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