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): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 16 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/2 5/8 3/4 5/6 1/1 5/4 3/2 5/3 2/1 5/2 3/1 10/3 4/1 5/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,10,0,1) (-5/1,1/0) -> (5/1,1/0) Parabolic Matrix(9,40,-16,-71) (-5/1,-4/1) -> (-4/7,-5/9) Hyperbolic Matrix(11,40,-8,-29) (-4/1,-3/1) -> (-7/5,-4/3) Hyperbolic Matrix(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(9,20,4,9) (-5/2,-2/1) -> (2/1,5/2) Hyperbolic Matrix(11,20,-16,-29) (-2/1,-5/3) -> (-5/7,-2/3) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(69,100,20,29) (-3/2,-10/7) -> (10/3,7/2) Hyperbolic Matrix(71,100,-120,-169) (-10/7,-7/5) -> (-3/5,-10/17) Hyperbolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(9,10,8,9) (-5/4,-1/1) -> (1/1,5/4) Hyperbolic 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(51,40,-88,-69) (-4/5,-3/4) -> (-7/12,-4/7) Hyperbolic Matrix(69,50,40,29) (-3/4,-5/7) -> (5/3,7/4) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(341,200,104,61) (-10/17,-7/12) -> (13/4,10/3) Hyperbolic Matrix(91,50,20,11) (-5/9,-1/2) -> (9/2,5/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(71,-40,16,-9) (1/2,4/7) -> (4/1,9/2) Hyperbolic Matrix(69,-40,88,-51) (4/7,3/5) -> (7/9,4/5) Hyperbolic Matrix(29,-20,16,-11) (2/3,3/4) -> (7/4,2/1) Hyperbolic Matrix(129,-100,40,-31) (3/4,7/9) -> (3/1,13/4) Hyperbolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Since the preimage of every curve is trivial, the pure modular group virtual endomorphism is trivial. ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0, lambda1 DeckMod(f) is isomorphic to Z/2Z. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0, lambda1 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): 36 Minimal number of generators: 7 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: 8 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 5/4 3/2 5/3 2/1 5/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/2 1/0 -5/3 -1/2 1/0 -3/2 -1/1 0/1 -1/1 -1/2 1/0 0/1 -1/2 1/0 1/1 -1/2 1/0 5/4 -1/2 1/0 4/3 -1/2 1/0 3/2 -1/1 0/1 5/3 -1/2 1/0 2/1 -1/2 1/0 5/2 -1/2 1/0 3/1 -1/2 1/0 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,5,0,1) (-2/1,1/0) -> (3/1,1/0) Parabolic Matrix(11,20,6,11) (-2/1,-5/3) -> (5/3,2/1) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(11,15,8,11) (-3/2,-1/1) -> (4/3,3/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(21,-25,16,-19) (1/1,5/4) -> (5/4,4/3) Parabolic Matrix(11,-25,4,-9) (2/1,5/2) -> (5/2,3/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL MULTI-ENDOMORPHISM This map is 2-valued. Matrix(1,5,0,1) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(11,20,6,11) -> Matrix(1,1,-2,-1) -> Matrix(1,0,0,1) Matrix(19,30,12,19) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(11,15,8,11) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(1,0,2,1) -> Matrix(1,1,-2,-1) -> Matrix(1,0,0,1) Matrix(21,-25,16,-19) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(11,-25,4,-9) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(1,0,0,1) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-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 0/1 (-1/1,0/1).(-1/2,1/0) 1/1 (-1/1,0/1).(-1/2,1/0) 5/4 (-1/1,0/1).(-1/2,1/0) 3/2 (-1/1,0/1).(-1/2,1/0) 5/3 (-1/1,0/1).(-1/2,1/0) 2/1 (-1/1,0/1).(-1/2,1/0) 5/2 (-1/1,0/1).(-1/2,1/0) 1/0 (-1/1,0/1).(-1/2,1/0) 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(9,-10,8,-9) (1/1,5/4) -> (1/1,5/4) Reflection Matrix(11,-15,8,-11) (5/4,3/2) -> (5/4,3/2) Reflection Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(11,-20,6,-11) (5/3,2/1) -> (5/3,2/1) Reflection Matrix(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) Reflection Matrix(-1,5,0,1) (5/2,1/0) -> (5/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL MULTI-ENDOMORPHISM FIXED POINT OF IMAGE This map is 2-valued. Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(1,0,2,-1) -> Matrix(1,1,0,-1) (0/1,1/1) -> (-1/2,1/0) -> Matrix(-1,0,2,1) -> (-1/1,0/1) Matrix(9,-10,8,-9) -> Matrix(-1,0,2,1) (1/1,5/4) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(11,-15,8,-11) -> Matrix(-1,0,2,1) (5/4,3/2) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(19,-30,12,-19) -> Matrix(-1,0,2,1) (3/2,5/3) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(11,-20,6,-11) -> Matrix(1,1,0,-1) (5/3,2/1) -> (-1/2,1/0) -> Matrix(-1,0,2,1) -> (-1/1,0/1) Matrix(9,-20,4,-9) -> Matrix(-1,0,2,1) (2/1,5/2) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(-1,5,0,1) -> Matrix(-1,0,2,1) (5/2,1/0) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(1,0,0,1) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0)