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 -2/1 -3/2 -5/4 -6/5 -1/1 -3/4 0/1 1/2 3/5 3/4 1/1 3/2 2/1 5/2 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 -2/1 0/1 -5/2 -1/1 -12/5 0/1 -7/3 -2/1 0/1 -2/1 0/1 -3/2 1/0 -4/3 -2/1 -5/4 1/0 -6/5 -2/1 -1/1 -2/1 0/1 -3/4 -1/1 -5/7 -2/3 0/1 -2/3 0/1 -5/8 1/0 -3/5 -2/1 0/1 -1/2 -1/1 0/1 0/1 1/2 1/1 3/5 0/1 2/1 5/8 1/0 2/3 0/1 3/4 1/1 4/5 2/1 1/1 0/1 2/1 3/2 1/0 5/3 -2/1 0/1 12/7 0/1 7/4 1/0 2/1 0/1 7/3 0/1 2/1 12/5 0/1 5/2 1/1 3/1 0/1 2/1 1/0 1/0 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(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(61,144,36,85) (-12/5,-7/3) -> (5/3,12/7) Hyperbolic Matrix(13,30,16,37) (-7/3,-2/1) -> (4/5,1/1) Hyperbolic Matrix(11,18,-8,-13) (-2/1,-3/2) -> (-3/2,-4/3) Parabolic Matrix(23,30,36,47) (-4/3,-5/4) -> (5/8,2/3) Hyperbolic Matrix(83,102,48,59) (-5/4,-6/5) -> (12/7,7/4) Hyperbolic Matrix(47,54,20,23) (-6/5,-1/1) -> (7/3,12/5) Hyperbolic Matrix(23,18,-32,-25) (-1/1,-3/4) -> (-3/4,-5/7) Parabolic Matrix(35,24,16,11) (-5/7,-2/3) -> (2/1,7/3) Hyperbolic Matrix(37,24,20,13) (-2/3,-5/8) -> (7/4,2/1) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(11,6,20,11) (-3/5,-1/2) -> (1/2,3/5) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(25,-18,32,-23) (2/3,3/4) -> (3/4,4/5) Parabolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,2,0,1) Matrix(11,30,4,11) -> Matrix(1,2,0,1) Matrix(49,120,20,49) -> Matrix(1,0,2,1) Matrix(61,144,36,85) -> Matrix(1,0,0,1) Matrix(13,30,16,37) -> Matrix(1,2,0,1) Matrix(11,18,-8,-13) -> Matrix(1,-2,0,1) Matrix(23,30,36,47) -> Matrix(1,2,0,1) Matrix(83,102,48,59) -> Matrix(1,2,0,1) Matrix(47,54,20,23) -> Matrix(1,2,0,1) Matrix(23,18,-32,-25) -> Matrix(1,2,-2,-3) Matrix(35,24,16,11) -> Matrix(1,0,2,1) Matrix(37,24,20,13) -> Matrix(1,0,0,1) Matrix(49,30,80,49) -> Matrix(1,2,0,1) Matrix(11,6,20,11) -> Matrix(1,2,0,1) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(25,-18,32,-23) -> Matrix(3,-2,2,-1) Matrix(13,-18,8,-11) -> Matrix(1,-2,0,1) 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): 4 Degree of the the map X: 4 Degree of the the map Y: 16 Permutation triple for Y: ((1,2)(3,9,14,6,13,10)(4,11,8,7,12,5)(15,16); (1,5,12,16,13,6)(2,8,11,15,9,3)(4,10)(7,14); (1,3,4)(2,6,7)(10,16,11)(12,14,15)) ----------------------------------------------------------------------- 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): 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 1/1 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 -2/1 0/1 -2/1 0/1 -3/2 1/0 -1/1 -2/1 0/1 0/1 0/1 1/1 0/1 2/1 3/2 1/0 2/1 0/1 3/1 0/1 2/1 1/0 1/0 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(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(5,6,4,5) (-3/2,-1/1) -> (1/1,3/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,2,0,1) Matrix(5,12,2,5) -> Matrix(1,0,1,1) Matrix(7,12,4,7) -> Matrix(1,0,0,1) Matrix(5,6,4,5) -> Matrix(1,2,0,1) Matrix(1,0,2,1) -> Matrix(1,0,1,1) 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): 2 ----------------------------------------------------------------------- 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 1 2 1/1 (0/1,2/1).(1/1,1/0) 0 6 3/2 1/0 2 2 2/1 0/1 1 6 3/1 (0/1,2/1).(1/1,1/0) 0 2 1/0 1/0 2 6 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(5,-6,4,-5) (1/1,3/2) -> (1/1,3/2) Reflection Matrix(7,-12,4,-7) (3/2,2/1) -> (3/2,2/1) Reflection Matrix(5,-12,2,-5) (2/1,3/1) -> (2/1,3/1) Reflection Matrix(-1,6,0,1) (3/1,1/0) -> (3/1,1/0) 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(1,0,2,-1) -> Matrix(1,0,1,-1) (0/1,1/1) -> (0/1,2/1) Matrix(5,-6,4,-5) -> Matrix(-1,2,0,1) (1/1,3/2) -> (1/1,1/0) Matrix(7,-12,4,-7) -> Matrix(1,0,0,-1) (3/2,2/1) -> (0/1,1/0) Matrix(5,-12,2,-5) -> Matrix(1,0,1,-1) (2/1,3/1) -> (0/1,2/1) Matrix(-1,6,0,1) -> Matrix(-1,2,0,1) (3/1,1/0) -> (1/1,1/0) ----------------------------------------------------------------------- The pullback map seems to have extra symmetries. ONE REPRESENTATIVE FROM EVERY RIGHT COSET OF THE MODULAR GROUP LIFTABLES IN THE GROUP OF POSSIBLE SYMMETRIES OF THE PULLBACK MAP AND ITS IMAGE UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,0,1) -> Matrix(1,0,0,1) Matrix(2,-3,1,-2) -> Matrix(0,2,1,0) Matrix(0,3,1,0) -> Matrix(0,2,1,0) Matrix(3,-6,2,-3) -> Matrix(1,0,0,1) THE REFLECTIONS WHICH APPEAR IN THE POSTSCRIPT FILE FOR THE PULLBACK MAP THE REFLECTION AND ITS IMAGE MAP ON REFLECTION AXES UNDER THE VIRTUAL ENDOMORPHISM Matrix(2,-3,1,-2) -> Matrix(0,2,1,0) (1/1,3/1) -> (-sqrt(2),sqrt(2)) Matrix(0,3,1,0) -> Matrix(0,2,1,0) (-sqrt(3),sqrt(3)) -> (-sqrt(2),sqrt(2))