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): 72 Minimal number of generators: 13 Number of equivalence classes of cusps: 12 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/10 0/1 1/5 1/4 1/3 2/5 1/2 3/5 2/3 4/5 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -4/5 -1/1 -3/4 -1/1 -1/2 -2/3 0/1 1/0 -3/5 1/0 -1/2 -1/1 1/0 -2/5 -1/1 -1/3 -3/4 -3/10 -2/3 -2/7 -2/3 -5/8 -1/4 -3/5 -1/2 -1/5 -1/2 0/1 -1/2 0/1 1/5 -1/2 1/4 -1/2 -3/7 1/3 -3/8 2/5 -1/3 1/2 -1/3 -1/4 3/5 -1/4 2/3 -1/4 0/1 7/10 0/1 5/7 1/0 3/4 -1/2 -1/3 4/5 -1/3 1/1 -1/4 1/0 0/1 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(9,8,10,9) (-1/1,-4/5) -> (4/5,1/1) Hyperbolic Matrix(31,24,40,31) (-4/5,-3/4) -> (3/4,4/5) Hyperbolic Matrix(11,8,-40,-29) (-3/4,-2/3) -> (-2/7,-1/4) Hyperbolic Matrix(19,12,30,19) (-2/3,-3/5) -> (3/5,2/3) Hyperbolic Matrix(11,6,20,11) (-3/5,-1/2) -> (1/2,3/5) Hyperbolic Matrix(9,4,20,9) (-1/2,-2/5) -> (2/5,1/2) Hyperbolic Matrix(11,4,30,11) (-2/5,-1/3) -> (1/3,2/5) Hyperbolic Matrix(71,22,100,31) (-1/3,-3/10) -> (7/10,5/7) Hyperbolic Matrix(69,20,100,29) (-3/10,-2/7) -> (2/3,7/10) Hyperbolic Matrix(9,2,40,9) (-1/4,-1/5) -> (1/5,1/4) Hyperbolic Matrix(1,0,10,1) (-1/5,0/1) -> (0/1,1/5) Parabolic Matrix(29,-8,40,-11) (1/4,1/3) -> (5/7,3/4) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(9,8,10,9) -> Matrix(1,2,-4,-7) Matrix(31,24,40,31) -> Matrix(3,2,-8,-5) Matrix(11,8,-40,-29) -> Matrix(5,2,-8,-3) Matrix(19,12,30,19) -> Matrix(1,0,-4,1) Matrix(11,6,20,11) -> Matrix(1,2,-4,-7) Matrix(9,4,20,9) -> Matrix(1,2,-4,-7) Matrix(11,4,30,11) -> Matrix(7,6,-20,-17) Matrix(71,22,100,31) -> Matrix(3,2,4,3) Matrix(69,20,100,29) -> Matrix(3,2,-20,-13) Matrix(9,2,40,9) -> Matrix(11,6,-24,-13) Matrix(1,0,10,1) -> Matrix(1,0,0,1) Matrix(29,-8,40,-11) -> Matrix(5,2,-8,-3) 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): 3 Degree of the the map X: 6 Degree of the the map Y: 12 Permutation triple for Y: ((1,4,11,5,2)(3,10,12,8,7); (1,2,8,9,3)(4,6,5,12,10); (2,6,4,3,7)(5,11,10,9,8)) ----------------------------------------------------------------------- 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): 36 Minimal number of generators: 7 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: 8 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/5 1/3 2/5 1/2 3/5 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -1/2 -1/1 1/0 -2/5 -1/1 -1/3 -3/4 0/1 -1/2 0/1 1/5 -1/2 1/4 -1/2 -3/7 1/3 -3/8 2/5 -1/3 1/2 -1/3 -1/4 3/5 -1/4 2/3 -1/4 0/1 1/1 -1/4 1/0 0/1 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(4,3,5,4) (-1/1,-1/2) -> (2/3,1/1) Hyperbolic Matrix(9,4,20,9) (-1/2,-2/5) -> (2/5,1/2) Hyperbolic Matrix(11,4,30,11) (-2/5,-1/3) -> (1/3,2/5) Hyperbolic Matrix(4,1,15,4) (-1/3,0/1) -> (1/4,1/3) Hyperbolic Matrix(6,-1,25,-4) (0/1,1/5) -> (1/5,1/4) Parabolic Matrix(16,-9,25,-14) (1/2,3/5) -> (3/5,2/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(4,3,5,4) -> Matrix(1,1,-4,-3) Matrix(9,4,20,9) -> Matrix(1,2,-4,-7) Matrix(11,4,30,11) -> Matrix(7,6,-20,-17) Matrix(4,1,15,4) -> Matrix(5,3,-12,-7) Matrix(6,-1,25,-4) -> Matrix(5,3,-12,-7) Matrix(16,-9,25,-14) -> Matrix(3,1,-16,-5) 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 elliptic points of order 2: 0 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): 3 ----------------------------------------------------------------------- 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 (-1/2,0/1) 0 5 1/5 -1/2 3 1 1/3 -3/8 1 5 2/5 -1/3 4 1 1/2 (-1/3,-1/4) 0 5 3/5 -1/4 1 1 1/1 -1/4 1 5 1/0 0/1 2 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(1,0,10,-1) (0/1,1/5) -> (0/1,1/5) Reflection Matrix(4,-1,15,-4) (1/5,1/3) -> (1/5,1/3) Reflection Matrix(11,-4,30,-11) (1/3,2/5) -> (1/3,2/5) Reflection Matrix(9,-4,20,-9) (2/5,1/2) -> (2/5,1/2) Reflection Matrix(11,-6,20,-11) (1/2,3/5) -> (1/2,3/5) Reflection Matrix(4,-3,5,-4) (3/5,1/1) -> (3/5,1/1) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,4,1) (0/1,1/0) -> (-1/2,0/1) Matrix(1,0,10,-1) -> Matrix(-1,0,4,1) (0/1,1/5) -> (-1/2,0/1) Matrix(4,-1,15,-4) -> Matrix(7,3,-16,-7) (1/5,1/3) -> (-1/2,-3/8) Matrix(11,-4,30,-11) -> Matrix(17,6,-48,-17) (1/3,2/5) -> (-3/8,-1/3) Matrix(9,-4,20,-9) -> Matrix(7,2,-24,-7) (2/5,1/2) -> (-1/3,-1/4) Matrix(11,-6,20,-11) -> Matrix(7,2,-24,-7) (1/2,3/5) -> (-1/3,-1/4) Matrix(4,-3,5,-4) -> Matrix(3,1,-8,-3) (3/5,1/1) -> (-1/2,-1/4) Matrix(-1,2,0,1) -> Matrix(-1,0,8,1) (1/1,1/0) -> (-1/4,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.