Extension of complexes of groups

Haefliger, André

Annales de l'Institut Fourier, …, Tome 42 (1992) no. 1-2, pp. 275-311.

Les complexes de groupes $G\left(X\right)$ 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 $BG\left(X\right)$. Puis nous développons quelques notions d’algèbre homologique pour ces complexes $G\left(X\right)$ qui généralisent les notions correspondantes pour les groupes. Nous définissons les groupes de cohomologie ou d’homologie de $G\left(X\right)$ à coefficients dans un $G\left(X\right)$-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.

Complexes of groups $G\left(X\right)$ 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 $BG\left(X\right)$. Then we develop their homological algebra aspects. We define the notions of homology and cohomology of a complex of groups $G\left(X\right)$ with coefficients in a $G\left(X\right)$-module and show the existence of free resolutions. We apply those notions to study extensions of complexes of groups with constant or abelian kernel.

[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). | Numdam | 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