Ziele Untersuchung
mit Columbo Integrität von
Datenbanken Interaktion und
Portierbarkeit Ergonomie der
Schnittstellen

Angebot Produkte Projekt Beratung

Mittel Analytik Modellierung Sprachen Algebra Logik Hardware Denken Kreativität

Zusammenhänge Gesellschaft Wirtschaft Branche Firma


products/sources/formale Sprachen/GAP/pkg/hap/www/SideLinks/About/ (Algebra von RWTH Aachen Version 4.15.1^©) Datei vom 19.6.2025 mit Größe 12 kB

Quelle aboutGraphsOfGroups.html Sprache: HTML

products/sources/formale Sprachen/GAP/pkg/hap/www/SideLinks/About/aboutGraphsOfGroups.html

<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
  <meta http-equiv="content-type"
content="text/html; charset=ISO-8859-1">
  <title>AboutHap</title>
</head>
<body
style="color: rgb(0, 0, 153); background-color: rgb(204, 255, 255);"
alink="#000066" link="#000066" vlink="#000066">
<br>
<table
style="text-align: left; margin-left: auto; margin-right: auto; color: rgb(0, 0, 102);"
border="0" cellpadding="20" cellspacing="10">
  <tbody>
    <tr align="center">
      <th style="vertical-align: top;">
      <table style="width: 100%; text-align: left;" cellpadding="2"
cellspacing="2">
        <tbody>
          <tr>
            <td style="vertical-align: top;"><a
href="aboutTwistedCoefficients.html"><small
style="color: rgb(0, 0, 102);">Previous</small></a><br>
            </td>
            <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>
            </td>
          </tr>
        </tbody>
      </table>
      <big><span style="font-weight: bold;"></span></big><br>
      </th>
    </tr>
    <tr>
      <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">
      <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>→G<sub>v</sub> and t:G<sub>e</sub>→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"
style="width: 353px; height: 92px;"> <br>
      <br>
      <br>
and<br>
      <br>
      <br>
      </div>
      <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>
      </div>
      </div>
      </div>
      </td>
    </tr>
    <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>
      </td>
    </tr>
    <tr>
      <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>
    </tr>
    <tr>
      <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>
    </tr>
    <tr>
      <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, "
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>
      <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="10" cellspacing="0">
        <tbody>
          <tr>
            <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>
            </td>
          </tr>
        </tbody>
      </table>
      <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>
    </tr>
    <tr>
      <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>
      </div>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <table
style="margin-left: auto; margin-right: auto; width: 100%; text-align: left;"
border="0" cellpadding="2" cellspacing="2">
        <tbody>
          <tr>
            <td style="vertical-align: top;"><a
style="color: rgb(0, 0, 102);" href="aboutTwistedCoefficients.html">Previous
Page</a><br>
            </td>
            <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>
            </a> </td>
          </tr>
        </tbody>
      </table>
      <a href="aboutTopology.html"><br>
      </a> </td>
    </tr>
  </tbody>
</table>
<br>
<br>
</body>
</html>

Messung V0.5

¤ Dauer der Verarbeitung: 0.13 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.