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: 16 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -2/5 -1/3 0/1 1/5 1/3 1/2 1/1 7/5 3/2 2/1 3/1 7/2 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 1/4 -3/7 1/3 -2/5 0/1 1/3 1/2 -3/8 1/2 -1/3 0/1 -3/10 1/4 -2/7 1/4 1/3 -1/4 1/4 1/2 -1/5 1/3 0/1 0/1 1/2 1/5 1/2 1/4 1/2 2/7 1/2 2/3 1/1 1/3 1/1 1/2 1/2 1/0 3/5 1/1 5/8 1/2 2/3 1/1 1/0 1/1 0/1 4/3 1/3 1/2 7/5 1/2 10/7 1/2 1/1 3/2 1/2 2/1 0/1 1/2 1/1 5/2 1/2 13/5 1/2 8/3 1/2 2/3 3/1 1/1 7/2 1/2 1/0 4/1 1/2 1/1 5/1 1/1 6/1 1/1 3/2 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(41,18,66,29) (-1/2,-3/7) -> (3/5,5/8) Hyperbolic Matrix(19,8,64,27) (-3/7,-2/5) -> (2/7,1/3) Hyperbolic Matrix(21,8,76,29) (-2/5,-3/8) -> (1/4,2/7) Hyperbolic Matrix(17,6,-54,-19) (-3/8,-1/3) -> (-1/3,-3/10) Parabolic Matrix(121,36,84,25) (-3/10,-2/7) -> (10/7,3/2) Hyperbolic Matrix(65,18,18,5) (-2/7,-1/4) -> (7/2,4/1) Hyperbolic Matrix(47,10,14,3) (-1/4,-1/5) -> (3/1,7/2) Hyperbolic Matrix(43,8,16,3) (-1/5,0/1) -> (8/3,3/1) Hyperbolic Matrix(53,-8,20,-3) (0/1,1/5) -> (13/5,8/3) Hyperbolic Matrix(77,-18,30,-7) (1/5,1/4) -> (5/2,13/5) Hyperbolic Matrix(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic Matrix(51,-32,8,-5) (5/8,2/3) -> (6/1,1/0) Hyperbolic Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic Matrix(71,-98,50,-69) (4/3,7/5) -> (7/5,10/7) Parabolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(41,18,66,29) -> Matrix(1,0,-2,1) Matrix(19,8,64,27) -> Matrix(5,-2,8,-3) Matrix(21,8,76,29) -> Matrix(5,-2,8,-3) Matrix(17,6,-54,-19) -> Matrix(1,0,2,1) Matrix(121,36,84,25) -> Matrix(1,0,-2,1) Matrix(65,18,18,5) -> Matrix(1,0,-2,1) Matrix(47,10,14,3) -> Matrix(1,0,-2,1) Matrix(43,8,16,3) -> Matrix(5,-2,8,-3) Matrix(53,-8,20,-3) -> Matrix(5,-2,8,-3) Matrix(77,-18,30,-7) -> Matrix(1,0,0,1) Matrix(9,-4,16,-7) -> Matrix(1,0,0,1) Matrix(51,-32,8,-5) -> Matrix(3,-2,2,-1) Matrix(7,-6,6,-5) -> Matrix(1,0,2,1) Matrix(71,-98,50,-69) -> Matrix(5,-2,8,-3) Matrix(9,-16,4,-7) -> Matrix(1,0,0,1) Matrix(11,-50,2,-9) -> Matrix(5,-4,4,-3) 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: 16 Permutation triple for Y: ((1,6,15,11,7,2)(3,4)(5,12,16,10,9,13)(8,14); (1,4,5)(3,10,11)(6,13,14)(7,16,8); (1,2,8,13,9,3)(4,11,15,14,16,12)(5,6)(7,10)) ----------------------------------------------------------------------- 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 -1/4 0/1 1/2 1/1 7/5 3/2 2/1 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 1/4 -1/3 0/1 -1/4 1/4 1/2 -1/5 1/3 0/1 0/1 1/2 1/2 1/2 1/0 2/3 1/1 1/0 1/1 0/1 4/3 1/3 1/2 7/5 1/2 3/2 1/2 2/1 0/1 1/2 1/1 5/2 1/2 13/5 1/2 8/3 1/2 2/3 3/1 1/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(9,4,2,1) (-1/2,-1/3) -> (3/1,1/0) Hyperbolic Matrix(7,2,-32,-9) (-1/3,-1/4) -> (-1/4,-1/5) Parabolic Matrix(43,8,16,3) (-1/5,0/1) -> (8/3,3/1) Hyperbolic Matrix(5,-2,8,-3) (0/1,1/2) -> (1/2,2/3) Parabolic Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic Matrix(79,-108,30,-41) (4/3,7/5) -> (13/5,8/3) Hyperbolic Matrix(51,-74,20,-29) (7/5,3/2) -> (5/2,13/5) Hyperbolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(9,4,2,1) -> Matrix(3,-1,4,-1) Matrix(7,2,-32,-9) -> Matrix(3,-1,10,-3) Matrix(43,8,16,3) -> Matrix(5,-2,8,-3) Matrix(5,-2,8,-3) -> Matrix(1,-1,2,-1) Matrix(7,-6,6,-5) -> Matrix(1,0,2,1) Matrix(79,-108,30,-41) -> Matrix(7,-3,12,-5) Matrix(51,-74,20,-29) -> Matrix(3,-1,4,-1) Matrix(9,-16,4,-7) -> Matrix(1,0,0,1) 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 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d -1/1 0/1 2 1 0/1 (0/1,1/2) 0 6 1/2 (0/1,1/1).(1/2,1/0) 0 2 2/3 (1/1,1/0) 0 6 1/1 0/1 1 3 4/3 (1/3,1/2) 0 6 7/5 1/2 1 1 3/2 1/2 1 6 2/1 (1/2,1/0) 0 2 1/0 1/0 1 6 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(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(5,-2,8,-3) (0/1,1/2) -> (1/2,2/3) Parabolic Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic Matrix(41,-56,30,-41) (4/3,7/5) -> (4/3,7/5) Reflection Matrix(29,-42,20,-29) (7/5,3/2) -> (7/5,3/2) 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(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(-1,0,2,1) -> Matrix(1,0,4,-1) (-1/1,0/1) -> (0/1,1/2) Matrix(5,-2,8,-3) -> Matrix(1,-1,2,-1) (0/1,1/1).(1/2,1/0) Matrix(7,-6,6,-5) -> Matrix(1,0,2,1) 0/1 Matrix(41,-56,30,-41) -> Matrix(5,-2,12,-5) (4/3,7/5) -> (1/3,1/2) Matrix(29,-42,20,-29) -> Matrix(1,0,4,-1) (7/5,3/2) -> (0/1,1/2) Matrix(7,-12,4,-7) -> Matrix(-1,1,0,1) (3/2,2/1) -> (1/2,1/0) Matrix(-1,4,0,1) -> Matrix(-1,1,0,1) (2/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.