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 -2/3 -1/2 -5/12 -1/3 -1/4 0/1 1/6 1/5 1/4 1/3 2/5 1/2 2/3 5/6 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/2 -5/6 1/1 -4/5 1/1 1/0 -3/4 0/1 -2/3 1/3 1/1 -5/8 0/1 -3/5 1/2 -1/2 1/1 -3/7 1/0 -5/12 1/0 -2/5 -1/1 1/0 -1/3 0/1 -2/7 1/2 1/1 -1/4 0/1 -1/5 1/2 -1/6 1/1 0/1 0/1 1/1 1/6 1/1 1/5 1/0 1/4 0/1 1/3 0/1 3/8 0/1 2/5 1/3 1/2 1/2 1/1 4/7 3/1 1/0 7/12 1/0 3/5 1/0 2/3 -1/1 1/1 5/7 1/0 3/4 0/1 4/5 1/2 1/1 5/6 1/1 1/1 1/0 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(11,10,12,11) (-1/1,-5/6) -> (5/6,1/1) Hyperbolic Matrix(49,40,60,49) (-5/6,-4/5) -> (4/5,5/6) Hyperbolic Matrix(23,18,60,47) (-4/5,-3/4) -> (3/8,2/5) Hyperbolic Matrix(23,16,-36,-25) (-3/4,-2/3) -> (-2/3,-5/8) Parabolic Matrix(13,8,60,37) (-5/8,-3/5) -> (1/5,1/4) Hyperbolic Matrix(11,6,-24,-13) (-3/5,-1/2) -> (-1/2,-3/7) Parabolic Matrix(85,36,144,61) (-3/7,-5/12) -> (7/12,3/5) Hyperbolic Matrix(83,34,144,59) (-5/12,-2/5) -> (4/7,7/12) Hyperbolic Matrix(11,4,-36,-13) (-2/5,-1/3) -> (-1/3,-2/7) Parabolic Matrix(37,10,48,13) (-2/7,-1/4) -> (3/4,4/5) Hyperbolic Matrix(35,8,48,11) (-1/4,-1/5) -> (5/7,3/4) Hyperbolic Matrix(11,2,60,11) (-1/5,-1/6) -> (1/6,1/5) Hyperbolic Matrix(1,0,12,1) (-1/6,0/1) -> (0/1,1/6) Parabolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(13,-6,24,-11) (2/5,1/2) -> (1/2,4/7) Parabolic Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(11,10,12,11) -> Matrix(3,-2,2,-1) Matrix(49,40,60,49) -> Matrix(1,-2,2,-3) Matrix(23,18,60,47) -> Matrix(1,0,2,1) Matrix(23,16,-36,-25) -> Matrix(1,0,0,1) Matrix(13,8,60,37) -> Matrix(1,0,-2,1) Matrix(11,6,-24,-13) -> Matrix(3,-2,2,-1) Matrix(85,36,144,61) -> Matrix(1,-2,0,1) Matrix(83,34,144,59) -> Matrix(1,4,0,1) Matrix(11,4,-36,-13) -> Matrix(1,0,2,1) Matrix(37,10,48,13) -> Matrix(1,0,0,1) Matrix(35,8,48,11) -> Matrix(1,0,-2,1) Matrix(11,2,60,11) -> Matrix(3,-2,2,-1) Matrix(1,0,12,1) -> Matrix(1,0,0,1) Matrix(13,-4,36,-11) -> Matrix(1,0,2,1) Matrix(13,-6,24,-11) -> Matrix(3,-2,2,-1) Matrix(25,-16,36,-23) -> 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 cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 5 Degree of the the map X: 5 Degree of the the map Y: 16 Permutation triple for Y: ((1,4,13,14,5,2)(3,10)(6,9,16,12,11,15)(7,8); (1,2,8,16,9,3)(4,12)(5,6)(7,15,11,10,14,13); (2,6,7)(3,11,4)(5,10,9)(8,13,12)) ----------------------------------------------------------------------- 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 0/1 (0/1,1/1) 0 6 1/6 1/1 1 1 1/5 1/0 1 6 1/4 0/1 1 3 1/3 0/1 1 2 3/8 0/1 1 3 2/5 (1/3,1/2) 0 6 1/2 1/1 1 3 4/7 (3/1,1/0) 0 6 7/12 1/0 3 1 3/5 1/0 1 6 2/3 0 2 5/7 1/0 1 6 3/4 0/1 1 3 4/5 (1/2,1/1) 0 6 5/6 1/1 2 1 1/1 1/0 1 6 1/0 0/1 1 1 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(1,0,12,-1) (0/1,1/6) -> (0/1,1/6) Reflection Matrix(11,-2,60,-11) (1/6,1/5) -> (1/6,1/5) Reflection Matrix(35,-8,48,-11) (1/5,1/4) -> (5/7,3/4) Glide Reflection Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(47,-18,60,-23) (3/8,2/5) -> (3/4,4/5) Glide Reflection Matrix(13,-6,24,-11) (2/5,1/2) -> (1/2,4/7) Parabolic Matrix(97,-56,168,-97) (4/7,7/12) -> (4/7,7/12) Reflection Matrix(71,-42,120,-71) (7/12,3/5) -> (7/12,3/5) Reflection Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(49,-40,60,-49) (4/5,5/6) -> (4/5,5/6) Reflection Matrix(11,-10,12,-11) (5/6,1/1) -> (5/6,1/1) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,2,-1) (0/1,1/0) -> (0/1,1/1) Matrix(1,0,12,-1) -> Matrix(1,0,2,-1) (0/1,1/6) -> (0/1,1/1) Matrix(11,-2,60,-11) -> Matrix(-1,2,0,1) (1/6,1/5) -> (1/1,1/0) Matrix(35,-8,48,-11) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(13,-4,36,-11) -> Matrix(1,0,2,1) 0/1 Matrix(47,-18,60,-23) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(13,-6,24,-11) -> Matrix(3,-2,2,-1) 1/1 Matrix(97,-56,168,-97) -> Matrix(-1,6,0,1) (4/7,7/12) -> (3/1,1/0) Matrix(71,-42,120,-71) -> Matrix(1,0,0,-1) (7/12,3/5) -> (0/1,1/0) Matrix(25,-16,36,-23) -> Matrix(1,0,0,1) Matrix(49,-40,60,-49) -> Matrix(3,-2,4,-3) (4/5,5/6) -> (1/2,1/1) Matrix(11,-10,12,-11) -> Matrix(-1,2,0,1) (5/6,1/1) -> (1/1,1/0) Matrix(-1,2,0,1) -> Matrix(1,0,0,-1) (1/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.