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): 48 Minimal number of generators: 9 Number of equivalence classes of cusps: 10 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -2/1 -4/3 -6/5 0/1 1/1 2/1 4/1 6/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 1/0 -3/2 -1/1 1/1 -4/3 1/1 -5/4 1/1 5/3 -6/5 2/1 -7/6 1/1 3/1 -8/7 2/1 -1/1 1/1 3/1 0/1 1/0 1/1 -3/1 -1/1 2/1 0/1 3/1 3/5 1/1 4/1 1/1 5/1 1/1 9/7 6/1 3/2 7/1 5/3 9/5 8/1 2/1 1/0 1/1 3/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,8,-2,-5) (-2/1,1/0) -> (-2/1,-3/2) Parabolic Matrix(23,32,-18,-25) (-3/2,-4/3) -> (-4/3,-5/4) Parabolic Matrix(59,72,-50,-61) (-5/4,-6/5) -> (-6/5,-7/6) Parabolic 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(5,-8,2,-3) (1/1,2/1) -> (2/1,3/1) Parabolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(3,8,-2,-5) -> Matrix(1,-2,0,1) Matrix(23,32,-18,-25) -> Matrix(3,-2,2,-1) Matrix(59,72,-50,-61) -> Matrix(5,-8,2,-3) Matrix(55,64,6,7) -> Matrix(1,0,0,1) Matrix(57,64,8,9) -> Matrix(7,-16,4,-9) Matrix(1,0,2,1) -> Matrix(1,-4,0,1) Matrix(5,-8,2,-3) -> Matrix(1,0,2,1) Matrix(9,-32,2,-7) -> Matrix(7,-6,6,-5) Matrix(13,-72,2,-11) -> Matrix(13,-18,8,-11) 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: 8 Permutation triple for Y: (id;(1,3,5,7,8,6,4,2);(1,2,4,6,8,7,5,3)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0, lambda1+lambda2 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 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 -2/1 -4/3 0/1 2/1 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 1/0 -3/2 -1/1 1/1 -4/3 1/1 -1/1 1/1 3/1 0/1 1/0 1/1 -3/1 -1/1 2/1 0/1 3/1 3/5 1/1 4/1 1/1 1/0 1/1 3/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,8,-2,-5) (-2/1,1/0) -> (-2/1,-3/2) Parabolic Matrix(11,16,-9,-13) (-3/2,-4/3) -> (-4/3,-1/1) Parabolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(5,-8,2,-3) (1/1,2/1) -> (2/1,3/1) Parabolic Matrix(5,-16,1,-3) (3/1,4/1) -> (4/1,1/0) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(3,8,-2,-5) -> Matrix(1,-2,0,1) Matrix(11,16,-9,-13) -> Matrix(2,-1,1,0) Matrix(1,0,2,1) -> Matrix(1,-4,0,1) Matrix(5,-8,2,-3) -> Matrix(1,0,2,1) Matrix(5,-16,1,-3) -> Matrix(4,-3,3,-2) 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 This is a reflection group. CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d -4/1 3/1 2 2 -2/1 1/0 4 4 0/1 1/0 4 2 2/1 0/1 4 4 4/1 1/1 6 2 1/0 (1/1,3/1) 0 8 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,8,0,-1) (-4/1,1/0) -> (-4/1,1/0) Reflection Matrix(3,8,-1,-3) (-4/1,-2/1) -> (-4/1,-2/1) Reflection Matrix(-1,0,1,1) (-2/1,0/1) -> (-2/1,0/1) Reflection Matrix(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(3,-8,1,-3) (2/1,4/1) -> (2/1,4/1) Reflection Matrix(-1,8,0,1) (4/1,1/0) -> (4/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,8,0,-1) -> Matrix(2,-3,1,-2) (-4/1,1/0) -> (1/1,3/1) Matrix(3,8,-1,-3) -> Matrix(-1,6,0,1) (-4/1,-2/1) -> (3/1,1/0) Matrix(-1,0,1,1) -> Matrix(-1,4,0,1) (-2/1,0/1) -> (2/1,1/0) Matrix(1,0,1,-1) -> Matrix(1,0,0,-1) (0/1,2/1) -> (0/1,1/0) Matrix(3,-8,1,-3) -> Matrix(1,0,2,-1) (2/1,4/1) -> (0/1,1/1) Matrix(-1,8,0,1) -> Matrix(2,-3,1,-2) (4/1,1/0) -> (1/1,3/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.