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: 12 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/1 -1/1 0/1 1/2 3/4 1/1 3/2 2/1 3/1 4/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -2/1 -3/1 -1/1 -2/1 -1/1 -3/2 -1/2 1/0 -4/3 0/1 -5/4 -1/2 -6/5 -1/2 -1/1 0/1 -4/5 0/1 -3/4 -1/2 1/0 -2/3 -1/1 -3/5 -1/1 -4/7 -2/3 -5/9 -2/3 -6/11 -1/2 -1/2 -1/2 0/1 0/1 1/2 1/2 2/3 1/1 3/4 1/2 1/0 4/5 0/1 1/1 1/1 4/3 0/1 3/2 1/2 1/0 2/1 1/1 5/2 1/0 3/1 1/0 7/2 1/0 11/3 -1/1 4/1 0/1 5/1 1/1 11/2 1/0 6/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,24,-4,-19) (-4/1,1/0) -> (-4/3,-5/4) Hyperbolic Matrix(7,24,-12,-41) (-4/1,-3/1) -> (-3/5,-4/7) Hyperbolic Matrix(5,12,-8,-19) (-3/1,-2/1) -> (-2/3,-3/5) Hyperbolic Matrix(7,12,4,7) (-2/1,-3/2) -> (3/2,2/1) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(29,36,4,5) (-5/4,-6/5) -> (6/1,1/0) Hyperbolic Matrix(31,36,-56,-65) (-6/5,-1/1) -> (-5/9,-6/11) Hyperbolic Matrix(29,24,-52,-43) (-1/1,-4/5) -> (-4/7,-5/9) Hyperbolic Matrix(31,24,40,31) (-4/5,-3/4) -> (3/4,4/5) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(133,72,24,13) (-6/11,-1/2) -> (11/2,6/1) 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(59,-48,16,-13) (4/5,1/1) -> (11/3,4/1) Hyperbolic Matrix(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(43,-156,8,-29) (7/2,11/3) -> (5/1,11/2) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,24,-4,-19) -> Matrix(1,2,-2,-3) Matrix(7,24,-12,-41) -> Matrix(3,4,-4,-5) Matrix(5,12,-8,-19) -> Matrix(1,0,0,1) Matrix(7,12,4,7) -> Matrix(1,0,2,1) Matrix(17,24,12,17) -> Matrix(1,0,2,1) Matrix(29,36,4,5) -> Matrix(1,0,2,1) Matrix(31,36,-56,-65) -> Matrix(5,2,-8,-3) Matrix(29,24,-52,-43) -> Matrix(1,2,-2,-3) Matrix(31,24,40,31) -> Matrix(1,0,2,1) Matrix(17,12,24,17) -> Matrix(1,0,2,1) Matrix(133,72,24,13) -> Matrix(1,0,2,1) Matrix(1,0,4,1) -> Matrix(1,0,4,1) Matrix(19,-12,8,-5) -> Matrix(3,-2,2,-1) Matrix(59,-48,16,-13) -> Matrix(1,0,-2,1) Matrix(19,-24,4,-5) -> Matrix(1,0,0,1) Matrix(13,-36,4,-11) -> Matrix(1,-2,0,1) Matrix(43,-156,8,-29) -> 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): 4 Degree of the the map X: 4 Degree of the the map Y: 16 Permutation triple for Y: ((1,2)(3,5,4,6,8,7)(9,10,14,13,12,11)(15,16); (1,5,11,16,13,6)(2,8,14,15,9,3)(4,12)(7,10); (1,3,10,15,11,4)(2,6,12,16,14,7)(5,9)(8,13)) ----------------------------------------------------------------------- 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): 48 Minimal number of generators: 9 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: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 1/1 3/2 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/1 -3/2 -1/2 1/0 -1/1 0/1 -1/2 -1/2 0/1 0/1 1/2 1/2 2/3 1/1 3/4 1/2 1/0 1/1 1/1 4/3 0/1 3/2 1/2 1/0 2/1 1/1 5/2 1/0 3/1 1/0 7/2 1/0 4/1 0/1 5/1 1/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,3,-2,-5) (-2/1,1/0) -> (-1/1,-1/2) Hyperbolic Matrix(7,12,4,7) (-2/1,-3/2) -> (3/2,2/1) Hyperbolic Matrix(7,9,10,13) (-3/2,-1/1) -> (2/3,3/4) 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(19,-15,14,-11) (3/4,1/1) -> (4/3,3/2) Hyperbolic Matrix(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(11,-39,2,-7) (7/2,4/1) -> (5/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,3,-2,-5) -> Matrix(1,1,-2,-1) Matrix(7,12,4,7) -> Matrix(1,0,2,1) Matrix(7,9,10,13) -> Matrix(1,1,0,1) Matrix(1,0,4,1) -> Matrix(1,0,4,1) Matrix(19,-12,8,-5) -> Matrix(3,-2,2,-1) Matrix(19,-15,14,-11) -> Matrix(1,-1,2,-1) Matrix(19,-24,4,-5) -> Matrix(1,0,0,1) Matrix(13,-36,4,-11) -> Matrix(1,-2,0,1) Matrix(11,-39,2,-7) -> 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): 4 ----------------------------------------------------------------------- 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 -3/2 (-1/2,1/0) 0 2 -1/1 0/1 1 6 0/1 0/1 2 2 1/1 1/1 1 6 3/2 (1/2,1/0) 0 2 2/1 1/1 1 6 3/1 1/0 1 2 4/1 0/1 1 6 1/0 1/0 1 6 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,3,0,-1) (-3/2,1/0) -> (-3/2,1/0) Reflection Matrix(7,9,4,5) (-3/2,-1/1) -> (3/2,2/1) Glide 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(17,-21,4,-5) (1/1,4/3) -> (4/1,5/1) Glide Reflection Matrix(11,-15,8,-11) (5/4,3/2) -> (5/4,3/2) 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(-1,9,0,1) (9/2,1/0) -> (9/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,3,0,-1) -> Matrix(1,1,0,-1) (-3/2,1/0) -> (-1/2,1/0) Matrix(7,9,4,5) -> Matrix(1,1,2,1) Matrix(-1,0,2,1) -> Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Matrix(1,0,2,-1) -> Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Matrix(17,-21,4,-5) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(11,-15,8,-11) -> Matrix(-1,1,0,1) (5/4,3/2) -> (1/2,1/0) Matrix(5,-12,2,-5) -> Matrix(-1,2,0,1) (2/1,3/1) -> (1/1,1/0) Matrix(7,-24,2,-7) -> Matrix(1,0,0,-1) (3/1,4/1) -> (0/1,1/0) Matrix(-1,9,0,1) -> Matrix(-1,1,0,1) (9/2,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.