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.

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