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 -1/1 0/1 1/2 2/3 3/4 1/1 4/3 3/2 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/1 -3/2 -1/1 -1/2 -4/3 -1/2 -1/1 -1/2 1/0 -4/5 -1/2 -3/4 -1/2 0/1 -2/3 0/1 -1/2 -1/1 0/1 0/1 -1/2 1/0 1/2 -1/1 0/1 2/3 0/1 3/4 0/1 1/0 1/1 -1/2 1/0 5/4 0/1 1/0 4/3 1/0 3/2 -1/1 1/0 2/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(7,12,4,7) (-2/1,-3/2) -> (3/2,2/1) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(7,8,-8,-9) (-4/3,-1/1) -> (-1/1,-4/5) Parabolic Matrix(41,32,32,25) (-4/5,-3/4) -> (5/4,4/3) 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 Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,0,1) Matrix(7,12,4,7) -> Matrix(3,2,-2,-1) Matrix(17,24,12,17) -> Matrix(3,2,-2,-1) Matrix(7,8,-8,-9) -> Matrix(1,0,0,1) Matrix(41,32,32,25) -> Matrix(1,0,2,1) Matrix(17,12,24,17) -> Matrix(1,0,2,1) Matrix(7,4,12,7) -> Matrix(1,0,0,1) Matrix(1,0,4,1) -> Matrix(1,0,0,1) Matrix(9,-8,8,-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): 1 Degree of the the map X: 1 Degree of the the map Y: 8 Permutation triple for Y: ((1,2)(3,4)(5,6)(7,8); (1,4,8,5)(2,6,7,3); (1,3)(2,5)(4,7)(6,8)) ----------------------------------------------------------------------- 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 1/1 4/3 3/2 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/1 -3/2 -1/1 -1/2 -4/3 -1/2 -1/1 -1/2 1/0 0/1 -1/2 1/0 1/1 -1/2 1/0 4/3 1/0 3/2 -1/1 1/0 2/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(7,12,4,7) (-2/1,-3/2) -> (3/2,2/1) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(7,8,6,7) (-4/3,-1/1) -> (1/1,4/3) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,0,1) Matrix(7,12,4,7) -> Matrix(3,2,-2,-1) Matrix(17,24,12,17) -> Matrix(3,2,-2,-1) Matrix(7,8,6,7) -> Matrix(1,1,-2,-1) Matrix(1,0,2,1) -> Matrix(1,1,-2,-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 (-1/1,0/1).(-1/2,1/0) 0 2 1/1 (-1/2,1/0) 0 4 4/3 1/0 1 2 3/2 (-1/1,1/0) 0 4 2/1 -1/1 1 2 1/0 (-1/1,0/1) 0 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(7,-8,6,-7) (1/1,4/3) -> (1/1,4/3) Reflection Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(7,-12,4,-7) (3/2,2/1) -> (3/2,2/1) Reflection Matrix(-1,4,0,1) (2/1,1/0) -> (2/1,1/0) 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(1,1,0,-1) (0/1,1/1) -> (-1/2,1/0) Matrix(7,-8,6,-7) -> Matrix(1,1,0,-1) (1/1,4/3) -> (-1/2,1/0) Matrix(17,-24,12,-17) -> Matrix(1,2,0,-1) (4/3,3/2) -> (-1/1,1/0) Matrix(7,-12,4,-7) -> Matrix(1,2,0,-1) (3/2,2/1) -> (-1/1,1/0) Matrix(-1,4,0,1) -> Matrix(-1,0,2,1) (2/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.