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 -3/5 -1/2 -1/3 0/1 1/4 1/3 1/2 1/1 5/3 2/1 3/1 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 0/1 2/1 -3/1 1/1 -8/3 4/3 2/1 -5/2 2/1 -2/1 2/1 -1/1 1/0 -2/3 0/1 -5/8 0/1 2/1 -3/5 1/0 -7/12 -2/1 0/1 -4/7 -2/1 0/1 -1/2 0/1 -2/5 0/1 -1/3 1/1 -2/7 2/1 -1/4 0/1 2/1 0/1 0/1 2/1 1/4 0/1 2/1 1/3 1/1 3/8 4/3 2/1 2/5 2/1 1/2 2/1 1/1 1/0 3/2 0/1 8/5 0/1 2/1 5/3 1/0 12/7 -2/1 0/1 7/4 -2/1 0/1 2/1 0/1 5/2 0/1 3/1 1/1 7/2 2/1 4/1 0/1 2/1 1/0 0/1 2/1 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(11,36,-4,-13) (-4/1,-3/1) -> (-3/1,-8/3) Parabolic Matrix(25,64,16,41) (-8/3,-5/2) -> (3/2,8/5) Hyperbolic Matrix(5,12,12,29) (-5/2,-2/1) -> (2/5,1/2) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(25,16,64,41) (-2/3,-5/8) -> (3/8,2/5) Hyperbolic Matrix(59,36,-100,-61) (-5/8,-3/5) -> (-3/5,-7/12) Parabolic Matrix(193,112,112,65) (-7/12,-4/7) -> (12/7,7/4) Hyperbolic Matrix(57,32,16,9) (-4/7,-1/2) -> (7/2,4/1) Hyperbolic Matrix(29,12,12,5) (-1/2,-2/5) -> (2/1,5/2) Hyperbolic Matrix(11,4,-36,-13) (-2/5,-1/3) -> (-1/3,-2/7) Parabolic 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(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(5,-4,4,-3) (1/2,1/1) -> (1/1,3/2) Parabolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,8,0,1) -> Matrix(1,0,0,1) Matrix(11,36,-4,-13) -> Matrix(3,-2,2,-1) Matrix(25,64,16,41) -> Matrix(1,-2,2,-3) Matrix(5,12,12,29) -> Matrix(1,0,0,1) Matrix(3,4,-4,-5) -> Matrix(1,-2,0,1) Matrix(25,16,64,41) -> Matrix(3,-2,2,-1) Matrix(59,36,-100,-61) -> Matrix(1,-2,0,1) Matrix(193,112,112,65) -> Matrix(1,0,0,1) Matrix(57,32,16,9) -> Matrix(1,2,0,1) Matrix(29,12,12,5) -> Matrix(1,0,0,1) Matrix(11,4,-36,-13) -> Matrix(3,-2,2,-1) Matrix(57,16,32,9) -> Matrix(1,-2,0,1) Matrix(1,0,8,1) -> Matrix(1,0,0,1) Matrix(13,-4,36,-11) -> Matrix(3,-2,2,-1) Matrix(5,-4,4,-3) -> Matrix(1,-2,0,1) Matrix(61,-100,36,-59) -> Matrix(1,-2,0,1) Matrix(13,-36,4,-11) -> Matrix(3,-2,2,-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: ((1,6,7,2)(3,9,15,10)(4,8,13,5)(11,14,16,12); (1,5)(2,3)(4,12)(6,15)(7,8)(9,16)(10,11)(13,14); (1,3,11,4)(2,8,12,9)(5,14,10,6)(7,15,16,13)) ----------------------------------------------------------------------- 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): 12 Minimal number of generators: 4 Number of equivalence classes of elliptic points of order 2: 2 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 1/1 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 2/1 1/2 2/1 1/1 1/0 3/2 0/1 2/1 0/1 1/0 0/1 2/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,0) (0/1,1/0) -> (0/1,1/0) Elliptic Matrix(4,-1,1,0) (0/1,1/2) -> (2/1,1/0) Hyperbolic Matrix(5,-4,4,-3) (1/2,1/1) -> (1/1,3/2) Parabolic Matrix(8,-13,5,-8) (3/2,2/1) -> (3/2,2/1) Elliptic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,0) -> Matrix(1,0,0,1) Matrix(4,-1,1,0) -> Matrix(1,-2,1,-1) Matrix(5,-4,4,-3) -> Matrix(1,-2,0,1) Matrix(8,-13,5,-8) -> Matrix(1,0,0,1) 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): 1 ----------------------------------------------------------------------- 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 This is a reflection group. CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 1/1 1/0 2 4 2/1 0/1 1 4 1/0 (0/1,2/1).(1/1,1/0) 0 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(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(4,-5,3,-4) (1/1,5/3) -> (1/1,5/3) Reflection Matrix(7,-12,4,-7) (3/2,2/1) -> (3/2,2/1) 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(-1,2,0,1) (0/1,1/0) -> (1/1,1/0) Matrix(0,1,1,0) -> Matrix(-1,2,0,1) (-1/1,1/1) -> (1/1,1/0) Matrix(4,-5,3,-4) -> Matrix(1,0,0,-1) (1/1,5/3) -> (0/1,1/0) Matrix(7,-12,4,-7) -> Matrix(1,0,0,-1) (3/2,2/1) -> (0/1,1/0) Matrix(-1,4,0,1) -> Matrix(1,0,1,-1) (2/1,1/0) -> (0/1,2/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.