This is an entry on the notion of complex of groups introduced by André Haefliger and Jon Corson? as a higher dimensional generalisation of the Bass-Serre theory of graphs of groups. (It does not refer to the idea of chain complexes of groups, i.e., chain complexes in the (more or less) usual sense.)
A complex of groups is a diagram of groups, homomorphisms and conjugations, corresponding, abstractly, to the system of inclusions of the stabiliser subgroups of an action of a group on a simplicial cell complex? or equivalently on a small category without loops?. If the complex is 1-dimensional one obtains a graph of groups - note however, the category of 1-complexes of groups is not equivalent as a category to the category of graphs of groups (see A. Thomas? Proposition 2.1).
We will initially give the definition in its ‘bare hands’ form. Here $K$ is a simplicial complex
A complex of groups, $G(K)$, on $K$ is specified by the data, $(\{G_\sigma\}, \{\psi_a\}, \{g_{a, b}\})$ given by
a group, $G_{\sigma}$, for each simplex, $\sigma$, of $K$;
an injective homomorphism,
for each edge, $a \in E_K$, of the barycentric subdivision of $K$;
and such that the ‘cocycle condition’
holds.
(to come later)
(to come later)
(to come later)
see paper by Tom Fiore et al (below)
M. Bridson and A. Haefliger, 1999, Metric Spaces of Non-Positive Curvature, number 31 in
Grundlehren der Math. Wiss, Springer.
A. Haefliger, 1991, Complexes of Groups and Orbihedra, in Group Theory from a Geometric
viewpoint , 504 – 540, ICTP, Trieste, 26 March- 6 April 1990, World Scientific.
J. M. Corson, Complexes of Groups, Proc. London Math. Soc., 65, (1992), 199–224.
A. Thomas?, 2006 Lattices acting on right-angled buildings, Alg. Geom. Top., 6, 1215-1238.
See also: