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?. If the complex is 1-dimensional one gets a graph of groups?.

Definition

We will initially give the definition in its ‘bare hands’ form. Here KK is a simplicial complex

A complex of groups, G(K)G(K), on KK is specified by the data, ({G σ},{ψ a},{g a,b})(\{G_\sigma\}, \{\psi_a\}, \{g_{a, b}\}) given by

  • a group, G σG_{\sigma}, for each simplex, σ\sigma, of KK;

  • an injective homomorphism,

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

for each edge, aE Ka \in E_K, of the barycentric subdivision of KK;

  • for each pair of composable edges, aa and bb, in E KE_K, an element g a,bG t(a)g_{a, b} \in G_{t(a)} is given such that
g a,b 1ψ ba( )g a,b=ψ aψ bg^{- 1}_{a, b} \psi_{ba} (_-) g_{a, b} = \psi_a \psi_b

and such that the ‘cocycle condition’

g a,cbψ a(g b,c)=g ab,cg a,bg_{a, cb} \psi_a (g_{b, c}) = g_{ab, c} g_{a, b}

holds.

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.

See also:

Revised on March 15, 2013 16:08:35 by Tim Porter (95.147.236.94)