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.)

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.

Developability and the universal cover

(to come later)

The universal group and fundamental group

(to come later)

Examples

(to come later)

Complexes of groups as pseudofunctors.

see paper by Tom Fiore et al (below)

References

• 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:

• Thomas M. Fiore, Wolfgang Lück, and Roman Sauer, Euler Characteristics of Categories and Homotopy Colimits, (arXiv: 1007.3868)