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 -5/1 -4/1 -3/1 -7/3 -2/1 -1/1 0/1 1/2 1/1 4/3 2/1 3/1 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -1/1 -4/1 -1/1 0/1 -3/1 0/1 -8/3 0/1 1/0 -5/2 -2/1 -1/1 1/0 -7/3 -1/1 -2/1 -1/1 0/1 -1/1 0/1 -2/3 0/1 1/1 -7/11 1/1 -5/8 1/1 2/1 1/0 -3/5 0/1 -4/7 0/1 1/1 -1/2 0/1 1/1 1/0 0/1 0/1 1/0 1/2 0/1 1/1 1/0 2/3 -1/1 1/0 3/4 -1/1 0/1 1/0 1/1 0/1 5/4 0/1 1/2 1/1 4/3 0/1 1/1 11/8 0/1 1/2 1/1 7/5 1/1 3/2 0/1 1/1 1/0 8/5 0/1 1/1 5/3 1/1 7/4 1/1 2/1 1/0 2/1 1/1 1/0 5/2 2/1 3/1 1/0 3/1 1/0 7/2 -4/1 -3/1 1/0 4/1 -2/1 1/0 5/1 -2/1 1/0 -1/1 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,40,4,23) (-5/1,1/0) -> (5/3,7/4) Hyperbolic Matrix(13,60,8,37) (-5/1,-4/1) -> (8/5,5/3) Hyperbolic Matrix(7,24,-12,-41) (-4/1,-3/1) -> (-3/5,-4/7) Hyperbolic Matrix(19,52,4,11) (-3/1,-8/3) -> (4/1,5/1) Hyperbolic Matrix(29,76,8,21) (-8/3,-5/2) -> (7/2,4/1) Hyperbolic Matrix(23,56,16,39) (-5/2,-7/3) -> (7/5,3/2) Hyperbolic Matrix(13,28,-20,-43) (-7/3,-2/1) -> (-2/3,-7/11) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(177,112,128,81) (-7/11,-5/8) -> (11/8,7/5) Hyperbolic Matrix(45,28,8,5) (-5/8,-3/5) -> (5/1,1/0) Hyperbolic Matrix(37,20,24,13) (-4/7,-1/2) -> (3/2,8/5) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(19,-12,8,-5) (1/2,2/3) -> (2/1,5/2) Hyperbolic Matrix(29,-20,16,-11) (2/3,3/4) -> (7/4,2/1) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,40,4,23) -> Matrix(1,2,0,1) Matrix(13,60,8,37) -> Matrix(1,0,2,1) Matrix(7,24,-12,-41) -> Matrix(1,0,2,1) Matrix(19,52,4,11) -> Matrix(1,-2,0,1) Matrix(29,76,8,21) -> Matrix(1,-2,0,1) Matrix(23,56,16,39) -> Matrix(1,2,0,1) Matrix(13,28,-20,-43) -> Matrix(1,0,2,1) Matrix(3,4,-4,-5) -> Matrix(1,0,2,1) Matrix(177,112,128,81) -> Matrix(1,-2,2,-3) Matrix(45,28,8,5) -> Matrix(1,-2,0,1) Matrix(37,20,24,13) -> Matrix(1,0,0,1) Matrix(1,0,4,1) -> Matrix(1,0,0,1) Matrix(19,-12,8,-5) -> Matrix(1,2,0,1) Matrix(29,-20,16,-11) -> Matrix(1,2,0,1) Matrix(9,-8,8,-7) -> Matrix(1,0,2,1) Matrix(49,-64,36,-47) -> Matrix(1,0,0,1) Matrix(13,-36,4,-11) -> Matrix(1,-6,0,1) 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: ((1,2)(3,7,16,8)(4,6,12,5)(9,13)(10,11)(14,15); (1,5,15,6)(2,3)(4,11)(7,9,8,10)(12,13)(14,16); (1,3,9,12,15,16,10,4)(2,6,11,8,14,5,13,7)) ----------------------------------------------------------------------- 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 -4/1 (-1/1,0/1) 0 8 -3/1 0/1 1 4 -5/2 0 8 -7/3 -1/1 2 2 -2/1 (-1/1,0/1) 0 8 -1/1 0/1 1 2 0/1 (0/1,1/0) 0 8 1/1 0/1 1 4 4/3 (0/1,1/1) 0 8 7/5 1/1 2 2 3/2 0 8 8/5 (0/1,1/1) 0 8 5/3 1/1 1 2 2/1 (1/1,1/0) 0 8 3/1 1/0 3 2 4/1 (-2/1,1/0) 0 8 5/1 -2/1 1 4 1/0 0 8 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,20,2,13) (-4/1,1/0) -> (3/2,8/5) Glide Reflection Matrix(7,24,-2,-7) (-4/1,-3/1) -> (-4/1,-3/1) Reflection Matrix(11,28,2,5) (-3/1,-5/2) -> (5/1,1/0) Glide Reflection Matrix(23,56,16,39) (-5/2,-7/3) -> (7/5,3/2) Hyperbolic Matrix(13,28,-6,-13) (-7/3,-2/1) -> (-7/3,-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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(7,-8,6,-7) (1/1,4/3) -> (1/1,4/3) Reflection Matrix(41,-56,30,-41) (4/3,7/5) -> (4/3,7/5) Reflection Matrix(49,-80,30,-49) (8/5,5/3) -> (8/5,5/3) 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 Matrix(9,-40,2,-9) (4/1,5/1) -> (4/1,5/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(3,20,2,13) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(7,24,-2,-7) -> Matrix(-1,0,2,1) (-4/1,-3/1) -> (-1/1,0/1) Matrix(11,28,2,5) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(23,56,16,39) -> Matrix(1,2,0,1) 1/0 Matrix(13,28,-6,-13) -> Matrix(-1,0,2,1) (-7/3,-2/1) -> (-1/1,0/1) 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(1,0,2,-1) -> Matrix(1,0,0,-1) (0/1,1/1) -> (0/1,1/0) Matrix(7,-8,6,-7) -> Matrix(1,0,2,-1) (1/1,4/3) -> (0/1,1/1) Matrix(41,-56,30,-41) -> Matrix(1,0,2,-1) (4/3,7/5) -> (0/1,1/1) Matrix(49,-80,30,-49) -> Matrix(1,0,2,-1) (8/5,5/3) -> (0/1,1/1) Matrix(11,-20,6,-11) -> Matrix(-1,2,0,1) (5/3,2/1) -> (1/1,1/0) Matrix(5,-12,2,-5) -> Matrix(-1,2,0,1) (2/1,3/1) -> (1/1,1/0) Matrix(7,-24,2,-7) -> Matrix(1,4,0,-1) (3/1,4/1) -> (-2/1,1/0) Matrix(9,-40,2,-9) -> Matrix(1,4,0,-1) (4/1,5/1) -> (-2/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.