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?. If the complex is 1-dimensional one gets a graph of groups?.
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)
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.
See also: