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: 12 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -4/1 0/1 1/1 3/2 2/1 3/1 10/3 24/7 4/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 0/1 1/0 -5/1 1/0 -4/1 -1/1 0/1 1/0 -7/2 1/0 -10/3 -1/1 0/1 -3/1 -1/2 1/0 -2/1 -1/1 0/1 -3/2 -1/2 1/0 -10/7 -1/1 0/1 -17/12 1/0 -24/17 -1/1 -7/5 -1/2 -4/3 -1/1 -1/2 0/1 -5/4 -1/2 -6/5 -1/1 -1/2 0/1 -1/1 -1/2 0/1 0/1 1/1 1/2 6/5 0/1 1/2 1/1 5/4 1/2 4/3 0/1 1/2 1/1 7/5 1/2 10/7 0/1 1/1 3/2 1/2 1/0 2/1 0/1 1/1 3/1 1/2 1/0 10/3 0/1 1/1 17/5 1/2 24/7 1/1 7/2 1/0 4/1 0/1 1/1 1/0 5/1 1/0 6/1 0/1 1/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,36,4,29) (-6/1,1/0) -> (6/5,5/4) Hyperbolic Matrix(7,36,6,31) (-6/1,-5/1) -> (1/1,6/5) Hyperbolic Matrix(11,48,8,35) (-5/1,-4/1) -> (4/3,7/5) Hyperbolic Matrix(13,48,10,37) (-4/1,-7/2) -> (5/4,4/3) Hyperbolic Matrix(71,240,-50,-169) (-7/2,-10/3) -> (-10/7,-17/12) Hyperbolic Matrix(19,60,6,19) (-10/3,-3/1) -> (3/1,10/3) Hyperbolic Matrix(5,12,2,5) (-3/1,-2/1) -> (2/1,3/1) Hyperbolic Matrix(7,12,4,7) (-2/1,-3/2) -> (3/2,2/1) Hyperbolic Matrix(41,60,28,41) (-3/2,-10/7) -> (10/7,3/2) Hyperbolic Matrix(407,576,118,167) (-17/12,-24/17) -> (24/7,7/2) Hyperbolic Matrix(409,576,120,169) (-24/17,-7/5) -> (17/5,24/7) Hyperbolic Matrix(35,48,8,11) (-7/5,-4/3) -> (4/1,5/1) Hyperbolic Matrix(37,48,10,13) (-4/3,-5/4) -> (7/2,4/1) Hyperbolic Matrix(29,36,4,5) (-5/4,-6/5) -> (6/1,1/0) Hyperbolic Matrix(31,36,6,7) (-6/5,-1/1) -> (5/1,6/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,36,4,29) -> Matrix(1,0,2,1) Matrix(7,36,6,31) -> Matrix(1,0,2,1) Matrix(11,48,8,35) -> Matrix(1,0,2,1) Matrix(13,48,10,37) -> Matrix(1,0,2,1) Matrix(71,240,-50,-169) -> Matrix(1,0,0,1) Matrix(19,60,6,19) -> Matrix(1,0,2,1) Matrix(5,12,2,5) -> Matrix(1,0,2,1) Matrix(7,12,4,7) -> Matrix(1,0,2,1) Matrix(41,60,28,41) -> Matrix(1,0,2,1) Matrix(407,576,118,167) -> Matrix(1,2,0,1) Matrix(409,576,120,169) -> Matrix(3,2,4,3) Matrix(35,48,8,11) -> Matrix(1,0,2,1) Matrix(37,48,10,13) -> Matrix(1,0,2,1) Matrix(29,36,4,5) -> Matrix(1,0,2,1) Matrix(31,36,6,7) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(169,-240,50,-71) -> 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): 2 Degree of the the map X: 2 Degree of the the map Y: 16 Permutation triple for Y: ((2,4,3)(5,8,7)(6,10,9)(12,13,15); (1,4,10,14,9,15,16,12,7,11,5,2)(3,8,13,6); (1,2,6,14,10,13,16,15,8,11,7,3)(4,5,12,9)) ----------------------------------------------------------------------- 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): 24 Minimal number of generators: 5 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: 6 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 3/1 4/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/2 4/3 0/1 1/2 1/1 3/2 1/2 1/0 2/1 0/1 1/1 3/1 1/2 1/0 4/1 0/1 1/1 1/0 5/1 1/0 6/1 0/1 1/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic Matrix(17,-24,5,-7) (4/3,3/2) -> (3/1,4/1) Hyperbolic Matrix(7,-12,3,-5) (3/2,2/1) -> (2/1,3/1) Parabolic Matrix(7,-36,1,-5) (5/1,6/1) -> (6/1,1/0) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,2,1) Matrix(19,-24,4,-5) -> Matrix(1,-1,2,-1) Matrix(17,-24,5,-7) -> Matrix(1,-1,2,-1) Matrix(7,-12,3,-5) -> Matrix(1,0,0,1) Matrix(7,-36,1,-5) -> 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 0/1 0/1 2 1 2/1 (0/1,1/1) 0 6 3/1 (0/1,1/1).(1/2,1/0) 0 4 4/1 (1/2,1/0) 0 3 6/1 (1/2,1/0) 0 2 1/0 1/0 1 12 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(5,-12,2,-5) (2/1,3/1) -> (2/1,3/1) Reflection Matrix(7,-24,2,-7) (3/1,4/1) -> (3/1,4/1) Reflection Matrix(5,-24,1,-5) (4/1,6/1) -> (4/1,6/1) Reflection Matrix(-1,12,0,1) (6/1,1/0) -> (6/1,1/0) 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(1,0,1,-1) -> Matrix(1,0,2,-1) (0/1,2/1) -> (0/1,1/1) Matrix(5,-12,2,-5) -> Matrix(1,0,2,-1) (2/1,3/1) -> (0/1,1/1) Matrix(7,-24,2,-7) -> Matrix(-1,1,0,1) (3/1,4/1) -> (1/2,1/0) Matrix(5,-24,1,-5) -> Matrix(-1,1,0,1) (4/1,6/1) -> (1/2,1/0) Matrix(-1,12,0,1) -> Matrix(-1,1,0,1) (6/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.