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 -3/1 -2/1 -1/1 0/1 1/2 1/1 2/1 3/1 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -3/1 -1/1 -3/1 -2/1 -1/1 1/0 -2/1 -3/1 -1/1 -1/1 -1/1 -2/3 -1/1 -1/3 -3/5 -1/1 -1/2 0/1 -4/7 -1/1 -1/3 -1/2 0/1 0/1 -1/1 1/1 1/2 0/1 1/1 -1/1 0/1 1/0 3/2 0/1 2/1 -1/1 1/1 5/2 0/1 3/1 -1/1 1/1 7/2 0/1 4/1 -1/1 1/1 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(7,24,-12,-41) (-4/1,-3/1) -> (-3/5,-4/7) Hyperbolic Matrix(5,12,-8,-19) (-3/1,-2/1) -> (-2/3,-3/5) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(57,32,16,9) (-4/7,-1/2) -> (7/2,4/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(5,-4,4,-3) (1/2,1/1) -> (1/1,3/2) Parabolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,8,0,1) -> Matrix(1,2,0,1) Matrix(7,24,-12,-41) -> Matrix(1,2,-2,-3) Matrix(5,12,-8,-19) -> Matrix(1,2,-2,-3) Matrix(3,4,-4,-5) -> Matrix(1,2,-2,-3) Matrix(57,32,16,9) -> Matrix(1,0,2,1) Matrix(1,0,4,1) -> Matrix(1,0,0,1) Matrix(5,-4,4,-3) -> Matrix(1,0,0,1) Matrix(9,-16,4,-7) -> Matrix(1,0,0,1) 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): 1 Degree of the the map X: 1 Degree of the the map Y: 8 Permutation triple for Y: ((1,2)(3,6)(4,5)(7,8); (1,5)(2,3)(4,8)(6,7); (1,3,7,4)(2,5,8,6)) ----------------------------------------------------------------------- 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 0/1 1/1 2/1 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 -1/2 0/1 0/1 -1/1 1/1 1/2 0/1 1/1 -1/1 0/1 1/0 3/2 0/1 2/1 -1/1 1/1 3/1 -1/1 1/1 4/1 -1/1 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(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(5,-4,4,-3) (1/2,1/1) -> (1/1,3/2) Parabolic Matrix(9,-14,2,-3) (3/2,2/1) -> (4/1,1/0) Hyperbolic 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(0,-1,1,2) Matrix(1,0,4,1) -> Matrix(1,0,0,1) Matrix(5,-4,4,-3) -> Matrix(1,0,0,1) Matrix(9,-14,2,-3) -> Matrix(0,-1,1,0) 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): 1 ----------------------------------------------------------------------- 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 1 2 0/1 (-1/1,1/1) 0 4 1/1 (-1/1,1/1) 0 2 2/1 (-1/1,1/1) 0 4 3/1 (-1/1,1/1).(0/1,1/0) 0 2 1/0 1/0 1 4 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(3,-4,2,-3) (1/1,2/1) -> (1/1,2/1) 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(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Matrix(-1,0,2,1) -> Matrix(0,1,1,0) (-1/1,0/1) -> (-1/1,1/1) Matrix(1,0,2,-1) -> Matrix(0,1,1,0) (0/1,1/1) -> (-1/1,1/1) Matrix(3,-4,2,-3) -> Matrix(0,1,1,0) (1/1,2/1) -> (-1/1,1/1) 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.