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 -3/1 -5/2 -2/1 -1/1 -1/2 -2/5 0/1 1/2 4/5 1/1 2/1 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -1/1 0/1 1/0 -7/2 -1/1 0/1 1/0 -3/1 -1/1 0/1 1/0 -8/3 -1/1 1/0 -5/2 -1/1 -2/1 -1/1 0/1 -1/1 0/1 -2/3 0/1 1/1 -5/8 1/1 -3/5 0/1 1/1 1/0 -4/7 0/1 1/1 1/0 -1/2 0/1 1/1 1/0 -2/5 1/0 -3/8 -2/1 -1/1 1/0 -4/11 -1/1 1/0 -1/3 -1/1 0/1 1/0 0/1 0/1 1/0 1/3 -1/1 0/1 1/0 1/2 0/1 3/5 0/1 1/2 1/1 5/8 0/1 1/2 1/1 2/3 0/1 1/1 5/7 1/1 3/4 0/1 1/1 1/0 4/5 0/1 1/1 1/0 1/1 0/1 1/1 1/0 2/1 1/0 3/1 -2/1 -1/1 1/0 7/2 -1/1 4/1 -1/1 1/0 9/2 -2/1 -1/1 1/0 5/1 -1/1 6/1 -1/1 0/1 1/0 -1/1 0/1 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(11,36,18,59) (-7/2,-3/1) -> (3/5,5/8) Hyperbolic Matrix(7,20,-20,-57) (-3/1,-8/3) -> (-4/11,-1/3) Hyperbolic Matrix(29,76,8,21) (-8/3,-5/2) -> (7/2,4/1) Hyperbolic Matrix(9,20,-14,-31) (-5/2,-2/1) -> (-2/3,-5/8) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(59,36,18,11) (-5/8,-3/5) -> (3/1,7/2) Hyperbolic Matrix(27,16,32,19) (-3/5,-4/7) -> (4/5,1/1) Hyperbolic Matrix(29,16,38,21) (-4/7,-1/2) -> (3/4,4/5) Hyperbolic Matrix(19,8,-50,-21) (-1/2,-2/5) -> (-2/5,-3/8) Parabolic Matrix(131,48,30,11) (-3/8,-4/11) -> (4/1,9/2) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic Matrix(51,-32,8,-5) (5/8,2/3) -> (6/1,1/0) Hyperbolic Matrix(57,-40,10,-7) (2/3,5/7) -> (5/1,6/1) Hyperbolic Matrix(83,-60,18,-13) (5/7,3/4) -> (9/2,5/1) Hyperbolic Matrix(5,-8,2,-3) (1/1,2/1) -> (2/1,3/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,32,-2,-9) -> Matrix(1,0,0,1) Matrix(11,36,18,59) -> Matrix(1,0,2,1) Matrix(7,20,-20,-57) -> Matrix(1,0,0,1) Matrix(29,76,8,21) -> Matrix(1,0,0,1) Matrix(9,20,-14,-31) -> Matrix(1,0,2,1) Matrix(3,4,-4,-5) -> Matrix(1,0,2,1) Matrix(59,36,18,11) -> Matrix(1,-2,0,1) Matrix(27,16,32,19) -> Matrix(1,0,0,1) Matrix(29,16,38,21) -> Matrix(1,0,0,1) Matrix(19,8,-50,-21) -> Matrix(1,-2,0,1) Matrix(131,48,30,11) -> Matrix(1,0,0,1) Matrix(1,0,6,1) -> Matrix(1,0,0,1) Matrix(9,-4,16,-7) -> Matrix(1,0,2,1) Matrix(51,-32,8,-5) -> Matrix(1,0,-2,1) Matrix(57,-40,10,-7) -> Matrix(1,0,-2,1) Matrix(83,-60,18,-13) -> Matrix(1,-2,0,1) Matrix(5,-8,2,-3) -> 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: 16 Permutation triple for Y: ((1,6,2)(3,8,9)(5,14,15)(11,16,12); (1,4,12,8,13,5)(2,3)(6,15,10,9,16,7)(11,14); (1,3,10,15,11,4)(2,7,16,14,13,8)(5,6)(9,12)) ----------------------------------------------------------------------- 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 -4/1 -2/1 -1/1 0/1 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -1/1 0/1 1/0 -3/1 -1/1 0/1 1/0 -2/1 -1/1 0/1 -1/1 0/1 0/1 0/1 1/0 1/2 0/1 2/3 0/1 1/1 1/1 0/1 1/1 1/0 2/1 1/0 1/0 -1/1 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,16,-1,-5) (-4/1,1/0) -> (-4/1,-3/1) Parabolic Matrix(3,8,4,11) (-3/1,-2/1) -> (2/3,1/1) Hyperbolic Matrix(3,4,5,7) (-2/1,-1/1) -> (1/2,2/3) Hyperbolic Matrix(1,0,3,1) (-1/1,0/1) -> (0/1,1/2) Parabolic Matrix(3,-4,1,-1) (1/1,2/1) -> (2/1,1/0) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(3,16,-1,-5) -> Matrix(1,0,0,1) Matrix(3,8,4,11) -> Matrix(1,1,0,1) Matrix(3,4,5,7) -> Matrix(1,0,2,1) Matrix(1,0,3,1) -> Matrix(1,0,0,1) Matrix(3,-4,1,-1) -> Matrix(1,-1,0,1) 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): 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 -4/1 (-1/2,1/0) 0 2 -2/1 (-1/1,0/1).(-1/2,1/0) 0 6 -1/1 0/1 1 2 0/1 (0/1,1/0) 0 6 2/1 1/0 2 2 1/0 (-1/2,1/0) 0 6 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(3,4,-2,-3) (-2/1,-1/1) -> (-2/1,-1/1) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(-1,4,0,1) (2/1,1/0) -> (2/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,8,0,-1) -> Matrix(1,1,0,-1) (-4/1,1/0) -> (-1/2,1/0) Matrix(3,8,-1,-3) -> Matrix(1,1,0,-1) (-4/1,-2/1) -> (-1/2,1/0) Matrix(3,4,-2,-3) -> Matrix(-1,0,2,1) (-2/1,-1/1) -> (-1/1,0/1) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(1,0,1,-1) -> Matrix(1,0,0,-1) (0/1,2/1) -> (0/1,1/0) Matrix(-1,4,0,1) -> Matrix(1,1,0,-1) (2/1,1/0) -> (-1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.