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 -2/1 0/1 1/1 4/3 3/2 8/5 2/1 16/7 7/3 8/3 3/1 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -1/1 1/0 -4/1 1/0 -3/1 -2/1 1/0 -8/3 -3/1 -1/1 -5/2 1/0 -7/3 -2/1 1/0 -2/1 -2/1 -9/5 -2/1 -3/2 -16/9 -5/3 -1/1 -7/4 -3/2 -5/3 -3/2 -1/1 -8/5 -1/1 -3/2 1/0 -7/5 -2/1 1/0 -4/3 -2/1 -1/1 -2/1 -1/1 0/1 -1/1 1/1 -1/1 0/1 4/3 0/1 7/5 0/1 1/0 3/2 1/0 8/5 -1/1 5/3 -1/1 -1/2 7/4 -1/2 2/1 0/1 9/4 1/0 16/7 -1/1 1/1 7/3 0/1 1/0 5/2 1/0 8/3 -1/1 1/1 3/1 0/1 1/0 4/1 1/0 5/1 -1/1 1/0 1/0 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(9,40,2,9) (-5/1,-4/1) -> (4/1,5/1) Hyperbolic Matrix(7,24,2,7) (-4/1,-3/1) -> (3/1,4/1) Hyperbolic Matrix(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(23,56,16,39) (-5/2,-7/3) -> (7/5,3/2) Hyperbolic Matrix(15,32,-8,-17) (-7/3,-2/1) -> (-2/1,-9/5) Parabolic Matrix(143,256,62,111) (-9/5,-16/9) -> (16/7,7/3) Hyperbolic Matrix(145,256,64,113) (-16/9,-7/4) -> (9/4,16/7) Hyperbolic Matrix(23,40,4,7) (-7/4,-5/3) -> (5/1,1/0) Hyperbolic Matrix(49,80,30,49) (-5/3,-8/5) -> (8/5,5/3) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(39,56,16,23) (-3/2,-7/5) -> (7/3,5/2) Hyperbolic Matrix(41,56,30,41) (-7/5,-4/3) -> (4/3,7/5) Hyperbolic Matrix(7,8,6,7) (-4/3,-1/1) -> (1/1,4/3) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,40,4,23) -> Matrix(1,2,-2,-3) Matrix(9,40,2,9) -> Matrix(1,0,0,1) Matrix(7,24,2,7) -> Matrix(1,2,0,1) Matrix(17,48,6,17) -> Matrix(1,2,0,1) Matrix(31,80,12,31) -> Matrix(1,2,0,1) Matrix(23,56,16,39) -> Matrix(1,2,0,1) Matrix(15,32,-8,-17) -> Matrix(3,8,-2,-5) Matrix(143,256,62,111) -> Matrix(1,2,-2,-3) Matrix(145,256,64,113) -> Matrix(1,2,-2,-3) Matrix(23,40,4,7) -> Matrix(1,2,-2,-3) Matrix(49,80,30,49) -> Matrix(3,4,-4,-5) Matrix(31,48,20,31) -> Matrix(1,0,0,1) Matrix(39,56,16,23) -> Matrix(1,2,0,1) Matrix(41,56,30,41) -> Matrix(1,2,0,1) Matrix(7,8,6,7) -> Matrix(1,2,-2,-3) Matrix(1,0,2,1) -> Matrix(1,2,-2,-3) Matrix(17,-32,8,-15) -> Matrix(1,0,2,1) 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: ((2,6,15,7)(3,11,12,4)(5,14)(8,9); (1,4,5,2)(3,8,7,10)(6,13,12,9)(11,14,15,16); (1,2,8,12,16,15,9,3)(4,13,6,5,11,10,7,14)) ----------------------------------------------------------------------- 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 -1/1 2 2 1/1 (-1/1,0/1) 0 8 4/3 0/1 1 2 7/5 (0/1,1/0) 0 8 3/2 1/0 1 8 8/5 -1/1 4 2 5/3 (-1/1,-1/2) 0 8 7/4 -1/2 1 8 2/1 0/1 1 4 9/4 1/0 1 8 16/7 (0/1,1/0) 0 2 7/3 (0/1,1/0) 0 8 5/2 1/0 1 8 8/3 (0/1,1/0) 0 2 3/1 (0/1,1/0) 0 8 4/1 1/0 1 2 5/1 (-1/1,1/0) 0 8 1/0 1/0 1 8 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(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(39,-56,16,-23) (7/5,3/2) -> (7/3,5/2) Glide Reflection Matrix(31,-48,20,-31) (3/2,8/5) -> (3/2,8/5) Reflection Matrix(49,-80,30,-49) (8/5,5/3) -> (8/5,5/3) Reflection Matrix(23,-40,4,-7) (5/3,7/4) -> (5/1,1/0) Glide Reflection Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(127,-288,56,-127) (9/4,16/7) -> (9/4,16/7) Reflection Matrix(97,-224,42,-97) (16/7,7/3) -> (16/7,7/3) Reflection Matrix(31,-80,12,-31) (5/2,8/3) -> (5/2,8/3) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) 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(1,0,0,-1) -> Matrix(1,2,0,-1) (0/1,1/0) -> (-1/1,1/0) Matrix(1,0,2,-1) -> Matrix(-1,0,2,1) (0/1,1/1) -> (-1/1,0/1) Matrix(7,-8,6,-7) -> Matrix(-1,0,2,1) (1/1,4/3) -> (-1/1,0/1) Matrix(41,-56,30,-41) -> Matrix(1,0,0,-1) (4/3,7/5) -> (0/1,1/0) Matrix(39,-56,16,-23) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(31,-48,20,-31) -> Matrix(1,2,0,-1) (3/2,8/5) -> (-1/1,1/0) Matrix(49,-80,30,-49) -> Matrix(3,2,-4,-3) (8/5,5/3) -> (-1/1,-1/2) Matrix(23,-40,4,-7) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(17,-32,8,-15) -> Matrix(1,0,2,1) 0/1 Matrix(127,-288,56,-127) -> Matrix(1,0,0,-1) (9/4,16/7) -> (0/1,1/0) Matrix(97,-224,42,-97) -> Matrix(1,0,0,-1) (16/7,7/3) -> (0/1,1/0) Matrix(31,-80,12,-31) -> Matrix(1,0,0,-1) (5/2,8/3) -> (0/1,1/0) Matrix(17,-48,6,-17) -> Matrix(1,0,0,-1) (8/3,3/1) -> (0/1,1/0) Matrix(7,-24,2,-7) -> Matrix(1,0,0,-1) (3/1,4/1) -> (0/1,1/0) Matrix(9,-40,2,-9) -> Matrix(1,2,0,-1) (4/1,5/1) -> (-1/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.