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 -4/1 -10/3 -2/1 -4/3 0/1 1/1 4/3 3/2 2/1 8/3 3/1 4/1 16/3 6/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 1/0 -7/2 1/0 -10/3 -4/1 -2/1 -3/1 -3/1 -2/1 -2/1 0/1 -5/3 -3/1 -8/5 -2/1 -3/2 -3/2 -4/3 -1/1 -5/4 1/0 -16/13 -1/1 1/0 -11/9 -1/1 -6/5 -2/1 0/1 -7/6 1/0 -8/7 -2/1 -1/1 -1/1 0/1 -1/1 1/0 1/1 -1/1 4/3 -1/1 7/5 -1/1 10/7 -2/3 0/1 3/2 -1/2 2/1 -2/1 0/1 5/2 -1/2 8/3 0/1 3/1 1/1 4/1 1/0 5/1 -1/1 16/3 -1/1 1/0 11/2 1/0 6/1 -2/1 0/1 7/1 -1/1 8/1 0/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,32,-2,-9) (-4/1,1/0) -> (-4/1,-7/2) Parabolic Matrix(33,112,-28,-95) (-7/2,-10/3) -> (-6/5,-7/6) Hyperbolic Matrix(39,128,-32,-105) (-10/3,-3/1) -> (-11/9,-6/5) Hyperbolic Matrix(7,16,-4,-9) (-3/1,-2/1) -> (-2/1,-5/3) Parabolic Matrix(39,64,14,23) (-5/3,-8/5) -> (8/3,3/1) Hyperbolic Matrix(41,64,16,25) (-8/5,-3/2) -> (5/2,8/3) Hyperbolic Matrix(23,32,-18,-25) (-3/2,-4/3) -> (-4/3,-5/4) Parabolic Matrix(207,256,38,47) (-5/4,-16/13) -> (16/3,11/2) Hyperbolic Matrix(209,256,40,49) (-16/13,-11/9) -> (5/1,16/3) Hyperbolic Matrix(55,64,6,7) (-7/6,-8/7) -> (8/1,1/0) Hyperbolic Matrix(57,64,8,9) (-8/7,-1/1) -> (7/1,8/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(25,-32,18,-23) (1/1,4/3) -> (4/3,7/5) Parabolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) Hyperbolic Matrix(89,-128,16,-23) (10/7,3/2) -> (11/2,6/1) Hyperbolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,32,-2,-9) -> Matrix(1,-2,0,1) Matrix(33,112,-28,-95) -> Matrix(1,2,0,1) Matrix(39,128,-32,-105) -> Matrix(1,2,0,1) Matrix(7,16,-4,-9) -> Matrix(1,0,0,1) Matrix(39,64,14,23) -> Matrix(1,2,2,5) Matrix(41,64,16,25) -> Matrix(1,2,-4,-7) Matrix(23,32,-18,-25) -> Matrix(1,2,-2,-3) Matrix(207,256,38,47) -> Matrix(1,0,0,1) Matrix(209,256,40,49) -> Matrix(1,0,0,1) Matrix(55,64,6,7) -> Matrix(1,2,0,1) Matrix(57,64,8,9) -> Matrix(1,2,-2,-3) Matrix(1,0,2,1) -> Matrix(1,0,0,1) Matrix(25,-32,18,-23) -> Matrix(1,2,-2,-3) Matrix(79,-112,12,-17) -> Matrix(3,2,-2,-1) Matrix(89,-128,16,-23) -> Matrix(3,2,-2,-1) Matrix(9,-16,4,-7) -> Matrix(1,0,0,1) Matrix(9,-32,2,-7) -> 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: ((2,6)(3,4)(11,12)(13,14); (1,4,10,13,15,11,5,2)(3,8,14,16,12,7,6,9); (1,2,7,12,15,13,8,3)(4,9,6,5,11,16,14,10)) ----------------------------------------------------------------------- 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 0/1 4/3 2/1 8/3 3/1 4/1 16/3 6/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/1 1/0 1/1 -1/1 4/3 -1/1 7/5 -1/1 10/7 -2/3 0/1 3/2 -1/2 2/1 -2/1 0/1 5/2 -1/2 8/3 0/1 3/1 1/1 4/1 1/0 5/1 -1/1 16/3 -1/1 1/0 11/2 1/0 6/1 -2/1 0/1 7/1 -1/1 8/1 0/1 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(25,-32,18,-23) (1/1,4/3) -> (4/3,7/5) Parabolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) Hyperbolic Matrix(89,-128,16,-23) (10/7,3/2) -> (11/2,6/1) Hyperbolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(25,-64,9,-23) (5/2,8/3) -> (8/3,3/1) Parabolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic Matrix(49,-256,9,-47) (5/1,16/3) -> (16/3,11/2) Parabolic Matrix(9,-64,1,-7) (7/1,8/1) -> (8/1,1/0) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,2,-1,-1) Matrix(25,-32,18,-23) -> Matrix(1,2,-2,-3) Matrix(79,-112,12,-17) -> Matrix(3,2,-2,-1) Matrix(89,-128,16,-23) -> Matrix(3,2,-2,-1) Matrix(9,-16,4,-7) -> Matrix(1,0,0,1) Matrix(25,-64,9,-23) -> Matrix(1,0,3,1) Matrix(9,-32,2,-7) -> Matrix(1,-2,0,1) Matrix(49,-256,9,-47) -> Matrix(1,2,-1,-1) Matrix(9,-64,1,-7) -> 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): 4 ----------------------------------------------------------------------- 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 (-2/1,0/1).(-1/1,1/0) 0 2 2/1 (-2/1,0/1) 0 4 8/3 0/1 6 2 3/1 1/1 1 8 4/1 1/0 2 2 5/1 -1/1 1 8 16/3 (-2/1,0/1).(-1/1,1/0) 0 2 6/1 (-2/1,0/1) 0 4 8/1 0/1 2 2 1/0 1/0 1 8 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(7,-16,3,-7) (2/1,8/3) -> (2/1,8/3) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) Reflection Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic Matrix(31,-160,6,-31) (5/1,16/3) -> (5/1,16/3) Reflection Matrix(17,-96,3,-17) (16/3,6/1) -> (16/3,6/1) Reflection Matrix(7,-48,1,-7) (6/1,8/1) -> (6/1,8/1) Reflection Matrix(-1,16,0,1) (8/1,1/0) -> (8/1,1/0) 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(1,0,1,-1) -> Matrix(-1,0,1,1) (0/1,2/1) -> (-2/1,0/1) Matrix(7,-16,3,-7) -> Matrix(-1,0,1,1) (2/1,8/3) -> (-2/1,0/1) Matrix(17,-48,6,-17) -> Matrix(1,0,2,-1) (8/3,3/1) -> (0/1,1/1) Matrix(9,-32,2,-7) -> Matrix(1,-2,0,1) 1/0 Matrix(31,-160,6,-31) -> Matrix(1,2,0,-1) (5/1,16/3) -> (-1/1,1/0) Matrix(17,-96,3,-17) -> Matrix(-1,0,1,1) (16/3,6/1) -> (-2/1,0/1) Matrix(7,-48,1,-7) -> Matrix(-1,0,1,1) (6/1,8/1) -> (-2/1,0/1) Matrix(-1,16,0,1) -> Matrix(1,0,0,-1) (8/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.