


Torsion Subcomplexes Subpackage by Alexander D. Rahm and Bui Anh Tuan, version 2.0 

Consider
a cell complex with a cellular action of a discrete group G on
it, and consider a prime number p. The goal for the usage of this
subpackage is to compute the homological ptorsion of G, by which we
mean the modulo p homology of G (i.e with nontwisted Z/pZ
coefficients) in degrees above the virtual cohomological dimension, or
the modulo p FarrellTate cohomology of G.
For the computation of the homological ptorsion of G, only the ptorsion subcomplex is relevant, consisting of all the cells the stabilizers in G of which contain elements of order p. 

For
instance, let us input Soulé's cell complex for SL_3(Z). 

gap>
S:= ContractibleGcomplex("SL3Z"); Nonfree resolution in characteristic 0 for matrix group with 65 generators. No contracting homotopy available. 

Rigid
Facets
Subdivision allows us to recover (essentially) Soulé's
subdivision of the above truncated cube, which is a fundamental domain
for a cell complex for SL_3(Z) such that each cell stabilizer fixes its
cell pointwise. 

gap>
R := RigidFacetsSubdivision(S); Nonfree resolution in characteristic 0 for matrix group with 65 generators. No contracting homotopy available. 

Now
that the cell stabilizers are "small" enough, it becomes useful to
extract the 3torsion subcomplex. 

gap>
TorsionSubcomplex(R,3); 

To
this 3torsion subcomplex, we can apply the torsion subcomplexes
reduction technique.
In fact, every time that two adjacent edges and their joining vertex satisfy the following conditions on their stabilizers, we can merge them without changing the equivariant modulo p Farrell homology of the ptorsion subcomplex [see the paper "Accessing the FarrellTate cohomology of discrete groups" on how torsion subcomplex reduction works in detail]. One of the sufficient conditions reads as follows. Let G_1 and G_2 be the stabilizers of the two adjacent edges, and let S be the stabilizer of their joining vertex. Then we require G_1 and G_2 to be isomorphic and either G_1 to be isomorphic to S or S to be pnormal and G_1 to be isomorphic to the normaliser in S of the center of a Sylowpsubgroup of S. 

gap>
ReduceTorsionSubcomplex(R,3); 

Then
we obtain the reduced system of stabilizer inclusion displayed in
Soulé's paper. 

Download
of the Torsion Subcomplexes Subpackage at:
http://math.uni.lu/~rahm/subpackagedocumentation/
TorsionSubcomplexesSubpackage.tar.gz. 

