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 1/2 -3/1 1/2 1/1 -2/1 1/1 -5/3 1/1 1/0 -8/5 1/1 -3/2 1/1 2/1 1/0 -4/3 1/0 -1/1 1/0 -4/5 1/0 -3/4 -2/1 -1/1 1/0 -2/3 -1/1 -5/8 -1/1 -1/2 0/1 -8/13 -1/1 -3/5 -1/1 -1/2 -4/7 -1/2 -1/2 -1/1 -1/2 0/1 0/1 0/1 1/2 0/1 1/3 1/2 3/5 1/2 2/3 0/1 5/7 1/4 3/4 0/1 1/3 1/2 1/1 0/1 1/2 5/4 0/1 1/3 1/2 4/3 1/2 3/2 0/1 1/2 1/1 2/1 0/1 5/2 0/1 1/4 1/3 8/3 1/3 11/4 1/3 3/8 2/5 3/1 1/2 7/2 0/1 1/3 1/2 4/1 1/2 1/0 0/1 1/2 1/1 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(1,0,0,1) Matrix(7,24,-12,-41) -> Matrix(3,-2,-4,3) Matrix(7,16,-4,-9) -> Matrix(3,-2,2,-1) Matrix(39,64,-64,-105) -> Matrix(1,0,-2,1) Matrix(41,64,16,25) -> Matrix(1,-2,4,-7) Matrix(17,24,12,17) -> Matrix(1,-2,2,-3) Matrix(7,8,-8,-9) -> Matrix(1,-2,0,1) Matrix(41,32,32,25) -> Matrix(1,2,2,5) Matrix(23,16,-36,-25) -> Matrix(1,2,-2,-3) Matrix(207,128,76,47) -> Matrix(1,2,2,5) Matrix(57,32,16,9) -> Matrix(1,0,4,1) Matrix(1,0,4,1) -> Matrix(1,0,4,1) Matrix(41,-24,12,-7) -> Matrix(1,0,0,1) Matrix(25,-16,36,-23) -> Matrix(1,0,2,1) Matrix(89,-64,32,-23) -> Matrix(7,-2,18,-5) Matrix(9,-8,8,-7) -> Matrix(1,0,0,1) Matrix(9,-16,4,-7) -> Matrix(1,0,2,1) 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): 4 Degree of the the map X: 4 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. ----------------------------------------------------------------------- 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): 48 Minimal number of generators: 9 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: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -2/1 -1/1 0/1 1/2 1/1 2/1 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 1/2 -3/1 1/2 1/1 -2/1 1/1 -3/2 1/1 2/1 1/0 -4/3 1/0 -1/1 1/0 -2/3 -1/1 -1/2 -1/1 -1/2 0/1 0/1 0/1 1/2 0/1 1/3 1/2 2/3 0/1 1/1 0/1 1/2 4/3 1/2 3/2 0/1 1/2 1/1 2/1 0/1 3/1 1/2 4/1 1/2 1/0 0/1 1/2 1/1 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(5,16,4,13) (-4/1,-3/1) -> (1/1,4/3) Hyperbolic Matrix(3,8,4,11) (-3/1,-2/1) -> (2/3,1/1) Hyperbolic Matrix(5,8,8,13) (-2/1,-3/2) -> (1/2,2/3) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(13,16,4,5) (-4/3,-1/1) -> (3/1,4/1) Hyperbolic Matrix(11,8,4,3) (-1/1,-2/3) -> (2/1,3/1) Hyperbolic Matrix(13,8,8,5) (-2/3,-1/2) -> (3/2,2/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,8,0,1) -> Matrix(1,0,0,1) Matrix(5,16,4,13) -> Matrix(1,-1,4,-3) Matrix(3,8,4,11) -> Matrix(1,-1,4,-3) Matrix(5,8,8,13) -> Matrix(1,-1,2,-1) Matrix(17,24,12,17) -> Matrix(1,-2,2,-3) Matrix(13,16,4,5) -> Matrix(1,-1,2,-1) Matrix(11,8,4,3) -> Matrix(1,1,2,3) Matrix(13,8,8,5) -> Matrix(1,1,0,1) Matrix(1,0,4,1) -> Matrix(1,0,4,1) 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): 4 ----------------------------------------------------------------------- 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 -2/1 1/1 1 2 -1/1 1/0 1 4 0/1 0/1 2 2 1/1 (0/1,1/2) 0 4 4/3 1/2 1 2 3/2 0 4 2/1 0/1 1 2 3/1 1/2 1 4 4/1 1/2 1 2 1/0 0 4 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,8,2,5) (-2/1,1/0) -> (3/2,2/1) Glide Reflection Matrix(5,8,2,3) (-2/1,-1/1) -> (2/1,3/1) Glide Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(7,-8,6,-7) (1/1,4/3) -> (1/1,4/3) Reflection Matrix(11,-16,2,-3) (4/3,3/2) -> (4/1,1/0) Glide Reflection Matrix(7,-24,2,-7) (3/1,4/1) -> (3/1,4/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(3,8,2,5) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(5,8,2,3) -> Matrix(1,-1,2,-3) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(1,0,2,-1) -> Matrix(1,0,4,-1) (0/1,1/1) -> (0/1,1/2) Matrix(7,-8,6,-7) -> Matrix(1,0,4,-1) (1/1,4/3) -> (0/1,1/2) Matrix(11,-16,2,-3) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(7,-24,2,-7) -> Matrix(1,0,4,-1) (3/1,4/1) -> (0/1,1/2) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.