( ) 1.1. (2 ),,.,.,.,,,,,.,,,,.,,., K, K.

( ),.,,, 1, [17]. 1. 1.1. (2 ),,.,.,.,,,,,.,,,,.,,., K, K. 1.2. Σ g g. M g, Σ g. g 1 Σ g,, Σ g Σ g. Σ g, M g,, Σ g.. g = 1, M 1 M 1, SL(2, Z). Q. g = 2, 2000 M 2 (Korkmaz [20], Bigelow Budney [5])., Bigelow [4], Krammer [23], Artin B n, B n, M 2 6 Birman Hilden [7]. M 2,, g 3 M g M g,. 1.3. 1.3.1. Lawrence, B n Lawrence [24]. 2 D 2, n P n. D 2 P n 782-8502 e-mail: kasahara.yasushi@kochi-tech.ac.jp

D n. D n, n, P n D 2,.,, P n., D n, B n. C m (D n ) D 2 P n m ( ). C m (D n ) 1 B n,, H 1 (C m (D n ); Z) B n = { Z (m = 1); Z Z (m 2) ([24]). C m (D n ) m, Z (m = 1) Z Z (m 2) Z, B n. m = 1, Burau, n = 3 (Magnus Peluso [26]). Bigelow Krammer, m = 2., 1, g 1 Σ g,1, M g,1., M g,1 Σ g,1,. m 1, M g,1 H 1 (C m (Σ g,1 ); Z) = H 1 (Σ g,1 ; Z) Z/2Z ( M g,1 Z/2Z ) (c.f. Scott [29], Bellingeri [3]), M g,1, C m (Σ g,1 ) 2 m, C m (Σ g,1 ) m,., M g,1, m, M g,1 Jonhnson filtration (Moriyama [27]). 1.3.2. Aut (F n ) F n n, Aut (F n ) F n. Aut (F 2 ), B 4 (Krammer [22]). n 3, Aut (F n ) Formanek Procesi [11]., M g,1 F 2g Σ g,1, Dehn Nilsen, M g,1 Aut (F 2g ) : M g,1 = {f Aut (F2g ) ; f(ζ) = ζ}, ζ F 2g a 1, b 1,..., a g, b g, ζ = [a 1, b 1 ][a 2, b 2 ] [a g, b g ]., Formanek Procesi, g 2 M g,1 Brendle Tehrani-Hamidi [8]. 1.3.3.,, 2 1 Kida [19]. 1 G, G Γ, G/Γ, G, Borel.

, g 1, M 1 = M1, = SL(2, Z).,, 5,, Korkmaz [20], Bigelow Budney [5],. 1.4.,. B n, n 1, n = 4, 5, 6, 1, n 1 Burau composition factor,, Iwahori Hecke Formanek [9]. B n n, Formanek [10],. g 1,, Franks Handel [12], 2g 1,. Korkmaz [21], 2g, g symplectic., symplectic, M g Σ g 1 M g Sp(2g, Z),, 2, M g. Korkmaz, g 3 3g 3. 2. 2.1. M g, M g,,, ad-hoc,,.,.,,. Σ g,n g 1, n 0, M g,n. M g,n Σ g,n.,. S Σ g,n.,, 1,.,,. ι: S M g,n c S c ( )Dehn t c M g,n. S, Dehn

, S., ι.,, f M g,n, c S,. fι(c)f 1 = ι(f(c)) (2.1),, Iwahori Hecke M 2, 2 Jones ([15], [16]), : 2.1 ([17]). g 1 n 0. φ: M g,n G, Ker φ M g,n Z(M g,n ),. φ ι: S G, (2.1), S M g,n Z(M g,n ), Dehn S,. 2.2. g 1, Z(M g,n ), ([28]). n = 0 : g 3, ; g = 1, 2, 2. (g, n) = (1, 1) :. : Dehn n., K. 2.2. 2.1, M g,n,. K[S], Σ g,n S K. M g,n S, K[S] M g,n.,. 2.3. K M M g,n, M g,n Σ g,n (S ). p: K[S] M p, M.

2.4. (1) M g,n, M = K[S]/ Ker p, Ker p S,, K[S] ( ) M g,n. (2), Ker p M g,n, Luo [25]. (3) M g,n., n = 0, K 2, M g H 1 (Σ g ; K) Σ g., M g K[S] H 1 (Σ g ; Z) 0., M g,n, Σ g,n. V ρ : M g,n GL(V ), ρ ι(s) End (V ) K M ρ. ρ(f) End (V ) M g,n., (2.1), M ρ End (V ) M g,n., p ρ : K[S] M ρ ρ ι, M ρ Σ g,n., 2.1,. 2.5 ([17]). M g,n, K,, Σ g,n p : K[S] M, p S. 2.2, n = 0 g 3, M g,n.,. 2.6 ([17]). g 3 M g K, Σ g p : K[S] M, p S. 2.3. 1 g 2.,, M g,.,,, S π 1 (Σ g, )., Birman [6] : 1 π 1 (Σ g, ) i M g, j Mg 1 (2.2), j., i, Σ g,, Σ g,,. (2.1), f M g,, γ π 1 (Σ g, ), f i(γ) f 1 = i(f γ) (2.3)

., f f π 1 (Σ g, ). 2.1 : 2.7 ([17]). M g, ρ : M g, G, ρ i.. ρ π 1 (Σ g, ), Ker ρ = {1}. f Ker ρ, (2.3), γ π 1 (Σ g ), ρ(i(γ)) = ρ(f)ρ(i(γ))ρ(f 1 ) = ρ(fi(γ)f 1 ) = ρ(i(f γ)).,, f γ = γ γ π 1 (Σ g, ). Dehn Nilsen, M g, π 1 (Σ g, ), f = 1 M g,., i, π 1 (Σ g, ) M g,. 2.7, M g,, Σ g, M g,., π 1 (Σ g, ), M g,,.,, (c.f. Goldman [13]). π 1 (Σ g, ) G Hom (π 1 (Σ g, ), G). G, G. X G = Hom (π 1 (Σ g, ), G) /G, π 1 (Σ g, ). ϕ Hom (π 1 (Σ g, ), G) X G [ϕ]. f M g,. ϕ : π 1 (Σ g, ) G, ϕ f 1, M g, Hom (π 1 (Σ g, ), G)., M g, X G, π 1 (Σ g, ) X G, M g Birman (2.2) X G. 2.8 ([17])., ϕ : π 1 (Σ g, ) G M g, M g, G, [ϕ] X G M g X G.., (2.3)., ϕ Hom (π 1 (Σ g, ), G), ϕ(π 1 (Σ g, )) G, 2.8.,, G = GL(n, K), X GL(n,K) M g, M g,,.

, ϕ : π 1 (Σ g, ) GL(n, K), ϕ Ad ϕ : π 1 (Σ g, ) GL(End (n, K)). V ϕ, ϕ(π 1 (Σ g, )) End (n, K) K, Ad ϕ π 1 (Σ g, ). Ad ϕ V ϕ A ϕ : π 1 (Σ g, ) GL(V ϕ )., [ϕ] X GL(n,K) M g, f M g, V ϕ ϕ(γ) ϕ(f γ) (γ π 1 (Σ g, )) well-defined,, A ϕ., Ψ : M g, GL(V ϕ ) Ker A ϕ = {γ π 1 (Σ g, ) ; [γ, π 1 (Σ g, )] Ker ϕ}., π 1 (Σ g, ), 2.7, ϕ, Ψ., V ϕ End (n, K), dim V ϕ n 2. : 2.9 ([17]). g 2. 1 M g, K, n X GL(n,K) M g, π 1 (Σ g, ).,, M g, K n 2.. 2.10. Goldman [13],,, ( ),,. 2.9, M g,.,, 1. 2.11. 2.8 G = PSL(2, R). π 1 (Σ g, ) PSL(2, R),, Σ g., ϕ : π 1 (Σ g, ) PSL(2, R). X PSL(2,R), M g stabilizer,., 2.8, ϕ M g, PSL(2, R).

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