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 -5/2 -2/1 -1/1 -1/2 -2/5 -1/4 0/1 1/2 3/4 1/1 5/4 7/5 3/2 2/1 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 -1/2 -5/2 0/1 -7/3 1/0 -2/1 -1/2 1/0 -1/1 -1/2 1/0 -2/3 -1/2 1/0 -5/8 -1/1 -3/5 -1/2 -1/2 -1/1 0/1 -3/7 -1/2 -2/5 0/1 -3/8 0/1 1/0 -4/11 -1/2 1/0 -1/3 1/0 -1/4 -1/1 -1/5 -1/2 0/1 -1/2 1/0 1/2 0/1 2/3 1/2 1/0 5/7 1/0 3/4 0/1 1/0 4/5 0/1 1/1 1/0 5/4 -1/1 4/3 -1/2 1/0 11/8 0/1 1/0 7/5 1/0 3/2 -1/1 1/0 2/1 -1/1 5/2 -1/1 -1/2 13/5 -1/2 8/3 -1/2 1/0 3/1 -1/2 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,6,0,1) (-3/1,1/0) -> (3/1,1/0) Parabolic Matrix(19,50,-8,-21) (-3/1,-5/2) -> (-5/2,-7/3) Parabolic Matrix(13,30,-36,-83) (-7/3,-2/1) -> (-4/11,-1/3) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(47,30,36,23) (-2/3,-5/8) -> (5/4,4/3) Hyperbolic Matrix(33,20,28,17) (-5/8,-3/5) -> (1/1,5/4) Hyperbolic Matrix(11,6,-24,-13) (-3/5,-1/2) -> (-1/2,-3/7) Parabolic Matrix(33,14,40,17) (-3/7,-2/5) -> (4/5,1/1) Hyperbolic Matrix(47,18,60,23) (-2/5,-3/8) -> (3/4,4/5) Hyperbolic Matrix(153,56,112,41) (-3/8,-4/11) -> (4/3,11/8) Hyperbolic Matrix(7,2,-32,-9) (-1/3,-1/4) -> (-1/4,-1/5) Parabolic Matrix(43,8,16,3) (-1/5,0/1) -> (8/3,3/1) Hyperbolic Matrix(5,-2,8,-3) (0/1,1/2) -> (1/2,2/3) Parabolic Matrix(95,-66,36,-25) (2/3,5/7) -> (13/5,8/3) Hyperbolic Matrix(105,-76,76,-55) (5/7,3/4) -> (11/8,7/5) Hyperbolic Matrix(51,-74,20,-29) (7/5,3/2) -> (5/2,13/5) Hyperbolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,0,0,1) Matrix(19,50,-8,-21) -> Matrix(1,0,2,1) Matrix(13,30,-36,-83) -> Matrix(1,0,0,1) Matrix(3,4,-4,-5) -> Matrix(1,0,0,1) Matrix(47,30,36,23) -> Matrix(1,0,0,1) Matrix(33,20,28,17) -> Matrix(3,2,-2,-1) Matrix(11,6,-24,-13) -> Matrix(1,0,0,1) Matrix(33,14,40,17) -> Matrix(1,0,2,1) Matrix(47,18,60,23) -> Matrix(1,0,0,1) Matrix(153,56,112,41) -> Matrix(1,0,0,1) Matrix(7,2,-32,-9) -> Matrix(1,2,-2,-3) Matrix(43,8,16,3) -> Matrix(1,0,0,1) Matrix(5,-2,8,-3) -> Matrix(1,0,2,1) Matrix(95,-66,36,-25) -> Matrix(1,0,-2,1) Matrix(105,-76,76,-55) -> Matrix(1,0,0,1) Matrix(51,-74,20,-29) -> Matrix(1,2,-2,-3) Matrix(9,-16,4,-7) -> Matrix(1,2,-2,-3) 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: ((1,6,15,11,7,2)(3,9,13,12,16,10)(4,5)(8,14); (1,5,13,9,14,6)(2,3)(4,10,16,8,7,11)(12,15); (1,3,4)(2,8,9)(5,11,12)(14,16,15)) ----------------------------------------------------------------------- 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: 10 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -1/4 0/1 1/2 1/1 7/5 3/2 2/1 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/2 1/0 -1/2 -1/1 0/1 -1/3 1/0 -1/4 -1/1 -1/5 -1/2 0/1 -1/2 1/0 1/2 0/1 2/3 1/2 1/0 1/1 1/0 4/3 -1/2 1/0 7/5 1/0 3/2 -1/1 1/0 2/1 -1/1 5/2 -1/1 -1/2 13/5 -1/2 8/3 -1/2 1/0 3/1 -1/2 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(9,4,2,1) (-1/2,-1/3) -> (3/1,1/0) Hyperbolic Matrix(7,2,-32,-9) (-1/3,-1/4) -> (-1/4,-1/5) Parabolic Matrix(43,8,16,3) (-1/5,0/1) -> (8/3,3/1) Hyperbolic Matrix(5,-2,8,-3) (0/1,1/2) -> (1/2,2/3) Parabolic Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic Matrix(79,-108,30,-41) (4/3,7/5) -> (13/5,8/3) Hyperbolic Matrix(51,-74,20,-29) (7/5,3/2) -> (5/2,13/5) Hyperbolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,1,-2,-1) Matrix(9,4,2,1) -> Matrix(1,1,-2,-1) Matrix(7,2,-32,-9) -> Matrix(1,2,-2,-3) Matrix(43,8,16,3) -> Matrix(1,0,0,1) Matrix(5,-2,8,-3) -> Matrix(1,0,2,1) Matrix(7,-6,6,-5) -> Matrix(1,-1,0,1) Matrix(79,-108,30,-41) -> Matrix(1,1,-2,-1) Matrix(51,-74,20,-29) -> Matrix(1,2,-2,-3) Matrix(9,-16,4,-7) -> Matrix(1,2,-2,-3) 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): 3 ----------------------------------------------------------------------- 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).(-1/2,1/0) 0 2 0/1 (-1/2,1/0) 0 6 1/2 0/1 2 2 2/3 (1/2,1/0) 0 6 1/1 1/0 1 6 4/3 (-1/2,1/0) 0 6 7/5 1/0 1 2 3/2 (-1/1,1/0) 0 6 2/1 -1/1 1 2 1/0 (-1/1,0/1) 0 6 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(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(5,-2,8,-3) (0/1,1/2) -> (1/2,2/3) Parabolic Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic Matrix(41,-56,30,-41) (4/3,7/5) -> (4/3,7/5) Reflection Matrix(29,-42,20,-29) (7/5,3/2) -> (7/5,3/2) 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(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(-1,0,2,1) -> Matrix(1,1,0,-1) (-1/1,0/1) -> (-1/2,1/0) Matrix(5,-2,8,-3) -> Matrix(1,0,2,1) 0/1 Matrix(7,-6,6,-5) -> Matrix(1,-1,0,1) 1/0 Matrix(41,-56,30,-41) -> Matrix(1,1,0,-1) (4/3,7/5) -> (-1/2,1/0) Matrix(29,-42,20,-29) -> Matrix(1,2,0,-1) (7/5,3/2) -> (-1/1,1/0) Matrix(7,-12,4,-7) -> Matrix(1,2,0,-1) (3/2,2/1) -> (-1/1,1/0) Matrix(-1,4,0,1) -> Matrix(-1,0,2,1) (2/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.