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 -1/2 -5/12 -1/6 0/1 1/4 1/3 3/8 1/2 2/3 5/6 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/12 -3/4 1/9 -2/3 1/8 -5/8 2/15 -3/5 5/36 -1/2 1/6 -3/7 7/36 -5/12 1/5 -2/5 5/24 -5/13 13/60 -3/8 2/9 -1/3 1/4 -1/4 1/3 -1/5 5/12 -1/6 1/2 -1/7 7/12 0/1 1/0 1/4 -1/3 1/3 -1/4 3/8 -2/9 2/5 -5/24 1/2 -1/6 4/7 -7/48 7/12 -1/7 3/5 -5/36 8/13 -13/96 5/8 -2/15 2/3 -1/8 3/4 -1/9 4/5 -5/48 5/6 -1/10 6/7 -7/72 1/1 -1/12 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(5,4,-24,-19) (-1/1,-3/4) -> (-1/4,-1/5) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(55,34,-144,-89) (-5/8,-3/5) -> (-5/13,-3/8) Hyperbolic Matrix(11,6,-24,-13) (-3/5,-1/2) -> (-1/2,-3/7) Parabolic Matrix(85,36,144,61) (-3/7,-5/12) -> (7/12,3/5) Hyperbolic Matrix(83,34,144,59) (-5/12,-2/5) -> (4/7,7/12) Hyperbolic Matrix(5,2,-48,-19) (-2/5,-5/13) -> (-1/7,0/1) Hyperbolic Matrix(17,6,48,17) (-3/8,-1/3) -> (1/3,3/8) Hyperbolic Matrix(7,2,24,7) (-1/3,-1/4) -> (1/4,1/3) Hyperbolic Matrix(11,2,-72,-13) (-1/5,-1/6) -> (-1/6,-1/7) Parabolic Matrix(19,-4,24,-5) (0/1,1/4) -> (3/4,4/5) Hyperbolic Matrix(89,-34,144,-55) (3/8,2/5) -> (8/13,5/8) Hyperbolic Matrix(13,-6,24,-11) (2/5,1/2) -> (1/2,4/7) Parabolic Matrix(43,-26,48,-29) (3/5,8/13) -> (6/7,1/1) Hyperbolic Matrix(61,-50,72,-59) (4/5,5/6) -> (5/6,6/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-24,1) Matrix(5,4,-24,-19) -> Matrix(19,-2,48,-5) Matrix(17,12,24,17) -> Matrix(17,-2,-144,17) Matrix(31,20,48,31) -> Matrix(31,-4,-240,31) Matrix(55,34,-144,-89) -> Matrix(89,-12,408,-55) Matrix(11,6,-24,-13) -> Matrix(13,-2,72,-11) Matrix(85,36,144,61) -> Matrix(61,-12,-432,85) Matrix(83,34,144,59) -> Matrix(59,-12,-408,83) Matrix(5,2,-48,-19) -> Matrix(19,-4,24,-5) Matrix(17,6,48,17) -> Matrix(17,-4,-72,17) Matrix(7,2,24,7) -> Matrix(7,-2,-24,7) Matrix(11,2,-72,-13) -> Matrix(13,-6,24,-11) Matrix(19,-4,24,-5) -> Matrix(5,2,-48,-19) Matrix(89,-34,144,-55) -> Matrix(55,12,-408,-89) Matrix(13,-6,24,-11) -> Matrix(11,2,-72,-13) Matrix(43,-26,48,-29) -> Matrix(29,4,-312,-43) Matrix(61,-50,72,-59) -> Matrix(59,6,-600,-61) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 96 Minimal number of generators: 17 Number of equivalence classes of cusps: 12 Genus: 3 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 16 Degree of the the map Y: 16 Permutation triple for Y: ((1,4,9,16,12,11,15,6,14,13,5,2)(3,8,7,10); (1,2,8,13,14,7,15,11,10,16,9,3)(4,12,6,5); (2,6,7)(3,11,4)(5,8,9)(10,14,12)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0, lambda1 DeckMod(f) is isomorphic to Z/2Z. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0, lambda1, lambda2, lambda1+lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z. ----------------------------------------------------------------------- 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 1/4 1/3 1/2 5/6 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 1/0 1/4 -1/3 1/3 -1/4 1/2 -1/6 2/3 -1/8 3/4 -1/9 4/5 -5/48 5/6 -1/10 1/1 -1/12 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(19,-4,24,-5) (0/1,1/4) -> (3/4,4/5) Hyperbolic Matrix(17,-5,24,-7) (1/4,1/3) -> (2/3,3/4) Hyperbolic Matrix(7,-3,12,-5) (1/3,1/2) -> (1/2,2/3) Parabolic Matrix(31,-25,36,-29) (4/5,5/6) -> (5/6,1/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-12,1) Matrix(19,-4,24,-5) -> Matrix(5,2,-48,-19) Matrix(17,-5,24,-7) -> Matrix(7,2,-60,-17) Matrix(7,-3,12,-5) -> Matrix(5,1,-36,-7) Matrix(31,-25,36,-29) -> Matrix(29,3,-300,-31) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM 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 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 1/0 1 12 1/6 -1/2 6 2 1/4 -1/3 4 3 1/3 -1/4 3 4 1/2 -1/6 2 6 1/0 0/1 12 1 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(1,0,12,-1) (0/1,1/6) -> (0/1,1/6) Reflection Matrix(5,-1,24,-5) (1/6,1/4) -> (1/6,1/4) Reflection Matrix(7,-2,24,-7) (1/4,1/3) -> (1/4,1/3) Reflection Matrix(5,-2,12,-5) (1/3,1/2) -> (1/3,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) 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(1,0,12,-1) -> Matrix(1,1,0,-1) (0/1,1/6) -> (-1/2,1/0) Matrix(5,-1,24,-5) -> Matrix(5,2,-12,-5) (1/6,1/4) -> (-1/2,-1/3) Matrix(7,-2,24,-7) -> Matrix(7,2,-24,-7) (1/4,1/3) -> (-1/3,-1/4) Matrix(5,-2,12,-5) -> Matrix(5,1,-24,-5) (1/3,1/2) -> (-1/4,-1/6) Matrix(-1,1,0,1) -> Matrix(-1,0,12,1) (1/2,1/0) -> (-1/6,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.