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 -4/1 -2/1 -1/1 -1/2 0/1 1/3 1/2 2/3 1/1 4/3 3/2 2/1 3/1 4/1 6/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,12,0,1) (-6/1,1/0) -> (6/1,1/0) Parabolic Matrix(7,36,6,31) (-6/1,-5/1) -> (1/1,6/5) Hyperbolic Matrix(5,24,6,29) (-5/1,-4/1) -> (4/5,1/1) Hyperbolic Matrix(7,24,-12,-41) (-4/1,-3/1) -> (-3/5,-4/7) Hyperbolic Matrix(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) Hyperbolic Matrix(23,60,18,47) (-8/3,-5/2) -> (5/4,4/3) 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(37,60,-66,-107) (-5/3,-8/5) -> (-4/7,-5/9) Hyperbolic Matrix(23,36,30,47) (-8/5,-3/2) -> (3/4,4/5) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(47,60,18,23) (-4/3,-5/4) -> (5/2,8/3) Hyperbolic Matrix(29,36,-54,-67) (-5/4,-6/5) -> (-6/11,-1/2) Hyperbolic Matrix(31,36,6,7) (-6/5,-1/1) -> (5/1,6/1) Hyperbolic Matrix(29,24,6,5) (-1/1,-4/5) -> (4/1,5/1) Hyperbolic Matrix(47,36,30,23) (-4/5,-3/4) -> (3/2,8/5) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(19,12,30,19) (-2/3,-3/5) -> (3/5,2/3) Hyperbolic Matrix(283,156,78,43) (-5/9,-6/11) -> (18/5,11/3) 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(41,-24,12,-7) (1/2,3/5) -> (3/1,7/2) Hyperbolic Matrix(107,-132,30,-37) (6/5,5/4) -> (7/2,18/5) Hyperbolic Matrix(67,-108,18,-29) (8/5,5/3) -> (11/3,4/1) 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 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): 18 Minimal number of generators: 4 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: 3 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 3/2 1/0 -1/1 1/1 3/1 0/1 0/1 2/1 1/2 1/1 3/1 1/1 1/1 3/1 2/1 3/2 1/0 3/1 1/1 3/1 1/0 1/1 3/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,8,1,3) (-2/1,1/0) -> (2/1,3/1) Hyperbolic Matrix(3,4,2,3) (-2/1,-1/1) -> (1/1,2/1) Hyperbolic Matrix(1,0,3,1) (-1/1,0/1) -> (0/1,1/2) Parabolic Matrix(7,-4,2,-1) (1/2,1/1) -> (3/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(3,8,1,3) -> Matrix(1,0,0,1) Matrix(3,4,2,3) -> Matrix(1,0,0,1) Matrix(1,0,3,1) -> Matrix(1,0,0,1) Matrix(7,-4,2,-1) -> Matrix(1,0,0,1) ----------------------------------------------------------------------- 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 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE -1/1 (1/1,3/1) 0/1 (1/1,3/1) 1/1 (1/1,3/1) 2/1 (1/1,3/1) 1/0 (1/1,3/1) GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(3,2,2,1) (-1/1,0/1) -> (1/1,2/1) Glide Reflection Matrix(3,-2,1,-1) (0/1,1/1) -> (2/1,1/0) Glide Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(2,-3,1,-2) (-1/1,1/0) -> (1/1,3/1) Matrix(3,2,2,1) -> Matrix(2,-3,1,-2) *** -> (1/1,3/1) Matrix(3,-2,1,-1) -> Matrix(2,-3,1,-2) *** -> (1/1,3/1)