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< td
style= "text-align: center; vertical-align: top; color: rgb(0, 0, 102);">< big>< span
style= "font-weight: bold;">About HAP: Graphs of Groups, Fuchsian
groups and Kleinian groups< br>
</ span></ big></ td>
< td style= "text-align: right; vertical-align: top;">< a
href= "aboutSpaceGroup.html">< small style= "color: rgb(0, 0, 102);">next</ small></ a>< br>
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< div style= "text-align: center;">< big>< span
style= "font-weight: bold;">Graphs of groups< br>
< br>
</ span></ big> </ div>
A < span style= "font-style: italic;">graph of groups</ span> is a
connected graph Y together with< br>
< ul>
< li>groups G< sub>v</ sub> and G< sub>e</ sub> where v and e range
respectively over the vertices and edges of Y, < br>
</ li>
< li>and monomorphisms s:G< sub>e</ sub>&# 8594;G< sub>v</ sub> and t:G< sub>e</ sub>&# 8594;G< sub>w</ sub>
for every edge e, where v and w are the vertices of e.</ li>
</ ul>
The < span style= "font-style: italic;">fundamental group</ span> of a
graph of groups is defined using a notion of "path" where an element of
G< sub>v</ sub> is regarded as a path from v to v; one imposes the
relations< br>
< br>
< div style= "text-align: center;"> s( a) e = e t( a)< br>
</ div>
< br>
for every a in G< sub>e</ sub> . That is, the composite paths< br>
< br>
< div style= "text-align: center;">
< div style= "text-align: left;">
< div style= "text-align: center;">< img alt= "" src= "path.gif"
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< br>
< br>
and< br>
< br>
< br>
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< div style= "text-align: center;">< img alt= "" src= "pathop.gif"
style= "width: 336px; height: 95px;">< br>
< br>
< div style= "text-align: left;">< br>
are regarded as equal paths from v to w.< br>
</ div>
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< tr>
< td
style= "vertical-align: top; background-color: rgb(255, 255, 255);">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*< sub> A</ sub>K where H is
the symmetric group H= S< sub> 5</ sub> and K is the symmetric group K= S< sub> 4</ sub>
. The common subgroup A is the symmetric group A= S< sub> 3 </ sub>. < br>
< br>
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 < span
style= "font-family: helvetica,arial,sans-serif;">SetName()</ span>
command, and distinct groups must be given different names.< br>
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< td
style= "background-color: rgb(255, 255, 204); vertical-align: top;">gap>
S5:=SymmetricGroup( 5);SetName(S5, "S5");;< br>
gap> S4:=SymmetricGroup( 4);SetName(S4, "S4");;< br>
gap> A:=SymmetricGroup( 3);SetName( A, "S3");;< br>
gap> AS5:=GroupHomomorphismByFunction( A,S5,x->x);;< br>
gap> AS4:=GroupHomomorphismByFunction( A,S4,x->x);;< br>
gap> D:=[S5,S4,[AS5,AS4]];;< br>
gap> GraphOfGroupsDisplay(D);;< br>
< div style= "text-align: center;">< img alt= "" src= "graphgroups.gif"
style= "width: 172px; height: 90px;"> </ div>
</ td>
</ tr>
< tr>
< td
style= "background-color: rgb(255, 255, 255); vertical-align: top;">The
following additional commands create a resolution for the above
amalgamated product G and then calculate H< sub> 7</ sub>(G,Z) = (Z< sub> 2</ sub>)< sup> 3</ sup>+Z< sub> 4</ sub>+ Z<sub>60</sub><br>
</td>
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<td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
R:=ResolutionGraphOfGroups(D,8);;<br>
gap> Homology(TensorWithIntegers(R),7);<br>
[ 2, 2, 2, 4, 60 ]<br>
</td>
</tr>
<tr>
<td
style="text-align: left; background-color: rgb(255, 255, 255); vertical-align: top;">The
following commands create a graph of groups corresponding to the HNN
extension G=H*<sub>A</sub> where H is the symmetric group H=S<sub>5</sub>
and A is the subgroup A=S<sub>3</sub> and f:A→S<sub>5</sub> 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<sup>-1</sup>
a t = f(a) for a in S<sub>3</sub>.<br>
</td>
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<td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
S5:=SymmetricGroup(5);SetName(S5,"S5");<br>
gap> A:=SymmetricGroup(3);SetName(A,"S3");<br>
gap>
f:=GroupHomomorphismByImages(A,S5,[(1,2),(1,2,3)],[(3,4),(3,4,5)]);;<br>
gap> g:=GroupHomomorphismByFunction(A,S5,x->x);<br>
gap> D:=[S5,[f,g]];;<br>
gap> GraphOfGroupsDisplay(D);;<br>
<div style="text-align: center;"><img alt="" src="HNN.gif"
style="width: 118px; height: 76px;"><br>
</div>
</td>
</tr>
<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following additional commands create a resolution for the HNN extension
and calculate H<sub>7</sub>(G,Z) = (Z<sub>2</sub>)<sup>2</sup>+Z<sub>60</sub>
. </td>
</tr>
<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
R:=ResolutionGraphOfGroups(D,8);;<br>
gap> Homology(TensorWithIntegers(R),7);<br>
[ 2, 2, 60 ]<br>
</td>
</tr>
<tr align="center">
<td
style="vertical-align: top; background-color: rgb(255, 255, 255);"><big><span
style="font-weight: bold;">Fuchsian and Kleinian groups<br>
<br>
</span></big>
<div style="text-align: left;"><big><small>A <span
style="font-style: italic;">Kleinian group</span> is discrete subgroup
of PSL(2,<span style="font-family: helvetica,arial,sans-serif;">C</span>),
the full group of orientation preserving isometries of 3-dimensional
hyperbolic space. </small></big><big><small>A <span
style="font-style: italic;">Fuchsian group</span> is a discrete
subgroup of PSL(2,<span style="font-family: helvetica,arial,sans-serif;">R</span>)
and as such acts on the hyperbolic plane. A "fundamental domain" for a co-finite volume
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 (possibly a subdivision of) 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.<br>
<br>
The computational method described in [G. Ellis & A.G. Williams, "<a
href="http://hamilton.nuigalway.ie/">On the cohomology of generalized
triangle groups</a>", Comment. Math. Helv. 80 (2005), 1-21] can be
partially summarized as follows.<br>
<br>
</small></big>
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<td
style="vertical-align: top; background-color: rgb(204, 255, 255);">Let
G be a co-finite area Fuchsian group. Let Y be the graph of groups arising
from G and a fundamental domain. Let P be the fundamental group of Y.
Then the "obvious" quotient homomorphism P→G induces an isomorphism<br>
<div style="text-align: center;"><br>
H<sub>n</sub>(P,M) → H<sub>n</sub>(G,M)<br>
<br>
<div style="text-align: left;">for any ZG-module M and all
n>2 .<br>
</div>
</div>
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<big><small><br>
</small><span style="font-weight: bold;"> </span></big></div>
<div style="text-align: left;"> </div>
</td>
</tr>
<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 255);">As
an example consider the classical triangle group defined by the
presentation<br>
<br>
<div style="text-align: center;">T = T(l,m,n) = < a, b |
a<sup>l</sup>
= b<sup>m</sup> = (ab<sup>-1</sup>)<sup>n</sup> = 1 ><br>
<br>
<div style="text-align: left;">where 1/|l| + 1/|m| + 1/|m| <
1. This group acts on the hyperbolic plane as follows. Let v<sub>1</sub>
and v<sub>2</sub> be distinct point in the hyperbolic plane. Let the
generator a of T act as a clockwise rotation about v<sub>1</sub>
through an angle 2 pi/¦l| , and let the generator b of T act as
a clockwise rotation about v<sub>2</sub> through an angle 2
pi/¦m| . It follows that (ab<sup>-1</sup>) acts as an
anti-clockwise rotation about some point v<sub>3</sub> through an angle
2 pi/|n|. Let v<sub>4</sub> be the image of v<sub>3</sub> under a
reflection in the line v<sub>1</sub>v<sub>2</sub> . Then the points v<sub>1</sub>
, v<sub>2</sub> , v<sub>3</sub> , v<sub>4</sub> 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
v<sub>1</sub> , v<sub>2</sub> , v<sub>3 </sub>have cyclic stabilizer
subgroups A = <a>, B=<b> and C=<c> respectively. <br>
<br>
The associated graph of groups is <br>
<br>
<div style="text-align: center;"> <img alt=""
src="graphtri.gif" style="width: 242px; height: 63px;"><br>
<br>
<div style="text-align: left;">The fundamental group P of this
graph has the same homology as the triangle group T(l,m,n) in
dimensions greater than 3. <br>
<br>
So for example, the following commands show that the triangle group
T=T(2,3,4) has 5-dimensional integral homology H<sub>5</sub>(T,Z) = Z<sub>2</sub>+Z<sub>12</sub>
.<br>
</div>
</div>
</div>
</div>
</td>
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<td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
a:=(1,2);; b:=(1,2,3);; c:=(1,2,3,4);; f:=();; g:=();;<br>
gap> A:=Group(a);; SetName(A,"A");;<br>
gap> B:=Group(b);; SetName(B,"B");;<br>
gap> C:=Group(c);; SetName(C,"C");;<br>
gap> F:=Group(f);; SetName(F,"F");;<br>
gap> G:=Group(g);; SetName(G,"G");;<br>
gap> FA:=GroupHomomorphismByFunction(F,A,x->x);;<br>
gap> FC:=GroupHomomorphismByFunction(F,C,x->x);;<br>
gap> GC:=GroupHomomorphismByFunction(G,C,x->x);;<br>
gap> GB:=GroupHomomorphismByFunction(G,B,x->x);;<br>
gap> Graph:=[A,B,C,[FA,FC],[GC,GB]];;<br>
gap> GraphOfGroupsDisplay(Graph);;<br>
<div style="text-align: center;"><img alt="" src="T234.gif"
style="width: 258px; height: 84px;"><br>
<div style="text-align: left;">gap>
R:=ResolutionGraphOfGroups(Graph,6);;<br>
gap> Homology(TensorWithIntegers(R),5);<br>
[ 2, 12 ]<br>
</div>
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<td style="vertical-align: top;"><a
style="color: rgb(0, 0, 102);" href="aboutTwistedCoefficients.html">Previous
Page</a><br>
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<td style="text-align: center; vertical-align: top;"><a
href="aboutContents.html"><span style="color: rgb(0, 0, 102);">Contents</span></a><br>
</td>
<td style="text-align: right; vertical-align: top;"><a
href="aboutSpaceGroup.html"><span style="color: rgb(0, 0, 102);">Next
page</span><br>
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<a href="aboutTopology.html"><br>
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