Extension of complexes of groups
Annales de l'Institut Fourier, Volume 42 (1992) no. 1-2, pp. 275-311.

Complexes of groups G(X) over ordered simplicial complexes X are generalizations to higher dimensions of graphs of groups. We first relate them to complexes of spaces by considering their classifying space BG(X). Then we develop their homological algebra aspects. We define the notions of homology and cohomology of a complex of groups G(X) with coefficients in a G(X)-module and show the existence of free resolutions. We apply those notions to study extensions of complexes of groups with constant or abelian kernel.

Les complexes de groupes G(X) sur des complexes simpliciaux ordonnés X sont des généralisations des graphes de groupes. Nous les mettons d’abord en relation avec les complexes d’espaces en considérant leur espace classifiant BG(X). Puis nous développons quelques notions d’algèbre homologique pour ces complexes G(X) qui généralisent les notions correspondantes pour les groupes. Nous définissons les groupes de cohomologie ou d’homologie de G(X) à coefficients dans un G(X)-module et nous montrons l’existence de résolutions libres. Nous appliquons ces notions pour étudier les extensions de complexes de groupes avec noyau constant ou abélien.

     author = {Haefliger, Andr\'e},
     title = {Extension of complexes of groups},
     journal = {Annales de l'Institut Fourier},
     pages = {275--311},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {42},
     number = {1-2},
     year = {1992},
     doi = {10.5802/aif.1292},
     zbl = {0762.20018},
     mrnumber = {93j:20080},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1292/}
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Haefliger, André. Extension of complexes of groups. Annales de l'Institut Fourier, Volume 42 (1992) no. 1-2, pp. 275-311. doi : 10.5802/aif.1292. https://aif.centre-mersenne.org/articles/10.5802/aif.1292/

[1] H. Bass, Covering theory for graphs of groups, preprint, Columbia University. | Zbl

[2] C. Bonatti & A. Haefliger, Déformations de feuilletages, Topology, 29 (1990), 205-229. | MR | Zbl

[3] K.S. Brown, Cohomology of groups, Graduate texts in Math., 87 (1982), Springer Verlag. | MR | Zbl

[4] J.M. Corson, Complexes of groups, Proc. of the London Math. Soc., to appear. | Zbl

[5] A. Haefliger, Complexes of groups and orbihedra, in “Group theory from a geometrical viewpoint, 26 March-6 April 1990, ICTP, Trieste”, World Scientific (1991), 504-540. | MR | Zbl

[6] D. Quillen, Higher algebraic K-theory : I, in Algebraic K-theory I, Battelle Institute Conf., 1972, Springer LN in Mathematics, 341 (1973), 77-139.

[7] S. Mac Lane, Homology, Grundlehren der Math. Wiss., 114 (1967), Springer Verlag.

[8] J. Milnor, The geometric realization of a semi-simplicial complex, Ann. of Math., 65 (1957), 357-362. | MR | Zbl

[9] G.P. Scott & C.T.C. Wall, Topological methods in group theory, Homological group theory, LMS Lect. Notes 36, Cambridge University Press (1979), 137-203. | MR | Zbl

[10] G. Segal, Classifying space and spectral sequences, Publ. Math. IHES, 134 (1968), 105-112. | Numdam | MR | Zbl

[11] J.-P. Serre, Trees, Springer Verlag, Berlin (1980), Translation of “Arbres, Amalgames, Sl2”, Astérisque, 46 (1977). | Zbl

[12] J.R. Stallings, Non positively curved triangles of groups, “Group theory from a geometrical viewpoint, 26 March-6 April 1990, ICTP, Trieste”, World Scientific (1991), 491-503. | MR | Zbl

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