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): 72 Minimal number of generators: 13 Number of equivalence classes of cusps: 12 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 5/4 3/2 5/3 2/1 5/2 3/1 10/3 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 0/1 -4/1 0/1 1/1 -3/1 0/1 1/0 -5/2 -1/1 1/1 -2/1 0/1 1/0 -5/3 1/0 -3/2 -1/1 1/0 -10/7 1/0 -7/5 -2/1 1/0 -4/3 -2/1 -1/1 -5/4 -1/1 -1/1 -1/1 0/1 0/1 0/1 1/1 0/1 1/1 5/4 1/1 4/3 1/1 2/1 3/2 1/1 1/0 5/3 1/0 2/1 0/1 1/0 5/2 -1/1 1/1 3/1 0/1 1/0 10/3 1/0 7/2 -1/1 1/0 4/1 -1/1 0/1 5/1 0/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,10,0,1) (-5/1,1/0) -> (5/1,1/0) Parabolic Matrix(9,40,2,9) (-5/1,-4/1) -> (4/1,5/1) Hyperbolic Matrix(11,40,-8,-29) (-4/1,-3/1) -> (-7/5,-4/3) Hyperbolic Matrix(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(9,20,4,9) (-5/2,-2/1) -> (2/1,5/2) Hyperbolic Matrix(11,20,6,11) (-2/1,-5/3) -> (5/3,2/1) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(69,100,20,29) (-3/2,-10/7) -> (10/3,7/2) Hyperbolic Matrix(71,100,22,31) (-10/7,-7/5) -> (3/1,10/3) Hyperbolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(9,10,8,9) (-5/4,-1/1) -> (1/1,5/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,10,0,1) -> Matrix(1,0,0,1) Matrix(9,40,2,9) -> Matrix(1,0,-2,1) Matrix(11,40,-8,-29) -> Matrix(1,-2,0,1) Matrix(11,30,4,11) -> Matrix(1,0,0,1) Matrix(9,20,4,9) -> Matrix(1,0,0,1) Matrix(11,20,6,11) -> Matrix(1,0,0,1) Matrix(19,30,12,19) -> Matrix(1,2,0,1) Matrix(69,100,20,29) -> Matrix(1,0,0,1) Matrix(71,100,22,31) -> Matrix(1,2,0,1) Matrix(31,40,24,31) -> Matrix(3,4,2,3) Matrix(9,10,8,9) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,2,1) Matrix(29,-40,8,-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): 2 Degree of the the map X: 2 Degree of the the map Y: 12 Permutation triple for Y: ((2,6,4,3,7)(5,11,10,9,8); (1,4,11,5,2)(3,10,12,8,7); (1,2,8,9,3)(4,6,5,12,10)) ----------------------------------------------------------------------- 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): 36 Minimal number of generators: 7 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: 8 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 5/4 3/2 5/3 2/1 5/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 0/1 1/0 -5/3 1/0 -3/2 -1/1 1/0 -1/1 -1/1 0/1 0/1 0/1 1/1 0/1 1/1 5/4 1/1 4/3 1/1 2/1 3/2 1/1 1/0 5/3 1/0 2/1 0/1 1/0 5/2 -1/1 1/1 3/1 0/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,5,0,1) (-2/1,1/0) -> (3/1,1/0) Parabolic Matrix(11,20,6,11) (-2/1,-5/3) -> (5/3,2/1) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(11,15,8,11) (-3/2,-1/1) -> (4/3,3/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(21,-25,16,-19) (1/1,5/4) -> (5/4,4/3) Parabolic Matrix(11,-25,4,-9) (2/1,5/2) -> (5/2,3/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,5,0,1) -> Matrix(1,0,0,1) Matrix(11,20,6,11) -> Matrix(1,0,0,1) Matrix(19,30,12,19) -> Matrix(1,2,0,1) Matrix(11,15,8,11) -> Matrix(1,2,0,1) Matrix(1,0,2,1) -> Matrix(1,0,2,1) Matrix(21,-25,16,-19) -> Matrix(3,-2,2,-1) Matrix(11,-25,4,-9) -> 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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 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 1 1 1/1 (0/1,1/1) 0 5 5/4 1/1 2 1 3/2 (1/1,1/0) 0 5 5/3 1/0 1 1 2/1 (0/1,1/0) 0 5 5/2 (0/1,1/0) 0 1 1/0 (0/1,1/0) 0 5 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(9,-10,8,-9) (1/1,5/4) -> (1/1,5/4) Reflection Matrix(11,-15,8,-11) (5/4,3/2) -> (5/4,3/2) Reflection Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(11,-20,6,-11) (5/3,2/1) -> (5/3,2/1) Reflection Matrix(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) Reflection Matrix(-1,5,0,1) (5/2,1/0) -> (5/2,1/0) 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(1,0,2,-1) -> Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Matrix(9,-10,8,-9) -> Matrix(1,0,2,-1) (1/1,5/4) -> (0/1,1/1) Matrix(11,-15,8,-11) -> Matrix(-1,2,0,1) (5/4,3/2) -> (1/1,1/0) Matrix(19,-30,12,-19) -> Matrix(-1,2,0,1) (3/2,5/3) -> (1/1,1/0) Matrix(11,-20,6,-11) -> Matrix(1,0,0,-1) (5/3,2/1) -> (0/1,1/0) Matrix(9,-20,4,-9) -> Matrix(1,0,0,-1) (2/1,5/2) -> (0/1,1/0) Matrix(-1,5,0,1) -> Matrix(1,0,0,-1) (5/2,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.