On the group of substitutions of formal power series with integer coefficients

I. K. Babenko ; S. A. Bogatyi

Izvestiya. Mathematics , Tome 72 (2008) no. 2, pp. 241-264.

Résumé

We study certain properties of the group $\mathcal J(\mathbb Z)$ of substitutions of formal power series in one variable with integer coefficients. We show that $\mathcal J(\mathbb Z)$, regarded as a topological group, has four generators and cannot be generated by fewer elements. In particular, we show that the one-dimensional continuous homology of $\mathcal J(\mathbb Z)$ is isomorphic to $\mathbb Z\oplus\mathbb Z\oplus\mathbb Z_2\oplus\mathbb Z_2$. We study various topological and geometric properties of the coset space $\mathcal J(\mathbb R)/\mathcal J(\mathbb Z)$. We compute the real cohomology $\widetilde{H}^*\bigl(\mathcal J(\mathbb Z); \mathbb R\bigr)$ with uniformly locally constant supports and show that it is naturally isomorphic to the cohomology of the nilpotent part of the Lie algebra of formal vector fields on the line.

Bibliographie

[1] S. A. Jennings, “Substitution groups of formal power series”, Canadian J. Math., 6 (1954), 325–340 | MR | Zbl

[2] M. Gotô, “On the group of formal analytic transformations”, Kōdai Math. Sem. Rep., 1950, no. 3, 45–46 | DOI | MR | Zbl

[3] V. M. Bukhshtaber, A. V. Shokurov, “The Landweber–Novikov algebra and formal vector fields on the line”, Funkt. Anal. Appl., 12:3 (1978), 159–168 | DOI | MR | Zbl

[4] R. Camina, “The Nottingham group”, New horizons in pro-$p$ groups, Progr. Math., 184, eds. M. P. F. du Sautoy, D. Segal, and A. Shalev, Birkhäuser, Boston, MA, 2000, 205–221 | MR | Zbl

[5] C. R. Leedham-Green, S. McKay, The structure of groups of prime power order, London Mathematical Society Monographs. New Series, 27, Oxford University Press, Oxford, 2002 | MR | Zbl

[6] I. Fesenko, “On just infinite pro-$p$-groups and arithmetically profinite extensions of local fields”, J. Reine Angew. Math., 517 (1999), 61–80 | DOI | MR | Zbl

[7] L. S. Pontryagin, Topological groups, Gordon and Breach, New York–London–Paris, 1966 | MR | MR | Zbl

[8] G. Bredon, Vvedenie v teoriyu kompaktnykh grupp preobrazovanii, Nauka, M., 1980 ; G. E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, 46, Academic Press, New York–London, 1972 | MR | Zbl | MR | Zbl

[9] D. L. Johnson, “The group of formal power series under substitution”, J. Austral. Math. Soc. Ser. A, 45:3 (1988), 296–302 | DOI | MR | Zbl

[10] B. Szegedy, “Almost all finitely generated subgroups of the Nottingham group are free”, Bull. London Math. Soc., 37:1 (2005), 75–79 | DOI | MR | Zbl

[11] B. Ya. Levin, Raspredelenie kornei tselykh funktsii, GITTL, M., 1956 | MR | Zbl

[12] E. Spener, Algebraicheskaya topologiya, Mir, M., 1971 ; E. Spanier, Algebraic topology, McGraw-Hill Book, New York–Toronto, 1966 | MR | Zbl | MR | Zbl

[13] K. Nomizu, “On the cohomology of compact homogeneous spaces of nilpotent Lie groups”, Ann. of Math. (2), 59:3 (1954), 531–538 | DOI | MR | Zbl

[14] L. V. Goncharova, “The cohomologies of Lie algebras of formal vector fields on the line”, Funkt. Anal. Appl., 7:2 (1973), 91–97 | DOI | MR | Zbl

[15] B. Klopsch, “Normal subgroups in substitution groups of formal power series”, J. Algebra, 228:1 (2000), 91–106 | DOI | MR | Zbl

[16] J. Milnor, “Remarks on infinite-dimensional Lie groups”, Relativity, groups and topology, II (Les Houches, 1983), eds. B. S. de Witt, R. Stora, North-Holland, Amsterdam, 1984, 1007–1057 | MR | Zbl

[17] A. I. Mal'tsev, “On a class of homogeneous spaces”, Am. Math. Soc. Transl., 39 (1951) | Zbl

[18] P. S. Alexandrov, P. S. Urysohn, “Über nulldimensionale Punktmengen”, Ann. Math., 98 (1928), 89–106 | DOI | MR

[19] R. Engelking, Obschaya topologiya, Mir, M., 1986 ; R. Engelking, General topology, Monografie Matematyczne, 60, Polish Scientific Publishers, Warsaw, 1977 | MR | MR | Zbl

[20] S. Eilenberg, N. Steenrod, Foundations of algebraic topology, Princeton Math. Ser., 15, Princeton University Press, Princeton, NJ, 1952 | MR | Zbl

[21] K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, No 3, Acad. Press, New York–London, 1967 | MR | Zbl

[22] Sze-tsen Hu, “Cohomology theory in topological groups”, Michigan Math. J., 1:1 (1952), 11–59 | DOI | MR | Zbl

[23] Zh.-P. Serr, Kogomologii Galua, Mir, M., 1968 ; J.-P. Serre, Cohomologie galoisienne, Troisième édition, Lecture Notes in Math., 5, Springer-Verlag, Berlin–New York, 1965 | MR | Zbl | MR | Zbl

[24] H. Hopf, “Über die Bettischen Gruppen, die zu einer beliebigen Gruppe gehören”, Comment. Math. Helv., 17:1 (1944), 39–79 | DOI | MR | Zbl

[25] B. Eckmann, “Der Cohomologie-Ring einer beliebigen Gruppe”, Comment. Math. Helv., 18:1 (1945), 232–282 | DOI | MR

[26] S. Eilenberg, “Topological methods in abstract algebra. Cohomology theory of groups”, Bull. Amer. Math. Soc., 55 (1949), 3–37 | DOI | MR | Zbl

[27] D. B. Fuks, Cohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986 | MR | MR | Zbl | Zbl

[28] H. H. Schaefer, Topological vector spaces, The Macmillan, New York; Collier-Macmillan, London, 1966 | MR | Zbl

[29] V. M. Bukhshtaber, A. N. Kholodov, “Groups of formal diffeomorphisms of the superline, generating functions for sequences of polynomials, and functional equations”, Math. USSR-Izv., 35:2 (1990), 277–305 | DOI | MR | Zbl

[30] V. M. Bukhshtaber, “Groups of polynomial transformations of a line, non-formal symplectic manifolds, and the Landweber–Novikov algebra”, Russian Math. Surveys, 54:4 (1999), 837–838 | DOI | MR | Zbl

[31] V. M. Buchstaber, A. Lazarev, “The Gelfand transform in commutative algebra”, Mosc. Math. J., 5:2 (2005), 311–327 | MR | Zbl