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 -3/1 -2/1 -1/1 -2/3 0/1 1/2 2/3 3/4 1/1 4/3 3/2 2/1 8/3 3/1 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 1/0 -3/1 -3/1 -1/1 -2/1 -2/1 0/1 -5/3 -3/1 -1/1 -8/5 -2/1 -3/2 -3/2 1/0 -4/3 -1/1 -1/1 -1/1 -4/5 -1/1 -3/4 -1/2 -2/3 -1/1 -5/8 -3/4 -8/13 -2/3 -3/5 -1/1 -3/5 -4/7 -1/2 -1/2 -1/2 1/0 0/1 0/1 1/2 1/2 1/0 3/5 -1/1 2/3 0/1 5/7 1/1 3/4 1/2 1/1 -1/1 1/1 5/4 1/2 4/3 1/1 3/2 3/2 1/0 2/1 1/0 5/2 -1/2 1/0 8/3 0/1 11/4 1/2 3/1 1/1 7/2 5/2 1/0 4/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,8,0,1) (-4/1,1/0) -> (4/1,1/0) Parabolic Matrix(7,24,-12,-41) (-4/1,-3/1) -> (-3/5,-4/7) Hyperbolic Matrix(7,16,-4,-9) (-3/1,-2/1) -> (-2/1,-5/3) Parabolic Matrix(39,64,-64,-105) (-5/3,-8/5) -> (-8/13,-3/5) Hyperbolic Matrix(41,64,16,25) (-8/5,-3/2) -> (5/2,8/3) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) 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(23,16,-36,-25) (-3/4,-2/3) -> (-2/3,-5/8) Parabolic Matrix(207,128,76,47) (-5/8,-8/13) -> (8/3,11/4) Hyperbolic Matrix(57,32,16,9) (-4/7,-1/2) -> (7/2,4/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(41,-24,12,-7) (1/2,3/5) -> (3/1,7/2) Hyperbolic Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(89,-64,32,-23) (5/7,3/4) -> (11/4,3/1) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,8,0,1) -> Matrix(1,2,0,1) Matrix(7,24,-12,-41) -> Matrix(1,4,-2,-7) Matrix(7,16,-4,-9) -> Matrix(1,0,0,1) Matrix(39,64,-64,-105) -> Matrix(1,4,-2,-7) Matrix(41,64,16,25) -> Matrix(1,2,-2,-3) Matrix(17,24,12,17) -> Matrix(3,4,2,3) Matrix(7,8,-8,-9) -> Matrix(1,2,-2,-3) Matrix(41,32,32,25) -> Matrix(3,2,4,3) Matrix(23,16,-36,-25) -> Matrix(1,2,-2,-3) Matrix(207,128,76,47) -> Matrix(3,2,10,7) Matrix(57,32,16,9) -> Matrix(5,2,2,1) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(41,-24,12,-7) -> Matrix(1,2,0,1) Matrix(25,-16,36,-23) -> Matrix(1,0,2,1) Matrix(89,-64,32,-23) -> Matrix(1,0,0,1) Matrix(9,-8,8,-7) -> Matrix(1,0,0,1) Matrix(9,-16,4,-7) -> 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): 5 Degree of the the map X: 5 Degree of the the map Y: 16 Permutation triple for Y: ((1,2)(3,10)(4,5)(6,15)(7,8)(9,14)(11,12)(13,16); (1,5,14,6)(2,8,9,3)(4,12,15,13)(7,11,10,16); (1,3,11,4)(2,6,12,7)(5,13,10,9)(8,16,15,14)) ----------------------------------------------------------------------- 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 -4/1 1/0 2 2 -3/1 0 4 -8/3 -2/1 3 2 -2/1 (-2/1,0/1) 0 2 0/1 0/1 1 2 2/3 0/1 1 2 3/4 1/2 1 4 1/1 0 4 4/3 1/1 2 2 3/2 0 4 2/1 1/0 1 2 5/2 0 4 8/3 0/1 3 2 3/1 1/1 1 4 7/2 0 4 4/1 1/0 2 2 1/0 1/0 1 4 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,8,0,1) (-4/1,1/0) -> (4/1,1/0) Parabolic Matrix(7,24,5,17) (-4/1,-3/1) -> (4/3,3/2) Glide Reflection Matrix(23,64,9,25) (-3/1,-8/3) -> (5/2,8/3) Glide Reflection Matrix(7,16,-3,-7) (-8/3,-2/1) -> (-8/3,-2/1) Reflection Matrix(-1,0,1,1) (-2/1,0/1) -> (-2/1,0/1) Reflection Matrix(1,0,3,-1) (0/1,2/3) -> (0/1,2/3) Reflection Matrix(23,-16,33,-23) (2/3,8/11) -> (2/3,8/11) Reflection Matrix(65,-48,23,-17) (5/7,3/4) -> (11/4,3/1) Glide Reflection Matrix(31,-24,9,-7) (3/4,1/1) -> (3/1,7/2) Glide Reflection Matrix(25,-32,7,-9) (1/1,4/3) -> (7/2,4/1) Glide Reflection Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(41,-112,15,-41) (8/3,14/5) -> (8/3,14/5) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,8,0,1) -> Matrix(1,2,0,1) 1/0 Matrix(7,24,5,17) -> Matrix(1,4,1,3) Matrix(23,64,9,25) -> Matrix(1,2,-1,-3) Matrix(7,16,-3,-7) -> Matrix(-1,0,1,1) (-8/3,-2/1) -> (-2/1,0/1) Matrix(-1,0,1,1) -> Matrix(-1,0,1,1) (-2/1,0/1) -> (-2/1,0/1) Matrix(1,0,3,-1) -> Matrix(1,0,1,-1) (0/1,2/3) -> (0/1,2/1) Matrix(23,-16,33,-23) -> Matrix(1,0,3,-1) (2/3,8/11) -> (0/1,2/3) Matrix(65,-48,23,-17) -> Matrix(1,0,3,-1) *** -> (0/1,2/3) Matrix(31,-24,9,-7) -> Matrix(3,-2,1,-1) Matrix(25,-32,7,-9) -> Matrix(3,-2,1,-1) Matrix(9,-16,4,-7) -> Matrix(1,-2,0,1) 1/0 Matrix(41,-112,15,-41) -> Matrix(1,0,3,-1) (8/3,14/5) -> (0/1,2/3) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.