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 -1/1 -4/5 -1/1 -2/3 -3/4 -2/3 -2/3 -1/2 -3/5 0/1 -1/2 -1/1 -2/5 -1/1 -1/2 -1/3 -1/2 -3/10 -1/2 0/1 -2/7 -1/2 -1/4 0/1 -1/5 0/1 0/1 -1/2 1/0 1/5 0/1 1/4 0/1 1/3 1/0 2/5 -1/1 1/0 1/2 -1/1 3/5 0/1 2/3 1/0 7/10 -2/1 1/0 5/7 1/0 3/4 -2/1 4/5 -2/1 -1/1 1/1 -1/1 1/0 -1/1 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(1,0,0,1) Matrix(9,8,10,9) -> Matrix(5,4,-4,-3) Matrix(31,24,40,31) -> Matrix(5,4,-4,-3) Matrix(11,8,-40,-29) -> Matrix(3,2,-8,-5) Matrix(19,12,30,19) -> Matrix(1,0,2,1) Matrix(11,6,20,11) -> Matrix(1,0,0,1) Matrix(9,4,20,9) -> Matrix(3,2,-2,-1) Matrix(11,4,30,11) -> Matrix(3,2,-2,-1) Matrix(71,22,100,31) -> Matrix(3,2,-2,-1) Matrix(69,20,100,29) -> Matrix(3,2,-2,-1) Matrix(9,2,40,9) -> Matrix(1,0,2,1) Matrix(1,0,10,1) -> Matrix(1,0,0,1) Matrix(29,-8,40,-11) -> Matrix(1,-2,0,1) 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): 3 Degree of the the map X: 3 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. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- 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 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 (-1/1,0/1) 0 5 1/5 0/1 5 5 1/4 0/1 1 5 1/3 1/0 1 5 2/5 (-1/1,1/0) 0 5 1/2 -1/1 1 5 3/5 0/1 5 5 2/3 1/0 2 5 7/10 (-2/1,1/0) 0 5 5/7 1/0 1 5 3/4 -2/1 1 5 4/5 (-2/1,-1/1) 0 5 1/1 -1/1 2 5 1/0 (-1/1,0/1) 0 5 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(9,-2,40,-9) (1/5,1/4) -> (1/5,1/4) Reflection Matrix(29,-8,40,-11) (1/4,1/3) -> (5/7,3/4) Hyperbolic 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(19,-12,30,-19) (3/5,2/3) -> (3/5,2/3) Reflection Matrix(41,-28,60,-41) (2/3,7/10) -> (2/3,7/10) Reflection Matrix(99,-70,140,-99) (7/10,5/7) -> (7/10,5/7) Reflection Matrix(31,-24,40,-31) (3/4,4/5) -> (3/4,4/5) Reflection Matrix(9,-8,10,-9) (4/5,1/1) -> (4/5,1/1) 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(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,10,-1) -> Matrix(-1,0,2,1) (0/1,1/5) -> (-1/1,0/1) Matrix(9,-2,40,-9) -> Matrix(1,0,0,-1) (1/5,1/4) -> (0/1,1/0) Matrix(29,-8,40,-11) -> Matrix(1,-2,0,1) 1/0 Matrix(11,-4,30,-11) -> Matrix(1,2,0,-1) (1/3,2/5) -> (-1/1,1/0) Matrix(9,-4,20,-9) -> Matrix(1,2,0,-1) (2/5,1/2) -> (-1/1,1/0) Matrix(11,-6,20,-11) -> Matrix(-1,0,2,1) (1/2,3/5) -> (-1/1,0/1) Matrix(19,-12,30,-19) -> Matrix(1,0,0,-1) (3/5,2/3) -> (0/1,1/0) Matrix(41,-28,60,-41) -> Matrix(1,4,0,-1) (2/3,7/10) -> (-2/1,1/0) Matrix(99,-70,140,-99) -> Matrix(1,4,0,-1) (7/10,5/7) -> (-2/1,1/0) Matrix(31,-24,40,-31) -> Matrix(3,4,-2,-3) (3/4,4/5) -> (-2/1,-1/1) Matrix(9,-8,10,-9) -> Matrix(3,4,-2,-3) (4/5,1/1) -> (-2/1,-1/1) Matrix(-1,2,0,1) -> Matrix(-1,0,2,1) (1/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.