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 0/1 1/1 3/2 2/1 3/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 -3/2 1/0 -5/2 1/0 -2/1 -2/1 -1/1 1/0 -7/4 1/0 -12/7 -2/1 -5/3 -2/1 -1/1 -3/2 -3/2 1/0 -1/1 -2/1 -1/1 0/1 -1/1 1/1 -1/1 0/1 3/2 -1/2 1/0 5/3 -1/1 0/1 2/1 -1/1 0/1 1/0 3/1 -1/2 1/0 4/1 -1/1 0/1 1/0 5/1 -1/1 0/1 6/1 0/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,18,-2,-7) (-3/1,1/0) -> (-3/1,-5/2) Parabolic Matrix(11,24,-6,-13) (-5/2,-2/1) -> (-2/1,-7/4) Parabolic Matrix(31,54,4,7) (-7/4,-12/7) -> (6/1,1/0) Hyperbolic Matrix(53,90,10,17) (-12/7,-5/3) -> (5/1,6/1) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(5,6,4,5) (-3/2,-1/1) -> (1/1,3/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(17,-30,4,-7) (5/3,2/1) -> (4/1,5/1) Hyperbolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,18,-2,-7) -> Matrix(1,0,0,1) Matrix(11,24,-6,-13) -> Matrix(1,0,0,1) Matrix(31,54,4,7) -> Matrix(1,2,0,1) Matrix(53,90,10,17) -> Matrix(1,2,-2,-3) Matrix(19,30,12,19) -> Matrix(1,2,-2,-3) Matrix(5,6,4,5) -> Matrix(1,2,-2,-3) Matrix(1,0,2,1) -> Matrix(1,2,-2,-3) Matrix(17,-30,4,-7) -> Matrix(1,0,0,1) Matrix(7,-18,2,-5) -> 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: ((2,5,6)(3,7,4); (1,4,2)(5,7,8); (1,2,6,8,7,3)(4,5)) ----------------------------------------------------------------------- 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 1/1 3/2 2/1 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 -3/2 1/0 -2/1 -2/1 -1/1 1/0 -3/2 -3/2 1/0 -1/1 -2/1 -1/1 0/1 -1/1 1/1 -1/1 0/1 3/2 -1/2 1/0 2/1 -1/1 0/1 1/0 3/1 -1/2 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,6,0,1) (-3/1,1/0) -> (3/1,1/0) Parabolic Matrix(5,12,2,5) (-3/1,-2/1) -> (2/1,3/1) Hyperbolic Matrix(7,12,4,7) (-2/1,-3/2) -> (3/2,2/1) Hyperbolic Matrix(5,6,4,5) (-3/2,-1/1) -> (1/1,3/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,1,0,1) Matrix(5,12,2,5) -> Matrix(1,1,0,1) Matrix(7,12,4,7) -> Matrix(1,1,0,1) Matrix(5,6,4,5) -> Matrix(1,2,-2,-3) Matrix(1,0,2,1) -> Matrix(1,2,-2,-3) 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 -1/1 2 2 1/1 (-1/1,0/1) 0 6 3/2 (-1/1,0/1).(-1/2,1/0) 0 2 2/1 (-1/2,1/0) 0 6 3/1 (-1/2,1/0) 0 2 1/0 1/0 1 6 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(5,-6,4,-5) (1/1,3/2) -> (1/1,3/2) Reflection Matrix(7,-12,4,-7) (3/2,2/1) -> (3/2,2/1) 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(1,0,0,-1) -> Matrix(1,2,0,-1) (0/1,1/0) -> (-1/1,1/0) Matrix(1,0,2,-1) -> Matrix(-1,0,2,1) (0/1,1/1) -> (-1/1,0/1) Matrix(5,-6,4,-5) -> Matrix(-1,0,2,1) (1/1,3/2) -> (-1/1,0/1) Matrix(7,-12,4,-7) -> Matrix(1,1,0,-1) (3/2,2/1) -> (-1/2,1/0) Matrix(5,-12,2,-5) -> Matrix(1,1,0,-1) (2/1,3/1) -> (-1/2,1/0) Matrix(-1,6,0,1) -> Matrix(1,1,0,-1) (3/1,1/0) -> (-1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.