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 -2/1 -3/2 -6/5 0/1 1/1 3/2 2/1 5/2 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 -1/1 1/1 -5/2 0/1 1/0 -2/1 -1/1 -3/2 -1/1 -1/2 0/1 -4/3 -1/1 -5/4 -2/3 -1/2 -6/5 -1/2 -1/1 -1/3 0/1 0/1 1/1 1/3 3/2 0/1 1/2 1/1 5/3 1/3 2/1 1/1 7/3 3/1 12/5 1/0 5/2 0/1 1/0 3/1 -1/1 1/1 1/0 0/1 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(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(13,30,-10,-23) (-5/2,-2/1) -> (-4/3,-5/4) Hyperbolic Matrix(11,18,-8,-13) (-2/1,-3/2) -> (-3/2,-4/3) Parabolic Matrix(73,90,30,37) (-5/4,-6/5) -> (12/5,5/2) Hyperbolic Matrix(47,54,20,23) (-6/5,-1/1) -> (7/3,12/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,0,0,1) Matrix(11,30,4,11) -> Matrix(1,0,0,1) Matrix(13,30,-10,-23) -> Matrix(1,2,-2,-3) Matrix(11,18,-8,-13) -> Matrix(1,0,0,1) Matrix(73,90,30,37) -> Matrix(3,2,-2,-1) Matrix(47,54,20,23) -> Matrix(9,4,2,1) Matrix(1,0,2,1) -> Matrix(1,0,6,1) Matrix(13,-18,8,-11) -> Matrix(1,0,0,1) Matrix(13,-24,6,-11) -> Matrix(3,-2,2,-1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 18 Minimal number of generators: 4 Number of equivalence classes of cusps: 5 Genus: 0 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 3 Degree of the the map Y: 8 Permutation triple for Y: ((2,5,6)(3,7,4); (1,4,7,8,5,2)(3,6); (1,2,3)(6,8,7)) ----------------------------------------------------------------------- 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 -1/1 1/1 -2/1 -1/1 -3/2 -1/1 -1/2 0/1 -1/1 -1/3 0/1 0/1 1/1 1/3 3/2 0/1 1/2 1/1 2/1 1/1 3/1 -1/1 1/1 1/0 0/1 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(1,0,0,1) Matrix(5,12,2,5) -> Matrix(0,-1,1,0) Matrix(7,12,4,7) -> Matrix(2,1,3,2) Matrix(5,6,4,5) -> Matrix(2,1,3,2) Matrix(1,0,2,1) -> Matrix(1,0,6,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 9 Minimal number of generators: 3 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: 3 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 3 1 1/1 1/3 1 6 3/2 (1/3,1/1) 0 2 2/1 1/1 1 3 3/1 (-1/1,1/1).(0/1,1/0) 0 2 1/0 (0/1,1/0) 0 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(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,2,-1) -> Matrix(1,0,6,-1) (0/1,1/1) -> (0/1,1/3) Matrix(5,-6,4,-5) -> Matrix(2,-1,3,-2) (1/1,3/2) -> (1/3,1/1) Matrix(7,-12,4,-7) -> Matrix(2,-1,3,-2) (3/2,2/1) -> (1/3,1/1) 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.