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 1/0 -5/3 -3/1 -3/2 -3/1 -1/1 -7/5 -3/1 -4/3 -2/1 -1/1 -1/1 -4/5 0/1 -3/4 0/1 -2/3 1/0 -1/2 -1/1 1/1 -2/5 1/0 -3/8 1/0 -4/11 -2/1 -1/3 -1/1 -2/7 -1/2 -1/4 0/1 0/1 0/1 1/4 0/1 1/3 1/1 1/2 -1/1 1/1 3/5 1/1 5/8 1/0 2/3 1/0 3/4 0/1 1/1 1/1 5/4 2/1 4/3 2/1 3/2 1/1 3/1 8/5 2/1 13/8 5/2 5/3 3/1 7/4 4/1 2/1 1/0 1/0 1/0 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(1,2,0,1) Matrix(7,12,-24,-41) -> Matrix(1,4,-2,-7) Matrix(23,36,-16,-25) -> Matrix(1,0,0,1) Matrix(23,32,-64,-89) -> Matrix(3,8,-2,-5) Matrix(7,8,-8,-9) -> Matrix(1,2,-2,-3) Matrix(41,32,32,25) -> Matrix(1,2,0,1) Matrix(17,12,24,17) -> Matrix(1,0,0,1) Matrix(7,4,-16,-9) -> Matrix(1,0,0,1) Matrix(41,16,64,25) -> Matrix(1,2,0,1) Matrix(207,76,128,47) -> Matrix(5,12,2,5) Matrix(57,16,32,9) -> Matrix(9,4,2,1) Matrix(1,0,8,1) -> Matrix(1,0,2,1) Matrix(41,-12,24,-7) -> Matrix(7,-4,2,-1) Matrix(9,-4,16,-7) -> Matrix(1,0,0,1) Matrix(105,-64,64,-39) -> Matrix(5,-8,2,-3) Matrix(9,-8,8,-7) -> Matrix(3,-2,2,-1) Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) 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): 6 Degree of the the map X: 6 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. ----------------------------------------------------------------------- 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 0/1 1/4 1/3 1/2 1/1 3/2 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 1/0 -5/3 -3/1 -3/2 -3/1 -1/1 -1/1 -1/1 -1/2 -1/1 1/1 -1/3 -1/1 -2/7 -1/2 -1/4 0/1 0/1 0/1 1/4 0/1 1/3 1/1 1/2 -1/1 1/1 1/1 1/1 3/2 1/1 3/1 5/3 3/1 7/4 4/1 2/1 1/0 1/0 1/0 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(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(5,6,4,5) (-3/2,-1/1) -> (1/1,3/2) Hyperbolic Matrix(3,2,4,3) (-1/1,-1/2) -> (1/2,1/1) Hyperbolic Matrix(5,2,12,5) (-1/2,-1/3) -> (1/3,1/2) 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 IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,2,0,1) Matrix(7,12,-24,-41) -> Matrix(1,4,-2,-7) Matrix(19,30,12,19) -> Matrix(2,3,1,2) Matrix(5,6,4,5) -> Matrix(2,3,1,2) Matrix(3,2,4,3) -> Matrix(0,-1,1,0) Matrix(5,2,12,5) -> Matrix(0,-1,1,0) Matrix(57,16,32,9) -> Matrix(9,4,2,1) Matrix(1,0,8,1) -> Matrix(1,0,2,1) Matrix(41,-12,24,-7) -> Matrix(7,-4,2,-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): 6 ----------------------------------------------------------------------- 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 0/1 1 4 1/4 0/1 2 4 1/3 1/1 2 4 1/2 (-1/1,1/1) 0 4 1/1 1/1 1 4 3/2 (1/1,3/1) 0 4 5/3 3/1 2 4 7/4 4/1 2 4 2/1 1/0 3 4 1/0 1/0 2 4 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(1,0,8,-1) (0/1,1/4) -> (0/1,1/4) Reflection Matrix(41,-12,24,-7) (1/4,1/3) -> (5/3,7/4) Hyperbolic Matrix(5,-2,12,-5) (1/3,1/2) -> (1/3,1/2) Reflection Matrix(3,-2,4,-3) (1/2,1/1) -> (1/2,1/1) Reflection Matrix(5,-6,4,-5) (1/1,3/2) -> (1/1,3/2) Reflection Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(15,-28,8,-15) (7/4,2/1) -> (7/4,2/1) Reflection Matrix(-1,4,0,1) (2/1,1/0) -> (2/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,8,-1) -> Matrix(1,0,2,-1) (0/1,1/4) -> (0/1,1/1) Matrix(41,-12,24,-7) -> Matrix(7,-4,2,-1) Matrix(5,-2,12,-5) -> Matrix(0,1,1,0) (1/3,1/2) -> (-1/1,1/1) Matrix(3,-2,4,-3) -> Matrix(0,1,1,0) (1/2,1/1) -> (-1/1,1/1) Matrix(5,-6,4,-5) -> Matrix(2,-3,1,-2) (1/1,3/2) -> (1/1,3/1) Matrix(19,-30,12,-19) -> Matrix(2,-3,1,-2) (3/2,5/3) -> (1/1,3/1) Matrix(15,-28,8,-15) -> Matrix(-1,8,0,1) (7/4,2/1) -> (4/1,1/0) Matrix(-1,4,0,1) -> Matrix(-1,2,0,1) (2/1,1/0) -> (1/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.