Several remarks on groups of automorphisms of free groups

Yu. A. Neretin

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXV, Tome 436 (2015), pp. 189-198

Résumé

Let $\mathbb G$ be the group of automorphisms of a free group $F_\infty$ of infinite order. Let $\mathbb H$ be the stabilizer of the first $m$ generators of $F_\infty$. We show that the double cosets $\Gamma_m=\mathbb{H\setminus G/H}$ admit a natural semigroup structure. For any compact group $K$, the semigroup $\Gamma_m$ acts in the space $L^2$ on the product of $m$ copies of $K$.

Bibliographie

[1] W. Goldman, “An ergodic action of the outer automorphism group of a free group”, Geom. Funct. Anal., 17:3 (2007), 793–805 | DOI | MR | Zbl

[2] W. Goldman, E. Z. Xia, “Action of the Johnson–Torelli group on representation varieties”, Proc. Amer. Math. Soc., 140:4 (2012), 1449–1457 | DOI | MR | Zbl

[3] R. S. Ismagilov, “Elementarnye sfericheskie funktsii na gruppe $SL(2,P)$ nad polem $P$, ne yavlyayuschimsya lokalno kompaktnym, otnositelno podgruppy matrits s tselymi elementami”, Izv. AN SSSR. Ser. matem., 31:2 (1967), 361–390 | MR | Zbl

[4] R. S. Ismagilov, “Sfericheskie funktsii nad normirovannym polem, pole vychetov kotorogo beskonechno”, Funkts. anal. i ego pril., 4:1 (1970), 42–51 | MR | Zbl

[5] R. Lindon, P. Shupp, Kombinatornaya teoriya grupp, Mir, M., 1980 | MR

[6] Yu. A. Neretin, Kategorii simmetrii i beskonechnomernye gruppy, URSS, M., 1998

[7] Yu. A. Neretin, “Multi-operator colligations and multivariate spherical functions”, Anal. Math. Phys., 1:2–3 (2011), 121–138 | DOI | MR | Zbl

[8] Yu. A. Neretin, “Sferichnost i umnozhenie dvoinykh klassov smezhnosti dlya beskonechnomernykh klassicheskikh grupp”, Funkts. anal. i ego pril., 45:3 (2011), 79–96 | DOI | MR | Zbl

[9] Yu. Neretin, “Infinite tri-symmetric group, multiplication of double cosets, and checker topological field theories”, Int. Math. Res. Not. IMRN, 2012:3 (2012), 501–523 | MR | Zbl

[10] Yu. A. Neretin, “Symmetries of Gaussian measures and operator colligations”, J. Funct. Anal., 263:3 (2012), 782–802 | DOI | MR | Zbl

[11] Yu. A. Neretin, “Beskonechnomernye $p$-adicheskie gruppy, polugruppy dvoinykh klassov smezhnosti i vnutrennie funktsii na ansamblyakh Bryua–Titsa”, Izv. RAN. Ser. matem., 79:3 (2015), 87–130 | DOI | MR | Zbl

[12] Yu. A. Neretin, “Beskonechnaya simmetricheskaya gruppa i kombinatornye konstruktsii tipa topologicheskikh teorii polya”, Uspekhi mat. nauk, 70:4(424) (2015), 143–204 | DOI | MR

[13] G. I. Olshanski, “Unitary representations of the infinite symmetric group: a semigroup approach”, Representations of Lie Groups and Lie Algebras, ed. A. A. Kirillov, Akad. Kiado, Budapest, 1985, 181–197 | MR

[14] G. I. Olshanski, “On semigroups related to infinite-dimensional groups”, Topics in Representation Theory, Adv. Sov. Math., 2, ed. A. A. Kirillov, Amer. Math. Soc., Providence, RI, 1991, 67–101 | MR

[15] G. I. Olshanski, “Unitary representations of infinite-dimensional pairs $(G,K)$ and the formalism of R. Howe”, Representation of Lie Groups and Related Topics, Gordon and Breach, 1990, 269–463 | MR

[16] D. Pickrell, E. Z. Xia, “Ergodicity of mapping class group actions on representation varieties. II. Surfaces with boundary”, Transform. Groups, 8:4 (2003), 397–402 | DOI | MR | Zbl

[17] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki, v. 1, Funktsionalnyi analiz, Mir, M., 1977 | MR

[18] A. N. Shiryaev, Veroyatnost, Nauka, M., 1979 | MR

[19] A. M. Vershik, “Mnogoznachnye otobrazheniya s invariantnoi meroi (polimorfizmy) i markovskie operatory”, Zap. nauchn. semin. LOMI, 72, 1977, 26–61 | MR | Zbl

[20] K. Vogtmann, “Automorphisms of free groups and outer space”, Geom. Dedicata, 94 (2002), 1–31 | DOI | MR | Zbl