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 -1/2 -1/3 0/1 1/4 1/3 1/2 1/1 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 1/0 -1/1 -1/1 0/1 -2/3 1/0 -1/2 -1/1 -2/5 -5/6 -1/3 -3/4 -2/7 -7/10 -1/4 -2/3 0/1 -1/2 1/4 0/1 1/3 -1/1 0/1 1/2 -1/1 1/1 -1/2 3/2 -1/3 5/3 -1/3 0/1 7/4 0/1 2/1 -1/2 1/0 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(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(7,4,-16,-9) (-2/3,-1/2) -> (-1/2,-2/5) Parabolic Matrix(11,4,-36,-13) (-2/5,-1/3) -> (-1/3,-2/7) Parabolic Matrix(57,16,32,9) (-2/7,-1/4) -> (7/4,2/1) Hyperbolic Matrix(1,0,8,1) (-1/4,0/1) -> (0/1,1/4) Parabolic Matrix(41,-12,24,-7) (1/4,1/3) -> (5/3,7/4) Hyperbolic Matrix(19,-8,12,-5) (1/3,1/2) -> (3/2,5/3) Hyperbolic Matrix(5,-4,4,-3) (1/2,1/1) -> (1/1,3/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-2,1) Matrix(3,4,-4,-5) -> Matrix(1,0,0,1) Matrix(7,4,-16,-9) -> Matrix(5,6,-6,-7) Matrix(11,4,-36,-13) -> Matrix(23,18,-32,-25) Matrix(57,16,32,9) -> Matrix(3,2,4,3) Matrix(1,0,8,1) -> Matrix(3,2,-8,-5) Matrix(41,-12,24,-7) -> Matrix(1,0,-2,1) Matrix(19,-8,12,-5) -> Matrix(1,0,-2,1) Matrix(5,-4,4,-3) -> Matrix(3,2,-8,-5) 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: ((1,5,6,2)(3,7,8,4); (1,4)(2,3)(5,8)(6,7); (1,3)(2,7)(4,5)(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 -1/1 -1/2 -1/3 0/1 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 1/0 -1/1 -1/1 0/1 -2/3 1/0 -1/2 -1/1 -1/3 -3/4 -1/4 -2/3 0/1 -1/2 1/1 -1/2 2/1 -1/2 1/0 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(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(3,2,-14,-9) (-2/3,-1/2) -> (-1/4,0/1) Hyperbolic Matrix(5,2,-18,-7) (-1/2,-1/3) -> (-1/3,-1/4) Parabolic Matrix(3,-2,2,-1) (0/1,1/1) -> (1/1,2/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-2,1) Matrix(3,4,-4,-5) -> Matrix(1,0,0,1) Matrix(3,2,-14,-9) -> Matrix(1,3,-2,-5) Matrix(5,2,-18,-7) -> Matrix(11,9,-16,-13) Matrix(3,-2,2,-1) -> Matrix(1,1,-4,-3) 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 -1/1 (-1/1,0/1) 0 4 -1/2 -1/1 3 2 -1/3 -3/4 2 4 0/1 -1/2 1 4 1/1 -1/2 2 4 1/0 0/1 1 2 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(3,2,-4,-3) (-1/1,-1/2) -> (-1/1,-1/2) Reflection Matrix(5,2,-12,-5) (-1/2,-1/3) -> (-1/2,-1/3) Reflection Matrix(-1,0,6,1) (-1/3,0/1) -> (-1/3,0/1) Reflection Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) 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(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(3,2,-4,-3) -> Matrix(-1,0,2,1) (-1/1,-1/2) -> (-1/1,0/1) Matrix(5,2,-12,-5) -> Matrix(7,6,-8,-7) (-1/2,-1/3) -> (-1/1,-3/4) Matrix(-1,0,6,1) -> Matrix(5,3,-8,-5) (-1/3,0/1) -> (-3/4,-1/2) Matrix(1,0,2,-1) -> Matrix(1,1,0,-1) (0/1,1/1) -> (-1/2,1/0) 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.