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 -1/1 -2/5 0/1 1/3 1/2 1/1 2/1 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/1 -2/5 0/1 1/1 1/0 -3/8 0/1 1/1 -1/3 0/1 1/1 1/0 0/1 0/1 1/0 1/3 0/1 1/1 1/0 1/2 1/0 3/5 -2/1 -1/1 1/0 5/8 -2/1 -1/1 2/3 -1/1 1/0 1/1 -1/1 0/1 1/0 2/1 -1/1 0/1 1/0 3/1 -1/1 0/1 1/0 4/1 -1/1 1/0 5/1 -1/1 6/1 -1/1 -1/2 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(19,8,-50,-21) (-1/2,-2/5) -> (-2/5,-3/8) Parabolic Matrix(39,14,64,23) (-3/8,-1/3) -> (3/5,5/8) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic Matrix(51,-32,8,-5) (5/8,2/3) -> (6/1,1/0) Hyperbolic Matrix(13,-10,4,-3) (2/3,1/1) -> (3/1,4/1) Hyperbolic Matrix(5,-8,2,-3) (1/1,2/1) -> (2/1,3/1) Parabolic Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,2,1) Matrix(19,8,-50,-21) -> Matrix(1,0,0,1) Matrix(39,14,64,23) -> Matrix(1,-2,0,1) Matrix(1,0,6,1) -> Matrix(1,0,0,1) Matrix(9,-4,16,-7) -> Matrix(1,-2,0,1) Matrix(51,-32,8,-5) -> Matrix(1,2,-2,-3) Matrix(13,-10,4,-3) -> Matrix(1,0,0,1) Matrix(5,-8,2,-3) -> Matrix(1,0,0,1) Matrix(11,-50,2,-9) -> Matrix(1,2,-2,-3) 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: ((1,5,2)(4,7,8); (1,3,4)(5,8,6); (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 -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 -1/1 0/1 -1/2 0/1 1/1 -1/3 0/1 1/1 1/0 0/1 0/1 1/0 1/2 1/0 2/3 -1/1 1/0 1/1 -1/1 0/1 1/0 2/1 -1/1 0/1 1/0 3/1 -1/1 0/1 1/0 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(9,4,2,1) (-1/2,-1/3) -> (3/1,1/0) Hyperbolic Matrix(7,2,10,3) (-1/3,0/1) -> (2/3,1/1) Hyperbolic Matrix(5,-2,8,-3) (0/1,1/2) -> (1/2,2/3) Parabolic Matrix(5,-8,2,-3) (1/1,2/1) -> (2/1,3/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,2,1) Matrix(9,4,2,1) -> Matrix(1,-1,0,1) Matrix(7,2,10,3) -> Matrix(1,-1,0,1) Matrix(5,-2,8,-3) -> Matrix(1,-1,0,1) Matrix(5,-8,2,-3) -> 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 -1/1 0/1 1 1 0/1 (0/1,1/0) 0 6 1/2 1/0 1 2 1/1 (-1/2,1/0) 0 3 2/1 (-1/2,1/0) 0 2 1/0 (-1/1,0/1).(-1/2,1/0) 0 6 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(3,-2,4,-3) (1/2,1/1) -> (1/2,1/1) Reflection Matrix(3,-4,2,-3) (1/1,2/1) -> (1/1,2/1) Reflection Matrix(-1,4,0,1) (2/1,1/0) -> (2/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(1,0,4,-1) -> Matrix(1,0,0,-1) (0/1,1/2) -> (0/1,1/0) Matrix(3,-2,4,-3) -> Matrix(1,1,0,-1) (1/2,1/1) -> (-1/2,1/0) Matrix(3,-4,2,-3) -> Matrix(1,1,0,-1) (1/1,2/1) -> (-1/2,1/0) Matrix(-1,4,0,1) -> Matrix(1,1,0,-1) (2/1,1/0) -> (-1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.