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 -2/3 -1/2 0/1 1/3 2/5 1/2 2/3 5/6 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/1 -2/3 0/1 1/1 1/0 -3/5 1/1 -1/2 -1/1 1/1 -3/7 1/1 -5/12 1/0 -2/5 0/1 1/0 -1/3 -1/1 1/1 0/1 0/1 1/0 1/3 -1/1 1/1 2/5 0/1 1/0 1/2 -1/1 1/1 2/3 -1/1 0/1 1/0 3/4 -1/1 1/1 4/5 0/1 1/0 5/6 1/0 1/1 -1/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(11,8,-18,-13) (-1/1,-2/3) -> (-2/3,-3/5) Parabolic Matrix(11,6,-24,-13) (-3/5,-1/2) -> (-1/2,-3/7) Parabolic Matrix(47,20,54,23) (-3/7,-5/12) -> (5/6,1/1) Hyperbolic Matrix(73,30,90,37) (-5/12,-2/5) -> (4/5,5/6) Hyperbolic Matrix(11,4,30,11) (-2/5,-1/3) -> (1/3,2/5) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(23,-10,30,-13) (2/5,1/2) -> (3/4,4/5) Hyperbolic Matrix(13,-8,18,-11) (1/2,2/3) -> (2/3,3/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(11,8,-18,-13) -> Matrix(1,0,0,1) Matrix(11,6,-24,-13) -> Matrix(1,0,0,1) Matrix(47,20,54,23) -> Matrix(1,-2,0,1) Matrix(73,30,90,37) -> Matrix(1,0,0,1) Matrix(11,4,30,11) -> Matrix(1,0,0,1) Matrix(1,0,6,1) -> Matrix(1,0,0,1) Matrix(23,-10,30,-13) -> Matrix(1,0,0,1) Matrix(13,-8,18,-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: ((1,4,2)(5,7,8); (1,2,6,8,7,3)(4,5); (2,5,6)(3,7,4)) ----------------------------------------------------------------------- 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 0/1 1/3 1/2 2/3 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/1 -2/3 0/1 1/1 1/0 -1/2 -1/1 1/1 -1/3 -1/1 1/1 0/1 0/1 1/0 1/3 -1/1 1/1 1/2 -1/1 1/1 2/3 -1/1 0/1 1/0 1/1 -1/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(5,4,6,5) (-1/1,-2/3) -> (2/3,1/1) Hyperbolic Matrix(7,4,12,7) (-2/3,-1/2) -> (1/2,2/3) Hyperbolic Matrix(5,2,12,5) (-1/2,-1/3) -> (1/3,1/2) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(5,4,6,5) -> Matrix(0,-1,1,0) Matrix(7,4,12,7) -> Matrix(0,-1,1,0) Matrix(5,2,12,5) -> Matrix(0,-1,1,0) Matrix(1,0,6,1) -> 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): 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 0/1 (0/1,1/0) 0 6 1/3 (-1/1,1/1).(0/1,1/0) 0 2 1/2 (-1/1,1/1) 0 3 2/3 (-1/1,1/1) 0 2 1/1 -1/1 1 6 1/0 0/1 1 1 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(1,0,6,-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(7,-4,12,-7) (1/2,2/3) -> (1/2,2/3) Reflection Matrix(5,-4,6,-5) (2/3,1/1) -> (2/3,1/1) 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(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,6,-1) -> Matrix(1,0,0,-1) (0/1,1/3) -> (0/1,1/0) Matrix(5,-2,12,-5) -> Matrix(0,1,1,0) (1/3,1/2) -> (-1/1,1/1) Matrix(7,-4,12,-7) -> Matrix(0,1,1,0) (1/2,2/3) -> (-1/1,1/1) Matrix(5,-4,6,-5) -> Matrix(0,1,1,0) (2/3,1/1) -> (-1/1,1/1) Matrix(-1,2,0,1) -> Matrix(-1,0,2,1) (1/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.