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 -1/3 0/1 1/3 1/2 3/5 1/1 9/7 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 -1/1 0/1 -1/2 0/1 1/4 -2/5 0/1 1/4 -3/8 1/4 1/3 -1/3 1/3 -2/7 0/1 1/2 -1/4 0/1 1/3 0/1 0/1 1/2 1/5 1/1 1/4 1/2 1/1 1/3 1/1 2/5 0/1 1/0 1/2 0/1 1/1 3/5 0/1 5/8 0/1 1/3 2/3 0/1 1/2 3/4 0/1 1/2 1/1 1/1 5/4 2/1 1/0 9/7 1/0 4/3 0/1 1/0 7/5 1/1 3/2 1/1 1/0 5/3 1/0 7/4 -1/1 1/0 2/1 0/1 1/0 7/3 1/1 12/5 2/1 1/0 17/7 1/0 5/2 0/1 1/0 3/1 -1/1 1/1 7/2 0/1 1/0 4/1 0/1 1/0 1/0 0/1 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(13,6,2,1) (-1/2,-2/5) -> (4/1,1/0) Hyperbolic 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(9,2,22,5) (-1/4,0/1) -> (2/5,1/2) Hyperbolic Matrix(67,-12,28,-5) (0/1,1/5) -> (7/3,12/5) Hyperbolic Matrix(37,-10,26,-7) (1/4,1/3) -> (7/5,3/2) Hyperbolic Matrix(47,-18,34,-13) (1/3,2/5) -> (4/3,7/5) Hyperbolic Matrix(31,-18,50,-29) (1/2,3/5) -> (3/5,5/8) Parabolic Matrix(37,-26,10,-7) (2/3,3/4) -> (7/2,4/1) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(103,-130,42,-53) (5/4,9/7) -> (17/7,5/2) Hyperbolic Matrix(135,-176,56,-73) (9/7,4/3) -> (12/5,17/7) Hyperbolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(13,6,2,1) -> Matrix(1,0,-4,1) Matrix(51,20,28,11) -> Matrix(1,0,-4,1) Matrix(11,4,52,19) -> Matrix(1,0,-2,1) Matrix(55,16,24,7) -> Matrix(1,0,-2,1) Matrix(43,12,68,19) -> Matrix(1,0,0,1) Matrix(9,2,22,5) -> Matrix(1,0,-2,1) Matrix(67,-12,28,-5) -> Matrix(3,-2,2,-1) Matrix(37,-10,26,-7) -> Matrix(3,-2,2,-1) Matrix(47,-18,34,-13) -> Matrix(1,0,0,1) Matrix(31,-18,50,-29) -> Matrix(1,0,2,1) Matrix(37,-26,10,-7) -> Matrix(1,0,-2,1) Matrix(9,-8,8,-7) -> Matrix(3,-2,2,-1) Matrix(103,-130,42,-53) -> Matrix(1,-2,0,1) Matrix(135,-176,56,-73) -> Matrix(1,2,0,1) Matrix(31,-50,18,-29) -> Matrix(1,-2,0,1) Matrix(13,-36,4,-11) -> 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): 3 Degree of the the map X: 3 Degree of the the map Y: 16 Permutation triple for Y: ((1,6,15,16,13,12,7,2)(3,10,14,5,9,8,11,4); (1,4,13,5)(3,9)(6,12)(7,10,15,8); (1,2,8,3)(4,11,15,12)(5,14,7,6)(9,13,16,10)) ----------------------------------------------------------------------- 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 -1/3 0/1 1/3 1/2 1/1 5/3 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 0/1 1/4 -2/5 0/1 1/4 -1/3 1/3 -2/7 0/1 1/2 -1/4 0/1 1/3 0/1 0/1 1/2 1/3 1/1 2/5 0/1 1/0 1/2 0/1 1/1 1/1 1/1 3/2 1/1 1/0 5/3 1/0 7/4 -1/1 1/0 2/1 0/1 1/0 3/1 -1/1 1/1 4/1 0/1 1/0 1/0 0/1 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(13,6,2,1) (-1/2,-2/5) -> (4/1,1/0) Hyperbolic 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(9,2,22,5) (-1/4,0/1) -> (2/5,1/2) Hyperbolic Matrix(7,-2,18,-5) (0/1,1/3) -> (1/3,2/5) Parabolic Matrix(5,-4,4,-3) (1/2,1/1) -> (1/1,3/2) Parabolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(13,6,2,1) -> Matrix(1,0,-4,1) Matrix(11,4,-36,-13) -> Matrix(4,-1,9,-2) Matrix(57,16,32,9) -> Matrix(2,-1,1,0) Matrix(9,2,22,5) -> Matrix(1,0,-2,1) Matrix(7,-2,18,-5) -> Matrix(2,-1,1,0) Matrix(5,-4,4,-3) -> Matrix(2,-1,1,0) Matrix(31,-50,18,-29) -> Matrix(1,-2,0,1) Matrix(7,-18,2,-5) -> Matrix(0,-1,1,0) 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): 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 1 0/1 (0/1,1/2).(1/3,1/1) 0 8 1/3 1/1 1 2 1/2 (0/1,1/1) 0 8 1/1 1/1 1 4 3/2 (1/1,1/0) 0 8 5/3 1/0 1 1 2/1 (-1/1,1/1).(0/1,1/0) 0 8 3/1 (-1/1,1/1).(0/1,1/0) 0 2 1/0 (0/1,1/0) 0 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(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,6,-1) (0/1,1/3) -> (0/1,1/3) Reflection Matrix(5,-2,12,-5) (1/3,1/2) -> (1/3,1/2) Reflection Matrix(5,-4,4,-3) (1/2,1/1) -> (1/1,3/2) Parabolic Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(11,-20,6,-11) (5/3,2/1) -> (5/3,2/1) Reflection Matrix(5,-12,2,-5) (2/1,3/1) -> (2/1,3/1) Reflection Matrix(-1,6,0,1) (3/1,1/0) -> (3/1,1/0) 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(1,0,6,-1) -> Matrix(2,-1,3,-2) (0/1,1/3) -> (1/3,1/1) Matrix(5,-2,12,-5) -> Matrix(1,0,2,-1) (1/3,1/2) -> (0/1,1/1) Matrix(5,-4,4,-3) -> Matrix(2,-1,1,0) 1/1 Matrix(19,-30,12,-19) -> Matrix(-1,2,0,1) (3/2,5/3) -> (1/1,1/0) Matrix(11,-20,6,-11) -> Matrix(1,0,0,-1) (5/3,2/1) -> (0/1,1/0) Matrix(5,-12,2,-5) -> Matrix(0,1,1,0) (2/1,3/1) -> (-1/1,1/1) Matrix(-1,6,0,1) -> Matrix(1,0,0,-1) (3/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.