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1 ( ),.,,, 1, [17] (2 ),,.,.,.,,,,,.,,,,.,,., K, K 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, Lawrence, B n Lawrence [24]. 2 D 2, n P n. D 2 P n kasahara.yasushi@kochi-tech.ac.jp

, Bigelow [4], Krammer [23], Artin B n, B n, M 2 6 Birman Hilden [7]. M 2,, g 3 M g M g,. 1.

2 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]) 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] ,, 2 1 Kida [19]. 1 G, G Γ, G/Γ, G, Borel.

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]).

3 , 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 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

, 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.

4 , 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, g 1, Z(M g,n ), ([28]). n = 0 : g 3, ; g = 1, 2, 2. (g, n) = (1, 1) :. : Dehn n., K , M g,n,. K[S], Σ g,n S K. M g,n S, K[S] M g,n., K M M g,n, M g,n Σ g,n (S ). p: K[S] M p, M.

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,.

5 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 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)

, 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]).

6 ., 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,,.

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).

7 , 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 Goldman [13],,, ( ),,. 2.9, M g,.,, 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).

, 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.

8 , M g, 3g , n 3g 3, X GL(n,C) M g π 1 (Σ g, ).,., Birman (2.2), 2.9.,., residually finite,,,,,. M g, residually finite Baumslag [2]. M g residually finite, Grossmann [14] Baumslag,, Bass Lubotzky [1], M g, residually finite, M g X GL(n,K), X GL(n,K), n 1 M g. ( M g, 2 π 1 (Σ g, ).) , p : K[S] M K?, Ker p M g,n? 1. Birman (2.2) 1 F n Aut (F n ) Out (F n ) 1, 2.9. Out (F n ) F n. Aut (F n ) n = 2, n 3, Out (F n ). 3., 2.1, M g,n, Σ g,n, Z(M g,n ). [18]. [1] H. Bass and A. Lubotzky, Automorphisms of groups and of schemes of finite type, Israel J. Math. 44 (1983), no. 1, [2] G. Baumslag, Automorphism groups of residually finite groups, J. London Math. Soc. 38 (1963), [3] P. Bellingeri, On presentations of surface braid groups, J. Algebra 274 (2004), no. 2,

1. Birman (2.2) 1 F n Aut (F n ) Out (F n ) 1, 2.9. Out (F n ) F n. Aut (F n ) n = 2, n 3, Out (F n ). 3., 2.1, M g,n, Σ g,n, Z(M g,n ). [18]. [1] H. Bass and A.

9 [4] S. J. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), no. 2, [5] S. J. Bigelow and R. D. Budney, The mapping class group of a genus two surface is linear, Algebr. Geom. Topol. 1 (2001), (electronic). [6] J. S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969), [7] J. S. Birman and H. Hilden, Isotopies of homeomorphisms of Riemann surfaces and a theorem about Artin s braid group, Ann. of Math. 97 (1973), [8] T. E. Brendle and H. Hamidi-Tehrani, On the linearity problem for mapping class groups, Algebr. Geom. Topol. 1 (2001), [9] E. Formanek, Braid group representations of low degree, Proc. London Math. Soc. (3) 73 (1996), no. 2, [10] E. Formanek, W. Lee, I. Sysoeva, and M. Vazirani, The irreducible complex representations of the braid group on n strings of degree n, J. Algebra Appl. 2 (2003), no. 3, [11] E. Formanek and C. Procesi, The automorphism group of a free group is not linear, J. Algebra 149 (1992), no. 2, [12] J. Franks and M. Handel, Triviality of some representations of MCG(S g ) in GL(n, C), Diff(S 2 ) and Homeo(T 2 ), Proc. Amer. Math. Soc. 141 (2013), no. 9, [13] W. M. Goldman, Mapping class group dynamics on surface group representations, Problems on mapping class groups and related topics, Proc. Sympos. Pure Math., vol. 74, Amer. Math. Soc., Providence, RI, 2006, pp [14] E. K. Grossman, On the residual finiteness of certain mapping class groups, J. London Math. Soc. (2) 9 (1974/75), [15] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, [16] Y. Kasahara, An expansion of the Jones representation of genus 2 and the Torelli group, Algebr. Geom. Topol. 1 (2001), [17], On visualization of the linearity problem for mapping class groups of surfaces, Geom. Dedicata 176 (2015), [18], Geometric intersection in representations of mapping class groups of surfaces, 2016, preprint, arxiv: v3. [19] Y. Kida, Measure equivalence rigidity of the mapping class group, Ann. of Math. (2) 171 (2010), no. 3, [20] M. Korkmaz, On the linearity of certain mapping class groups, Turkish J. Math. 24 (2000), no. 4, [21], The symplectic representation of the mapping class group is unique, 2011, preprint, arxiv: [22] D. Krammer, The braid group B 4 is linear, Invent. Math. 142 (2000), no. 3, [23], Braid groups are linear, Ann. of Math. (2) 155 (2002), no. 1, [24] R. J. Lawrence, Homological representations of the Hecke algebra, Comm. Math. Phys. 135 (1990), no. 1, [25] F. Luo, On non-separating simple closed curves in a compact surface, Topology 36 (1997), no. 2, [26] W. Magnus and A. Peluso, On a theorem of V. I. Arnol d, Comm. Pure Appl. Math. 22 (1969), [27] T. Moriyama, The mapping class group action on the homology of the configuration spaces of surfaces, J. Lond. Math. Soc. (2) 76 (2007), no. 2,

Hilden, Isotopies of homeomorphisms of Riemann surfaces and a theorem about Artin s braid group, Ann. of Math. 97 (1973), 424 439. [8] T. E. Brendle and H.

10 [28] L. Paris and D. Rolfsen, Geometric subgroups of mapping class groups, J. Reine Angew. Math. 521 (2000), [29] G. P. Scott, Braid groups and the group of homeomorphisms of a surface, Proc. Cambridge Philos. Soc. 68 (1970),

Reine Angew. Math. 521 (2000), 47 83. [29] G. P.

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