nLab complex of groups

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.)

Contents

Idea

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).

Definition

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,

$\psi_a :G_{i (a)} \rightarrow G_{t(a)},$

for each edge, $a \in E_K$, of the barycentric subdivision of $K$;

• for each pair of composable edges, $a$ and $b$, in $E_K$, an element $g_{a, b} \in G_{t(a)}$ is given such that
$g^{- 1}_{a, b} \psi_{ba} (_-) g_{a, b} = \psi_a \psi_b$

and such that the ‘cocycle condition’

$g_{a, cb} \psi_a (g_{b, c}) = g_{ab, c} g_{a, b}$

holds.

(to come later)

(to come later)

(to come later)

Complexes of groups as pseudofunctors.

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.