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): 96 Minimal number of generators: 17 Number of equivalence classes of cusps: 14 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -1/2 -1/3 0/1 1/4 1/3 1/2 1/1 3/2 5/3 2/1 5/2 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -1/2 -3/1 1/0 -2/1 -1/2 -1/1 0/1 -2/3 1/2 -1/2 0/1 1/2 -2/5 1/2 -3/8 1/4 -1/3 1/2 -2/7 1/2 -1/4 1/0 -1/5 0/1 0/1 1/2 1/5 1/2 1/4 3/4 1/3 1/1 1/2 1/1 1/0 3/5 1/1 5/8 5/4 2/3 3/2 1/1 1/0 4/3 -1/2 3/2 0/1 1/0 8/5 -1/2 5/3 0/1 7/4 1/4 2/1 1/2 7/3 1/2 5/2 1/2 1/1 3/1 1/1 7/2 1/1 3/2 11/3 3/2 4/1 3/2 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,8,0,1) (-4/1,1/0) -> (4/1,1/0) Parabolic Matrix(5,16,4,13) (-4/1,-3/1) -> (1/1,4/3) Hyperbolic Matrix(3,8,4,11) (-3/1,-2/1) -> (2/3,1/1) Hyperbolic 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(51,20,28,11) (-2/5,-3/8) -> (7/4,2/1) Hyperbolic Matrix(11,4,52,19) (-3/8,-1/3) -> (1/5,1/4) Hyperbolic Matrix(55,16,24,7) (-1/3,-2/7) -> (2/1,7/3) Hyperbolic Matrix(43,12,68,19) (-2/7,-1/4) -> (5/8,2/3) Hyperbolic Matrix(55,12,32,7) (-1/4,-1/5) -> (5/3,7/4) Hyperbolic Matrix(45,8,28,5) (-1/5,0/1) -> (8/5,5/3) Hyperbolic Matrix(31,-4,8,-1) (0/1,1/5) -> (11/3,4/1) Hyperbolic Matrix(27,-8,44,-13) (1/4,1/3) -> (3/5,5/8) Hyperbolic Matrix(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(43,-104,12,-29) (7/3,5/2) -> (7/2,11/3) Hyperbolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,8,0,1) -> Matrix(1,2,0,1) Matrix(5,16,4,13) -> Matrix(1,0,0,1) Matrix(3,8,4,11) -> Matrix(1,2,0,1) Matrix(3,4,-4,-5) -> Matrix(1,0,4,1) Matrix(7,4,-16,-9) -> Matrix(1,0,0,1) Matrix(51,20,28,11) -> Matrix(1,0,0,1) Matrix(11,4,52,19) -> Matrix(5,-2,8,-3) Matrix(55,16,24,7) -> Matrix(1,0,0,1) Matrix(43,12,68,19) -> Matrix(5,-4,4,-3) Matrix(55,12,32,7) -> Matrix(1,0,4,1) Matrix(45,8,28,5) -> Matrix(1,0,-4,1) Matrix(31,-4,8,-1) -> Matrix(5,-4,4,-3) Matrix(27,-8,44,-13) -> Matrix(9,-8,8,-7) Matrix(9,-4,16,-7) -> Matrix(1,0,0,1) Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) Matrix(43,-104,12,-29) -> Matrix(5,-4,4,-3) Matrix(13,-36,4,-11) -> Matrix(5,-4,4,-3) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 3 Degree of the the map X: 6 Degree of the the map Y: 16 Permutation triple for Y: ((1,7,16,11,10,14,8,2)(3,9,13,5,4,12,15,6); (1,5,10,6)(2,3)(4,11)(7,15)(8,12,16,9)(13,14); (1,3,10,4)(2,9)(5,14)(6,7)(8,13,16,15)(11,12)) ----------------------------------------------------------------------- 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): 48 Minimal number of generators: 9 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: 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 -1/2 -1/1 0/1 -2/3 1/2 -1/2 0/1 1/2 -2/5 1/2 -1/3 1/2 -2/7 1/2 -1/4 1/0 0/1 1/2 1/4 3/4 1/3 1/1 1/2 1/1 1/0 1/1 1/0 3/2 0/1 1/0 5/3 0/1 7/4 1/4 2/1 1/2 1/0 1/0 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(1,1,0,1) Matrix(3,4,-4,-5) -> Matrix(1,0,4,1) Matrix(7,4,-16,-9) -> Matrix(1,0,0,1) Matrix(11,4,-36,-13) -> Matrix(3,-1,4,-1) Matrix(57,16,32,9) -> Matrix(1,-1,4,-3) Matrix(1,0,8,1) -> Matrix(3,-1,4,-1) Matrix(41,-12,24,-7) -> Matrix(1,-1,8,-7) Matrix(19,-8,12,-5) -> Matrix(1,-1,0,1) Matrix(5,-4,4,-3) -> Matrix(1,-1,0,1) 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): 3 ----------------------------------------------------------------------- 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 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d -1/1 0/1 2 2 -1/2 (0/1,1/2) 0 8 -1/3 1/2 1 4 -1/4 1/0 2 8 0/1 1/2 1 8 1/4 3/4 2 8 1/3 1/1 4 2 1/2 (1/1,1/0) 0 8 1/1 1/0 1 4 1/0 1/0 2 8 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(7,2,-24,-7) (-1/3,-1/4) -> (-1/3,-1/4) Reflection Matrix(1,0,8,1) (-1/4,0/1) -> (0/1,1/4) Parabolic Matrix(7,-2,24,-7) (1/4,1/3) -> (1/4,1/3) Reflection Matrix(5,-2,12,-5) (1/3,1/2) -> (1/3,1/2) Reflection Matrix(3,-2,4,-3) (1/2,1/1) -> (1/2,1/1) 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(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(3,2,-4,-3) -> Matrix(1,0,4,-1) (-1/1,-1/2) -> (0/1,1/2) Matrix(5,2,-12,-5) -> Matrix(1,0,4,-1) (-1/2,-1/3) -> (0/1,1/2) Matrix(7,2,-24,-7) -> Matrix(-1,1,0,1) (-1/3,-1/4) -> (1/2,1/0) Matrix(1,0,8,1) -> Matrix(3,-1,4,-1) 1/2 Matrix(7,-2,24,-7) -> Matrix(7,-6,8,-7) (1/4,1/3) -> (3/4,1/1) Matrix(5,-2,12,-5) -> Matrix(-1,2,0,1) (1/3,1/2) -> (1/1,1/0) Matrix(3,-2,4,-3) -> Matrix(-1,2,0,1) (1/2,1/1) -> (1/1,1/0) Matrix(-1,2,0,1) -> Matrix(-1,1,0,1) (1/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.