Let \( \mathbb{G} \) be the group of automorphisms of a free group F∞ of infinite order. Let ℍ be the stabilizer of the first m generators of F∞. We show that the double cosets Γm = ℍ \ \( \mathbb{G} \)/ℍ admit a natural semigroup structure. For any compact group K, the semigroup Γm acts in the space L on the product of m copies of K. Bibliography: 20 titles.
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References
W. Goldman, “An ergodic action of the outer automorphism group of a free group,” Geom. Funct. Anal., 17, No. 3, 793–805 (2007).
W. Goldman and E. Z. Xia, “Action of the Johnson–Torelli group on representation varieties,” Proc. Amer. Math. Soc., 140, No. 4, 1449–1457 (2012).
R. S. Ismagilov, “Elementary spherical functions on the group SL(2, P) over a field P, which is not locally compact, with respect to the subgroup of matrices with integral elements,” Math. USSR-Izvestiya, 3, No. 2, 349–380 (1967).
R. S. Ismagilov, “Spherical functions over a normed field whose residue field is infinite,” Funct. Anal. Appl., 4, 37–45 (1970).
R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Reprint of the 1977 edition, Springer-Verlag, Berlin (2001).
Yu. A. Neretin, Categories of Symmetries and Infinite-Dimensional Groups, Oxford Univ. Press, New York (1996).
Yu. A. Neretin, “Multi-operator colligations and multivariate spherical functions,” Anal. Math. Phys., 1, No. 2–3, 121–138 (2011).
Yu. A. Neretin, “Sphericity and multiplication of double cosets for infinite-dimensional classical groups,” Funct. Anal. Appl., 45, No. 3, 225–239 (2011).
Yu. A. Neretin, “Infinite tri-symmetric group, multiplication of double cosets, and checker topological field theories,” Int. Math. Res. Not. IMRN, No. 3, 501–523 (2012).
Yu. A. Neretin, “Symmetries of Gaussian measures and operator colligations,” J. Funct. Anal., 263, No. 3, 782–802 (2012).
Yu. A. Neretin, “Infinite-dimensional p-adic groups, semigroups of double cosets, and inner functions on Bruhat–Tits buildings,” Izv. Math., 79, No. 3, 512–553 (2015).
Yu. A. Neretin, “Infinite symmetric groups and combinatorial constructions of topological field theory type,” Russian Math. Surveys, 70, No. 4, 715–773 (2015).
G. I. Olshanski, “Unitary representations of the infinite symmetric group: a semigroup approach,” in: A. A. Kirillov (ed.), Representations of Lie groups and Lie algebras, Akad. Kiado, Budapest (1985), pp. 181–197.
G. I. Olshanski, “On semigroups related to infinite-dimensional groups,” in: A. A. Kirillov (ed.), Topics in Representation Theory, Adv. Sov. Math., 2, Amer. Math. Soc., Providence, Rhode Island (1991), pp. 67–101.
G. I. Olshanski, “Unitary representations of infinite dimensional pairs (G,K) and the formalism of R. Howe,” in: A. M. Vershik and D. P. Zhelobenko (eds.), Representation of Lie Groups and Related Topics, Gordon and Breach, Amsterdam (1990), pp. 269–463.
D. Pickrell and E. Z. Xia, “Ergodicity of mapping class group actions on representation varieties. II. Surfaces with boundary,” Transform. Groups, 8, No. 4, 397–402 (2003).
M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York–London (1972).
A. N. Shiryaev, Probability, 2nd edition, Springer-Verlag, New York (1996).
A. M. Vershik, “Multivalued mappings with invariant measure (polymorphisms) and Markov operators,” J. Sov. Math., 23, 2243–2266 (1983).
K. Vogtmann, “Automorphisms of free groups and outer space,” Geom. Dedicata, 94, 1–31 (2002).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 436, 2015, pp. 189–198.
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Neretin, Y.A. Several Remarks on Groups of Automorphisms of Free Groups. J Math Sci 215, 748–754 (2016). https://doi.org/10.1007/s10958-016-2880-4
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DOI: https://doi.org/10.1007/s10958-016-2880-4