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: 14 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 1/3 1/2 3/5 1/1 5/3 2/1 3/1 4/1 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 1/1 -2/5 0/1 1/0 -3/8 1/1 -1/3 -1/1 1/1 -2/7 1/1 1/0 -1/4 -1/1 0/1 0/1 1/0 1/4 -1/1 1/3 0/1 2/5 0/1 1/1 3/7 -1/1 1/1 1/2 1/1 3/5 1/0 5/8 -3/1 2/3 -1/1 1/0 3/4 -1/1 1/1 -1/1 1/1 5/4 -1/1 9/7 -1/1 4/3 -1/1 0/1 7/5 0/1 3/2 -1/1 5/3 0/1 7/4 1/1 2/1 0/1 1/0 5/2 1/1 3/1 1/0 7/2 -1/1 4/1 -1/1 1/0 5/1 -3/1 -1/1 6/1 -2/1 -1/1 7/1 -1/1 1/0 -1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(29,12,12,5) (-1/2,-2/5) -> (2/1,5/2) Hyperbolic Matrix(51,20,28,11) (-2/5,-3/8) -> (7/4,2/1) Hyperbolic Matrix(27,10,62,23) (-3/8,-1/3) -> (3/7,1/2) Hyperbolic Matrix(47,14,10,3) (-1/3,-2/7) -> (4/1,5/1) Hyperbolic Matrix(43,12,68,19) (-2/7,-1/4) -> (5/8,2/3) Hyperbolic Matrix(1,0,8,1) (-1/4,0/1) -> (0/1,1/4) Parabolic Matrix(37,-10,26,-7) (1/4,1/3) -> (7/5,3/2) Hyperbolic Matrix(47,-18,34,-13) (1/3,2/5) -> (4/3,7/5) Hyperbolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(31,-18,50,-29) (1/2,3/5) -> (3/5,5/8) Parabolic Matrix(37,-26,10,-7) (2/3,3/4) -> (7/2,4/1) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(35,-44,4,-5) (5/4,9/7) -> (7/1,1/0) Hyperbolic Matrix(63,-82,10,-13) (9/7,4/3) -> (6/1,7/1) Hyperbolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,2,1) Matrix(29,12,12,5) -> Matrix(1,0,0,1) Matrix(51,20,28,11) -> Matrix(1,0,0,1) Matrix(27,10,62,23) -> Matrix(1,0,0,1) Matrix(47,14,10,3) -> Matrix(1,-2,0,1) Matrix(43,12,68,19) -> Matrix(1,-2,0,1) Matrix(1,0,8,1) -> Matrix(1,0,0,1) Matrix(37,-10,26,-7) -> Matrix(1,0,0,1) Matrix(47,-18,34,-13) -> Matrix(1,0,-2,1) Matrix(67,-28,12,-5) -> Matrix(1,-2,0,1) Matrix(31,-18,50,-29) -> Matrix(1,-4,0,1) Matrix(37,-26,10,-7) -> Matrix(1,0,0,1) Matrix(9,-8,8,-7) -> Matrix(1,0,0,1) Matrix(35,-44,4,-5) -> Matrix(1,0,0,1) Matrix(63,-82,10,-13) -> Matrix(3,2,-2,-1) Matrix(31,-50,18,-29) -> Matrix(1,0,2,1) Matrix(13,-36,4,-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): 3 Degree of the the map X: 3 Degree of the the map Y: 16 Permutation triple for Y: ((1,6,7,2)(3,8,11,4)(5,10,9,14)(12,15,16,13); (1,4,13,5)(3,10)(6,12)(7,9,15,8); (1,2,8,10,13,16,9,3)(4,11,15,6,5,14,7,12)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- 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 -1/1 0/1 1 1 0/1 (0/1,1/0) 0 8 1/3 0/1 1 2 2/5 (0/1,1/1) 0 8 1/2 1/1 1 8 3/5 1/0 2 1 2/3 (-1/1,1/0) 0 8 3/4 -1/1 1 8 1/1 0 4 5/4 -1/1 1 8 9/7 -1/1 1 1 4/3 (-1/1,0/1) 0 8 3/2 -1/1 1 8 5/3 0/1 1 1 2/1 (0/1,1/0) 0 8 3/1 1/0 1 2 4/1 (-1/1,1/0) 0 8 1/0 -1/1 1 8 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,6,-1) (0/1,1/3) -> (0/1,1/3) Reflection Matrix(11,-4,30,-11) (1/3,2/5) -> (1/3,2/5) Reflection Matrix(23,-10,16,-7) (2/5,1/2) -> (4/3,3/2) Glide Reflection Matrix(11,-6,20,-11) (1/2,3/5) -> (1/2,3/5) Reflection Matrix(19,-12,30,-19) (3/5,2/3) -> (3/5,2/3) Reflection Matrix(19,-14,4,-3) (2/3,3/4) -> (4/1,1/0) Glide Reflection Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(71,-90,56,-71) (5/4,9/7) -> (5/4,9/7) Reflection Matrix(55,-72,42,-55) (9/7,4/3) -> (9/7,4/3) 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(5,-12,2,-5) (2/1,3/1) -> (2/1,3/1) Reflection Matrix(7,-24,2,-7) (3/1,4/1) -> (3/1,4/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(1,0,6,-1) -> Matrix(1,0,0,-1) (0/1,1/3) -> (0/1,1/0) Matrix(11,-4,30,-11) -> Matrix(1,0,2,-1) (1/3,2/5) -> (0/1,1/1) Matrix(23,-10,16,-7) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(11,-6,20,-11) -> Matrix(-1,2,0,1) (1/2,3/5) -> (1/1,1/0) Matrix(19,-12,30,-19) -> Matrix(1,2,0,-1) (3/5,2/3) -> (-1/1,1/0) Matrix(19,-14,4,-3) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(9,-8,8,-7) -> Matrix(1,0,0,1) Matrix(71,-90,56,-71) -> Matrix(1,2,0,-1) (5/4,9/7) -> (-1/1,1/0) Matrix(55,-72,42,-55) -> Matrix(-1,0,2,1) (9/7,4/3) -> (-1/1,0/1) Matrix(19,-30,12,-19) -> Matrix(-1,0,2,1) (3/2,5/3) -> (-1/1,0/1) Matrix(11,-20,6,-11) -> Matrix(1,0,0,-1) (5/3,2/1) -> (0/1,1/0) Matrix(5,-12,2,-5) -> Matrix(1,0,0,-1) (2/1,3/1) -> (0/1,1/0) Matrix(7,-24,2,-7) -> Matrix(1,2,0,-1) (3/1,4/1) -> (-1/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.