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): 72 Minimal number of generators: 13 Number of equivalence classes of cusps: 12 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -2/1 -1/1 -1/2 0/1 1/3 1/2 2/3 1/1 3/2 2/1 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 -3/1 -1/1 -5/2 -2/1 -1/1 1/0 -2/1 -2/1 0/1 -5/3 -3/2 1/0 -3/2 -1/1 -1/1 -1/1 0/1 1/0 -3/4 -1/1 -2/3 -1/2 1/0 -3/5 -1/1 -1/3 -1/2 0/1 -2/5 0/1 1/2 1/1 -1/3 1/2 1/0 0/1 -1/1 1/1 1/3 1/2 1/0 2/5 0/1 2/1 1/2 0/1 1/1 1/0 3/5 -1/1 1/1 2/3 1/2 1/0 1/1 0/1 2/1 4/3 1/2 1/0 3/2 1/1 5/3 3/2 1/0 2/1 1/1 2/1 1/0 5/2 2/1 3/1 1/1 3/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,6,0,1) (-3/1,1/0) -> (3/1,1/0) Parabolic Matrix(7,18,12,31) (-3/1,-5/2) -> (1/2,3/5) Hyperbolic Matrix(5,12,12,29) (-5/2,-2/1) -> (2/5,1/2) Hyperbolic Matrix(7,12,18,31) (-2/1,-5/3) -> (1/3,2/5) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(5,6,-6,-7) (-3/2,-1/1) -> (-1/1,-3/4) Parabolic Matrix(25,18,18,13) (-3/4,-2/3) -> (4/3,3/2) Hyperbolic Matrix(19,12,30,19) (-2/3,-3/5) -> (3/5,2/3) Hyperbolic Matrix(31,18,12,7) (-3/5,-1/2) -> (5/2,3/1) Hyperbolic Matrix(29,12,12,5) (-1/2,-2/5) -> (2/1,5/2) Hyperbolic Matrix(31,12,18,7) (-2/5,-1/3) -> (5/3,2/1) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,4,0,1) Matrix(7,18,12,31) -> Matrix(1,2,0,1) Matrix(5,12,12,29) -> Matrix(1,2,0,1) Matrix(7,12,18,31) -> Matrix(1,2,0,1) Matrix(19,30,12,19) -> Matrix(3,4,2,3) Matrix(5,6,-6,-7) -> Matrix(1,0,0,1) Matrix(25,18,18,13) -> Matrix(1,0,2,1) Matrix(19,12,30,19) -> Matrix(1,0,2,1) Matrix(31,18,12,7) -> Matrix(5,2,2,1) Matrix(29,12,12,5) -> Matrix(3,-2,2,-1) Matrix(31,12,18,7) -> Matrix(3,-2,2,-1) Matrix(1,0,6,1) -> Matrix(1,0,0,1) Matrix(7,-6,6,-5) -> 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): 2 Degree of the the map X: 2 Degree of the the map Y: 12 Permutation triple for Y: ((1,7,2)(3,11,12)(4,8,5)(6,10,9); (1,5,6)(2,10,3)(4,12,9)(7,11,8); (1,3,4)(2,8,9)(5,11,10)(6,12,7)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- 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): 12 Minimal number of generators: 3 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: 4 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 1/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 0/1 1/0 0/1 -1/1 1/1 1/2 0/1 1/1 1/0 1/1 0/1 2/1 2/1 1/1 2/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,3,0,1) (-1/1,1/0) -> (2/1,1/0) Parabolic Matrix(3,1,2,1) (-1/1,0/1) -> (1/1,2/1) Hyperbolic Matrix(3,-1,4,-1) (0/1,1/2) -> (1/2,1/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,3,0,1) -> Matrix(1,2,0,1) Matrix(3,1,2,1) -> Matrix(1,-1,1,0) Matrix(3,-1,4,-1) -> Matrix(1,-1,1,0) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 2 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 2 Number of equivalence classes of cusps: 1 Genus: 0 Degree of H/liftables -> H/(image of liftables): 1 ----------------------------------------------------------------------- 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 This is a reflection group. CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d -1/1 (-1/1,1/1).(-1/2,1/0).(-2/1,0/1) 0 3 0/1 (-1/1,1/1) 0 3 1/2 (0/1,2/1).(1/2,1/0).(-1/1,1/1) 0 3 1/0 1/0 2 3 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,1,0,-1) (-1/1,1/0) -> (-1/2,1/0) Matrix(-1,0,2,1) -> Matrix(0,1,1,0) (-1/1,0/1) -> (-1/1,1/1) Matrix(1,0,4,-1) -> Matrix(0,1,1,0) (0/1,1/2) -> (-1/1,1/1) Matrix(-1,1,0,1) -> Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.