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Graphs of groups
A graph of groups is a
connected graph Y together with
s(a) e = e t(a)
for every a in Ge . That is, the composite paths ![]() and ![]() are regarded as equal paths from v to w. |
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Amalgamated
products and HNN extensions can both be viewed as fundamental groups of
graphs of
groups. For example, the following commands create a graph of groups
corresponding to the amalgamated product G=H*AK where H is
the symmetric group H=S5 and K is the symmetric group K=S4
. The common subgroup A is the symmetric group A=S3 . A graph of groups is represented by a list consisting of the vertex groups and pairs of monomorphisms which define edges. Each group must be given a name using the SetName() command, and distinct groups must be given different names. |
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gap>
S5:=SymmetricGroup(5);SetName(S5,"S5");; gap> S4:=SymmetricGroup(4);SetName(S4,"S4");; gap> A:=SymmetricGroup(3);SetName(A,"S3");; gap> AS5:=GroupHomomorphismByFunction(A,S5,x->x);; gap> AS4:=GroupHomomorphismByFunction(A,S4,x->x);; gap> D:=[S5,S4,[AS5,AS4]];; gap> GraphOfGroupsDisplay(D);; ![]() |
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The
following additional commands create a resolution for the above
amalgamated product G and then calculate H7(G,Z) = (Z2)3+Z4+Z60 |
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gap>
R:=ResolutionGraphOfGroups(D,8);; gap> Homology(TensorWithIntegers(R),7); [ 2, 2, 2, 4, 60 ] |
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The
following commands create a graph of groups corresponding to the HNN
extension G=H*A where H is the symmetric group H=S5
and A is the subgroup A=S3 and f:A→S5 is the
homomorphism (1,2) → (3,4), (1,2,3) → (3,4,5). The HNN extension G is
obtained from H by adding an element t subject to the relations t-1
a t = f(a) for a in S3. |
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gap>
S5:=SymmetricGroup(5);SetName(S5,"S5"); gap> A:=SymmetricGroup(3);SetName(A,"S3"); gap> f:=GroupHomomorphismByImages(A,S5,[(1,2),(1,2,3)],[(3,4),(3,4,5)]);; gap> g:=GroupHomomorphismByFunction(A,S5,x->x); gap> D:=[S5,[f,g]];; gap> GraphOfGroupsDisplay(D);; ![]() |
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The following additional commands create a resolution for the HNN extension and calculate H7(G,Z) = (Z2)2+Z60 . | |||
gap>
R:=ResolutionGraphOfGroups(D,8);; gap> Homology(TensorWithIntegers(R),7); [ 2, 2, 60 ] |
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Fuchsian and Kleinian groups A Kleinian group is discrete subgroup
of PSL(2,C),
the full group of orientation preserving isometries of 3-dimensional
hyperbolic space. A Fuchsian group is a discrete
subgroup of PSL(2,R)
and as such acts on the hyperbolic plane. A "fundamental domain" for a
Kleinian or Fuchsian group G gives rise to a tessellation of hyperbolic
3- or 2-space. Let X denote the 1-skeleton of this tessellation. Then G
acts on X in such a way that no edge is inverted. The quotient graph
Y=X/G is thus a graph of groups in which vertices and edges are
labelled by
the subgroups of G stabilizing the corresponding vertices and edges in
Y.
The computational method described in [G. Ellis & A.G. Williams, "On the cohomology of generalized triangle groups", Comment. Math. Helv. 80 (2005), 1-21] can be partially summarized as follows.
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As
an example consider the classical triangle group defined by the
presentation T = T(l,m,n) = < a, b |
al
= bm = (ab-1)n = 1 >
where 1/|l| + 1/|m| + 1/|m| <
1. This group acts on the hyperbolic plane as follows. Let v1
and v2 be distinct point in the hyperbolic plane. Let the
generator a of T act as a clockwise rotation about v1
through an angle 2 pi/¦l| , and let the generator b of T act as
a clockwise rotation about v2 through an angle 2
pi/¦m| . It follows that (ab-1) acts as an
anti-clockwise rotation about some point v3 through an angle
2 pi/|n|. Let v4 be the image of v3 under a
reflection in the line v1v2 . Then the points v1
, v2 , v3 , v4 are the corners of a
quadrilateral fundamental region for the action of T. The edges of this
fundamental region have trivial stabilizer subgroups in G. The vertices
v1 , v2 , v3 have cyclic stabilizer
subgroups A = <a>, B=<b> and C=<c> respectively.
The associated graph of groups is ![]() The fundamental group P of this
graph has the same homology as the triangle group T(l,m,n) in
dimensions greater than 3.
So for example, the following commands show that the triangle group T=T(2,3,4) has 5-dimensional integral homology H5(T,Z) = Z2+Z12 . |
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gap>
a:=(1,2);; b:=(1,2,3);; c:=(1,2,3,4);; f:=();; g:=();; gap> A:=Group(a);; SetName(A,"A");; gap> B:=Group(b);; SetName(B,"B");; gap> C:=Group(c);; SetName(C,"C");; gap> F:=Group(f);; SetName(F,"F");; gap> G:=Group(g);; SetName(G,"G");; gap> FA:=GroupHomomorphismByFunction(F,A,x->x);; gap> FC:=GroupHomomorphismByFunction(F,C,x->x);; gap> GC:=GroupHomomorphismByFunction(G,C,x->x);; gap> GB:=GroupHomomorphismByFunction(G,B,x->x);; gap> Graph:=[A,B,C,[FA,FC],[GC,GB]];; gap> GraphOfGroupsDisplay(Graph);; ![]() gap>
R:=ResolutionGraphOfGroups(Graph,6);;
gap> Homology(TensorWithIntegers(R),5); [ 2, 12 ] |
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