2 2/3 3/4 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -3/4 -3/1 -2/3 -2/1 -1/2 -2/1 0/1 -2/5 -2/1 -3/8 -3/2 -1/3 -1/1 -1/4 -1/1 0/1 0/1 1/4 1/1 1/3 1/1 1/2 0/1 2/1 3/5 1/1 5/8 3/2 2/3 2/1 3/4 3/1 1/1 1/0 1/0 1/0 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(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(7,4,-16,-9) (-2/3,-1/2) -> (-1/2,-2/5) Parabolic Matrix(41,16,64,25) (-2/5,-3/8) -> (5/8,2/3) Hyperbolic Matrix(39,14,64,23) (-3/8,-1/3) -> (3/5,5/8) Hyperbolic Matrix(7,2,24,7) (-1/3,-1/4) -> (1/4,1/3) Hyperbolic Matrix(1,0,8,1) (-1/4,0/1) -> (0/1,1/4) Parabolic Matrix(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,2,0,1) Matrix(7,6,8,7) -> Matrix(1,6,0,1) Matrix(17,12,24,17) -> Matrix(5,12,2,5) Matrix(7,4,-16,-9) -> Matrix(1,0,0,1) Matrix(41,16,64,25) -> Matrix(7,12,4,7) Matrix(39,14,64,23) -> Matrix(5,6,4,5) Matrix(7,2,24,7) -> Matrix(1,2,0,1) Matrix(1,0,8,1) -> Matrix(1,0,2,1) Matrix(9,-4,16,-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): 4 Degree of the the map X: 4 Degree of the the map Y: 8 Permutation triple for Y: ((1,4,5,2)(3,7,6,8); (1,2,7,3)(4,8,6,5); (2,6)(3,4)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0, lambda2 DeckMod(f) is isomorphic to Z/2Z. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0, lambda2 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 2/3 3/4 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -3/4 -3/1 -2/3 -2/1 -1/2 -2/1 0/1 0/1 0/1 1/2 0/1 2/1 2/3 2/1 3/4 3/1 1/1 1/0 1/0 1/0 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(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(7,4,12,7) (-2/3,-1/2) -> (1/2,2/3) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,2,0,1) Matrix(7,6,8,7) -> Matrix(1,6,0,1) Matrix(17,12,24,17) -> Matrix(5,12,2,5) Matrix(7,4,12,7) -> Matrix(1,0,1,1) Matrix(1,0,4,1) -> Matrix(1,0,1,1) 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): 4 ----------------------------------------------------------------------- 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 4 1/2 (0/1,2/1) 0 4 2/3 2/1 3 4 3/4 3/1 4 4 1/1 1/0 2 4 1/0 1/0 4 4 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(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(7,-6,8,-7) (3/4,1/1) -> (3/4,1/1) 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(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,4,-1) -> Matrix(1,0,1,-1) (0/1,1/2) -> (0/1,2/1) Matrix(7,-4,12,-7) -> Matrix(1,0,1,-1) (1/2,2/3) -> (0/1,2/1) Matrix(17,-12,24,-17) -> Matrix(5,-12,2,-5) (2/3,3/4) -> (2/1,3/1) Matrix(7,-6,8,-7) -> Matrix(-1,6,0,1) (3/4,1/1) -> (3/1,1/0) Matrix(-1,2,0,1) -> Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.