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 0/1 1/2 4/7 2/3 3/4 4/5 1/1 5/4 4/3 3/2 7/4 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/1 1/1 -7/4 0/1 1/0 -5/3 -1/1 1/1 -8/5 1/0 -3/2 -1/1 -4/3 0/1 -5/4 0/1 1/1 -1/1 -1/1 1/1 -4/5 1/0 -3/4 -1/1 1/0 -2/3 -1/1 -5/8 -1/1 0/1 -3/5 -1/1 1/1 -7/12 0/1 1/0 -4/7 1/0 -1/2 -1/1 0/1 0/1 1/2 1/1 4/7 1/0 3/5 -1/1 1/1 5/8 0/1 1/1 2/3 1/1 3/4 1/1 1/0 4/5 1/0 1/1 -1/1 1/1 5/4 -1/1 0/1 4/3 0/1 3/2 1/1 8/5 1/0 5/3 -1/1 1/1 12/7 1/0 7/4 0/1 1/0 2/1 -1/1 1/1 1/0 0/1 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(15,28,8,15) (-2/1,-7/4) -> (7/4,2/1) Hyperbolic Matrix(33,56,-56,-95) (-7/4,-5/3) -> (-3/5,-7/12) Hyperbolic Matrix(17,28,20,33) (-5/3,-8/5) -> (4/5,1/1) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(17,20,28,33) (-5/4,-1/1) -> (3/5,5/8) Hyperbolic Matrix(33,28,20,17) (-1/1,-4/5) -> (8/5,5/3) Hyperbolic Matrix(31,24,40,31) (-4/5,-3/4) -> (3/4,4/5) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(33,20,28,17) (-5/8,-3/5) -> (1/1,5/4) Hyperbolic Matrix(193,112,112,65) (-7/12,-4/7) -> (12/7,7/4) Hyperbolic Matrix(15,8,28,15) (-4/7,-1/2) -> (1/2,4/7) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(95,-56,56,-33) (4/7,3/5) -> (5/3,12/7) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,0,1) Matrix(15,28,8,15) -> Matrix(1,0,0,1) Matrix(33,56,-56,-95) -> Matrix(1,0,0,1) Matrix(17,28,20,33) -> Matrix(1,0,0,1) Matrix(31,48,20,31) -> Matrix(1,2,0,1) Matrix(17,24,12,17) -> Matrix(1,0,2,1) Matrix(31,40,24,31) -> Matrix(1,0,-2,1) Matrix(17,20,28,33) -> Matrix(1,0,0,1) Matrix(33,28,20,17) -> Matrix(1,0,0,1) Matrix(31,24,40,31) -> Matrix(1,2,0,1) Matrix(17,12,24,17) -> Matrix(1,2,0,1) Matrix(31,20,48,31) -> Matrix(1,0,2,1) Matrix(33,20,28,17) -> Matrix(1,0,0,1) Matrix(193,112,112,65) -> Matrix(1,0,0,1) Matrix(15,8,28,15) -> Matrix(1,2,0,1) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(95,-56,56,-33) -> 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: ((1,2)(3,9,10,4)(5,7,6,14)(8,12)(11,13)(15,16); (1,4,12,6,16,9,13,5)(2,7,11,10,15,14,8,3); (1,3)(2,5,15,6)(4,11,9,8)(7,13)(10,16)(12,14)) ----------------------------------------------------------------------- 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 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 1/1 -3/2 -1/1 -4/3 0/1 -1/1 -1/1 1/1 -4/5 1/0 -3/4 -1/1 1/0 -2/3 -1/1 -1/2 -1/1 0/1 0/1 1/2 1/1 2/3 1/1 3/4 1/1 1/0 1/1 -1/1 1/1 5/4 -1/1 0/1 4/3 0/1 3/2 1/1 2/1 -1/1 1/1 1/0 0/1 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(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(0,-1,1,0) Matrix(17,24,12,17) -> Matrix(1,0,2,1) Matrix(7,8,-8,-9) -> Matrix(0,-1,1,0) Matrix(41,32,32,25) -> Matrix(0,-1,1,2) Matrix(17,12,24,17) -> Matrix(1,2,0,1) Matrix(7,4,12,7) -> Matrix(0,-1,1,0) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(9,-8,8,-7) -> 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 0/1 0/1 1 2 1/2 1/1 2 8 2/3 1/1 1 4 3/4 (1/1,1/0) 0 8 1/1 (-1/1,1/1).(0/1,1/0) 0 8 5/4 (-1/1,0/1) 0 8 4/3 0/1 2 2 3/2 1/1 2 8 2/1 (-1/1,1/1).(0/1,1/0) 0 4 1/0 (0/1,1/0) 0 8 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(7,-4,12,-7) (1/2,2/3) -> (1/2,2/3) Reflection Matrix(17,-12,24,-17) (2/3,3/4) -> (2/3,3/4) Reflection Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(31,-40,24,-31) (5/4,4/3) -> (5/4,4/3) 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(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,4,-1) -> Matrix(1,0,2,-1) (0/1,1/2) -> (0/1,1/1) Matrix(7,-4,12,-7) -> Matrix(0,1,1,0) (1/2,2/3) -> (-1/1,1/1) Matrix(17,-12,24,-17) -> Matrix(-1,2,0,1) (2/3,3/4) -> (1/1,1/0) Matrix(9,-8,8,-7) -> Matrix(0,-1,1,0) (-1/1,1/1).(0/1,1/0) Matrix(31,-40,24,-31) -> Matrix(-1,0,2,1) (5/4,4/3) -> (-1/1,0/1) Matrix(17,-24,12,-17) -> Matrix(1,0,2,-1) (4/3,3/2) -> (0/1,1/1) Matrix(7,-12,4,-7) -> Matrix(0,1,1,0) (3/2,2/1) -> (-1/1,1/1) Matrix(-1,4,0,1) -> Matrix(1,0,0,-1) (2/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.