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 -1/2 -3/8 0/1 1/4 1/3 1/2 2/3 3/4 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 0/1 -3/4 -1/1 -2/3 -1/1 -2/3 -1/2 -1/2 -2/5 -1/3 0/1 -3/8 -1/2 -1/4 -1/3 -1/3 0/1 -1/4 0/1 0/1 -1/1 0/1 1/4 0/1 1/3 0/1 1/1 1/2 1/0 3/5 -2/1 -1/1 5/8 -3/2 1/0 2/3 -2/1 -1/1 3/4 -1/1 1/1 -1/1 0/1 1/0 -1/2 1/0 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(7,6,8,7) (-1/1,-3/4) -> (3/4,1/1) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(7,4,-16,-9) (-2/3,-1/2) -> (-1/2,-2/5) Parabolic Matrix(41,16,64,25) (-2/5,-3/8) -> (5/8,2/3) Hyperbolic Matrix(39,14,64,23) (-3/8,-1/3) -> (3/5,5/8) Hyperbolic Matrix(7,2,24,7) (-1/3,-1/4) -> (1/4,1/3) Hyperbolic Matrix(1,0,8,1) (-1/4,0/1) -> (0/1,1/4) Parabolic Matrix(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,0,1) Matrix(7,6,8,7) -> Matrix(1,0,0,1) Matrix(17,12,24,17) -> Matrix(5,4,-4,-3) Matrix(7,4,-16,-9) -> Matrix(3,2,-8,-5) Matrix(41,16,64,25) -> Matrix(7,2,-4,-1) Matrix(39,14,64,23) -> Matrix(7,2,-4,-1) Matrix(7,2,24,7) -> Matrix(1,0,4,1) Matrix(1,0,8,1) -> Matrix(1,0,0,1) Matrix(9,-4,16,-7) -> Matrix(1,-2,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 2 Degree of the the map Y: 8 Permutation triple for Y: ((1,4,5,2)(3,7,6,8); (1,2,7,3)(4,8,6,5); (2,6)(3,4)) ----------------------------------------------------------------------- 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): 12 Minimal number of generators: 3 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: 4 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/2 3/4 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/1 0/1 1/2 1/0 2/3 -2/1 -1/1 3/4 -1/1 1/1 -1/1 0/1 1/0 -1/2 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(5,-2,8,-3) (0/1,1/2) -> (1/2,2/3) Parabolic Matrix(13,-9,16,-11) (2/3,3/4) -> (3/4,1/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,1,-2,-1) Matrix(5,-2,8,-3) -> Matrix(1,-1,0,1) Matrix(13,-9,16,-11) -> 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 (-1/1,0/1) 0 4 1/4 0/1 2 1 1/2 1/0 1 2 1/0 (-1/1,0/1).(-1/2,1/0) 0 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(1,0,8,-1) (0/1,1/4) -> (0/1,1/4) Reflection Matrix(3,-1,8,-3) (1/4,1/2) -> (1/4,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,8,-1) -> Matrix(-1,0,2,1) (0/1,1/4) -> (-1/1,0/1) Matrix(3,-1,8,-3) -> Matrix(1,0,0,-1) (1/4,1/2) -> (0/1,1/0) Matrix(-1,1,0,1) -> Matrix(1,1,0,-1) (1/2,1/0) -> (-1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.