Groupes fondamentaux des variétés de dimension $3$ et algèbres d’opérateurs

Nous proposons une caractérisation géométrique des variétés de dimension $3$ ayant des groupes fondamentaux dont toutes les classes de conjugaison autres que $\left\{1\right\}$ sont infinies, c’est-à-dire dont les algèbres de von Neumann sont des facteurs de type $I{I}_1$  : ce sont essentiellement les $3$-variétés à groupes fondamentaux infinis qui n’admettent pas de fibration de Seifert. Autrement dit et plus précisément, soient $M$ une $3$-variété connexe compacte et $\Gamma$ son groupe fondamental, qu’on suppose être infini et avec au moins une classe de conjugaison finie autre que $\left\{1\right\}$. Si $M$ est orientable, alors $\Gamma$ est groupe fondamental d’une variété de Seifert  ; si $M$ est non orientable, alors $\Gamma$ est groupe fondamental d’une variété de Seifert modulo $\mathbb{P}$ au sens de Heil et Whitten [HeWh-94].

Nous faisons un usage intensif de résultats concernant les $3$-variétés, autant classiques (comme on les trouve dans les livres de Hempel, Jaco et Shalen) que plus récents (solution de la conjecture des fibrés de Seifert).

We provide a geometric characterization of manifolds of dimension $3$ with fundamental groups of which all conjugacy classes except $\left\{1\right\}$ are infinite, namely of which the von Neumann algebras are factors of type $I{I}_1$: they are essentially the $3$-manifolds with infinite fundamental groups on which there does not exist any Seifert fibration.

Otherwise said and more precisely, let $M$ be a compact connected $3$-manifold and let $\Gamma$ be its fundamental group, supposed to be infinite and with at least one finite conjugacy class besides $\left\{1\right\}$. If $M$ is orientable, then $\Gamma$ is the fundamental group of a Seifert manifold; if $M$ is not orientable, then $\Gamma$ is the fundamental group of a Seifert manifold modulo $\mathbb{P}$ in the sense of Heil and Whitten [HeWh-94].

We make heavy use of results on $3$-manifolds, as well classical results (as can be found in the books of Hempel, Jaco, and Shalen), as more recent ones (solution of the Seifert fibred space conjecture)

[BeHa-94] Bekka (M.) et de la Harpe (P.).— Représentations d’un groupe faiblement équivalentes à la représentation régulière, Bull. Soc. math. France 122, p. 333-342 (1994). | Numdam | Zbl 0824.22011

[BoMP-03] Boileau (M.), Maillot (S.) et Porti (J.).— Three-dimensional orbifolds and their geometric structures, Panoramas et synthèses 15, Soc. Math. France (2003). | MR 2060653 | Zbl 1058.57009

[Bore-60] Borel (A.).— Density properties for certain subgroups of semisimple Lie groups without compact factors, Annals of Math. 72, p. 179-188 (1960) [Oeuvres, volume II, pages 125-134]. | MR 123639 | Zbl 0094.24901

[Bowd-04] Bowditch (B.).— Planar groups and the Seifert conjecture, J. reine angew. Math. 576, p. 11-62 (2004). | MR 2099199 | Zbl 1056.20029

[Brow-82] Brown (K.S.).— Cohomology of groups, Springer (1982). | MR 672956 | Zbl 0584.20036

[BuMu-70] Burde (G.) et Murasugi (K.).— Links and seifert fiber spaces, Duke Math. J. 37, p. 89-93 (1970). | MR 253313 | Zbl 0195.54003

[Dixm-69] Dixmier (J.).— Les C${}^{*}$-algèbres et leurs représentations, Gauthier-Villars (1969). | MR 246136 | Zbl 0174.18601

[Epst-72] Epstein (D.B.A.).— Periodic flows on $3$-manifolds, Annals of Math. 95, p. 66-82 (1972). | MR 288785 | Zbl 0231.58009

[Gaba-92] Gabai (D.).— Convergence groups are Fuchsian groups, Ann. of Math. 136, p. 447-510 (1992). | MR 1189862 | Zbl 0785.57004

[Harp-07] de la Harpe (P.).— On simplicity of reduced C${}^{*}$-algebras of groups, Bull. London Math. Soc., 39, p. 1-26 (2007). | MR 2303514 | Zbl 05139536

[HeWh-94] Heil (W.) et Whitten (W.).— The Seifert fiber space conjecture and torus theorem for non-orientable $3$-manifold, Canad. Math. Bull 37(4), p. 482-489 (1994). | MR 1303675 | Zbl 0829.57008

[HeJa-72] Hempel (J.) et Jaco (W.).— Fundamental groups of $3$-manifolds which are extensions, Annals of Math. 95, p. 86-98 (1972). | MR 287550 | Zbl 0226.57003

[Hemp-76] Hempel (J.).— $3$-manifolds, Princeton Univ. Press (1976). | MR 415619 | Zbl 0345.57001

[Hill-87] Hillman (J.A.).— Three-dimensional Poincaré duality groups which are extensions, Math. Z. 195, p. 89-9 (1987). | MR 888129 | Zbl 0597.57009

[Hopf-25] Hopf (H.).— Zum Clifford-Kleinnchen Raumproblem, Math. Ann. 95, p. 340-367 (1925).

[Jaco-77] Jaco (W.).— Lectures on three-manifold topology, CBMS 43, Amer. Math. Soc. (1977). | MR 565450 | Zbl 0433.57001

[JaSh-79] Jaco (W.), Shalen (P.).— Seifert fibre space in $3$-manifolds, Memoir 220, Amer. Math. Soc. (1979). | MR 537728 | Zbl 0415.57005

[Mail-03] Maillot (S.).— Open $3$-manifolds whose fundamental groups have infinite center, and a torus theorem for $3$-orbifolds, Trans. Amer. Math. Soc. 355, p. 4595-4638 (2003). | MR 1990764 | Zbl 1030.57029

[Miln-57] Milnor (J.).— Groups which act on ${𝕊}^n$ without fixed points, Amer. J. Math. 79, p. 623-630 (1957) [Collected Papers, Volume 2, pp. 93 et 97-104]. | MR 90056 | Zbl 0078.16304

[Mose-71] Moser (L.).— Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), p. 737-745. | MR 383406 | Zbl 0202.54701

[Neum-54] Neumann (B.H.).— Groups covered by permutable subsets, J. London Math. Soc. 29, p. 236-248 (1954). | MR 62122 | Zbl 0055.01604

[PaSa-79] W. Paschke (W.), Salinas (N.).— $C^{*}$-algebras associated with the free products of groups , Pacific J. Math. 82, p. 211-221 (1979). | MR 549845 | Zbl 0413.46049

[ROIV] Murray (F.J.), von Neumann (J.).— On rings of operators, IV, Annals of Math. 44, p. 716-808 (1943) [Collected Works, Volume III, p. 229-321]. | MR 9096 | Zbl 0060.26903

[Rotm-95] Rotman (J.J.).— An introduction to the theory of groups, fourth edition, Springer (1995) [First edition 1965]. | MR 1307623 | Zbl 0810.20001

[Rubi-95] Rubinstein (J.H.).— An algorithm to recognize the $3$-sphere, Proc. ICM Zurich 1994 Vol. 1 (Birkhäuser 1995), p. 601-611. | MR 1403961 | Zbl 0864.57009

[Saka-71] Sakai (S.).— C${}^{*}$-algebras and W${}^{*}$-algebras, Springer (1971). | MR 442701 | Zbl 0219.46042

[Sco-83a] Scott (P.).— There is no fake Seifert fibre space with infinite ${\pi }_1$, Annals of Math. 117, p. 35-70 (1983). | MR 683801 | Zbl 0516.57006

[Sco-83b] Scott (P.).— The geometries of $3$-manifolds, Bull. London Math. Soc. 15 :5, p. 401-487 (1983). | MR 705527 | Zbl 0561.57001

[SeTh-34] Seifert (H.) et Threlfall (W.).— A textbook of topology, Academic Press (1980) [traduit de  : Lehrbuch der Topology, Teubner, 1934]. | MR 575168

[Stal-06] Stalder (V.).— Moyennabilité intérieure et extensions HNN, Ann. Inst. Fourier 56, p. 309-323 (2006). | Numdam | MR 2226017

[Swar-73] Swarup (G.A.).— Projective planes in irreducible $3$-manifolds, Math. Z. 132, p. 305-317 (1973). | MR 322883 | Zbl 0249.57003

[Toll-70] Tollefson (J.).— Free involutions on non-prime $3$-manifolds, Osaka J. Math. 7, p. 161-164 (1970). | MR 266184 | Zbl 0198.28503

[Toll-78] Tollefson (J.).— Involutions on Seifert fiber spaces, Pacific J. Math. 74, p. 519-529 (1978). | MR 645400 | Zbl 0395.57022

[Whit-92] Whitten (W.).— Recognizing non-orientable Seifert Manifolds, J. Knot Theory and its ramifications 1, p. 471-475 (1992). | MR 1194999 | Zbl 0783.57005