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): 96 Minimal number of generators: 17 Number of equivalence classes of cusps: 16 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/2 -1/1 -1/2 -3/8 -1/3 0/1 1/4 1/3 1/2 2/3 3/4 1/1 4/3 3/2 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 -5/3 0/1 -3/2 -1/2 1/0 -7/5 0/1 -4/3 -2/1 0/1 -1/1 -1/1 0/1 -4/5 -2/1 0/1 -3/4 -1/1 -2/3 -2/3 0/1 -1/2 -1/2 1/0 -2/5 -2/3 0/1 -3/8 -1/2 -4/11 -2/5 0/1 -1/3 0/1 -2/7 -2/1 0/1 -1/4 -1/1 0/1 -2/3 0/1 1/4 -1/1 1/3 -1/1 -1/2 1/2 -1/2 3/5 -1/2 -1/3 5/8 -1/2 2/3 -2/5 0/1 3/4 -1/3 1/1 0/1 5/4 -1/1 4/3 -2/3 0/1 3/2 -1/2 8/5 -2/5 0/1 13/8 -1/2 5/3 -1/2 -1/3 7/4 -1/3 2/1 -2/5 0/1 1/0 0/1 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(7,12,-24,-41) (-2/1,-5/3) -> (-1/3,-2/7) Hyperbolic Matrix(23,36,-16,-25) (-5/3,-3/2) -> (-3/2,-7/5) Parabolic Matrix(23,32,-64,-89) (-7/5,-4/3) -> (-4/11,-1/3) Hyperbolic Matrix(7,8,-8,-9) (-4/3,-1/1) -> (-1/1,-4/5) Parabolic Matrix(41,32,32,25) (-4/5,-3/4) -> (5/4,4/3) 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(207,76,128,47) (-3/8,-4/11) -> (8/5,13/8) Hyperbolic Matrix(57,16,32,9) (-2/7,-1/4) -> (7/4,2/1) Hyperbolic Matrix(1,0,8,1) (-1/4,0/1) -> (0/1,1/4) Parabolic Matrix(41,-12,24,-7) (1/4,1/3) -> (5/3,7/4) Hyperbolic Matrix(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic Matrix(105,-64,64,-39) (3/5,5/8) -> (13/8,5/3) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-2,1) Matrix(7,12,-24,-41) -> Matrix(1,0,0,1) Matrix(23,36,-16,-25) -> Matrix(1,0,0,1) Matrix(23,32,-64,-89) -> Matrix(1,0,-2,1) Matrix(7,8,-8,-9) -> Matrix(1,0,0,1) Matrix(41,32,32,25) -> Matrix(1,2,-2,-3) Matrix(17,12,24,17) -> Matrix(3,2,-8,-5) Matrix(7,4,-16,-9) -> Matrix(1,0,0,1) Matrix(41,16,64,25) -> Matrix(3,2,-8,-5) Matrix(207,76,128,47) -> Matrix(1,0,0,1) Matrix(57,16,32,9) -> Matrix(1,0,-2,1) Matrix(1,0,8,1) -> Matrix(1,0,0,1) Matrix(41,-12,24,-7) -> Matrix(3,2,-8,-5) Matrix(9,-4,16,-7) -> Matrix(3,2,-8,-5) Matrix(105,-64,64,-39) -> Matrix(1,0,0,1) Matrix(9,-8,8,-7) -> Matrix(1,0,2,1) Matrix(25,-36,16,-23) -> Matrix(3,2,-8,-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 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: 16 Permutation triple for Y: ((1,6,7,2)(3,11,12,4)(5,9,8,15)(10,14,13,16); (1,4,14,5)(2,9,10,3)(6,15,13,12)(7,11,16,8); (1,3)(2,8)(4,13)(5,6)(7,12)(9,14)(10,11)(15,16)) ----------------------------------------------------------------------- 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 -1/1 (-1/1,0/1) 0 4 -3/4 -1/1 2 2 -2/3 0 4 -1/2 0 2 -2/5 0 4 -3/8 -1/2 1 2 -1/3 0/1 1 4 -1/4 -1/1 1 2 0/1 0 4 1/4 -1/1 1 2 1/3 (-1/1,-1/2) 0 4 1/2 -1/2 1 2 3/5 (-1/2,-1/3) 0 4 5/8 -1/2 1 2 2/3 0 4 3/4 -1/3 2 2 1/1 0/1 1 4 1/0 0/1 1 2 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(7,6,-8,-7) (-1/1,-3/4) -> (-1/1,-3/4) Reflection 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(17,6,-48,-17) (-3/8,-1/3) -> (-3/8,-1/3) Reflection Matrix(7,2,-24,-7) (-1/3,-1/4) -> (-1/3,-1/4) Reflection Matrix(1,0,8,1) (-1/4,0/1) -> (0/1,1/4) Parabolic Matrix(7,-2,24,-7) (1/4,1/3) -> (1/4,1/3) Reflection Matrix(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic Matrix(49,-30,80,-49) (3/5,5/8) -> (3/5,5/8) Reflection Matrix(7,-6,8,-7) (3/4,1/1) -> (3/4,1/1) 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(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(7,6,-8,-7) -> Matrix(-1,0,2,1) (-1/1,-3/4) -> (-1/1,0/1) Matrix(17,12,24,17) -> Matrix(3,2,-8,-5) -1/2 Matrix(7,4,-16,-9) -> Matrix(1,0,0,1) Matrix(41,16,64,25) -> Matrix(3,2,-8,-5) -1/2 Matrix(17,6,-48,-17) -> Matrix(-1,0,4,1) (-3/8,-1/3) -> (-1/2,0/1) Matrix(7,2,-24,-7) -> Matrix(-1,0,2,1) (-1/3,-1/4) -> (-1/1,0/1) Matrix(1,0,8,1) -> Matrix(1,0,0,1) Matrix(7,-2,24,-7) -> Matrix(3,2,-4,-3) (1/4,1/3) -> (-1/1,-1/2) Matrix(9,-4,16,-7) -> Matrix(3,2,-8,-5) -1/2 Matrix(49,-30,80,-49) -> Matrix(5,2,-12,-5) (3/5,5/8) -> (-1/2,-1/3) Matrix(7,-6,8,-7) -> Matrix(-1,0,6,1) (3/4,1/1) -> (-1/3,0/1) Matrix(-1,2,0,1) -> Matrix(-1,0,4,1) (1/1,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.