| Summary: | <p>The thesis is concerned with groups acting on three dimensional spaces. The groups are assumed to have a compact simply-connected fundamental region. The action of the group is given partially by its action on the boundary of the fundamental region. This boundary is naturally split up by its intersection with transforms of the fundamental region. We assume that each such intersection is a single proper face of the boundary unless the transforming element is of order two, in which case there can be two faces. We also assume that any point of the boundary has only a finite number of transforms under the group which lie on the boundary. This enables one to give generators and defining relations for the group. The generators correspond to faces of the boundary inequivalent under the group, and defining relations to inequivalent lines.</p> <p>In these circumstances two questions arise, <ol type="i"><li>Is the three-dimensional space a manifold,</li> <li>Is the group finite?</li></ol> <p>If the space is not a manifold, then the group cannot be finite. So an answer to the first question gives some information about the second.</p> <p>The thesis answers the first question as follows:-</p> <p><em>THEOREM</em> Given a group acting on a three-dimensional space with a fundamental region satisfying the conditions above, then the space is a manifold under the following condition.</p> <p>Let X be any point which is the intersection of lines of the boundary. Let &ell;<sub>1</sub>-, &ell;<sub>2</sub>, ... &ell;<sub>n</sub> be a set of lines such that (i) given a line &ell; only one end of which is a point equivalent to X then exactly one &ell;<sub>i</sub> is equivalent to &ell;, (ii) given a line both ends of which are points equivalent to X then exactly two &ell;<sub>i</sub> are equivalent to &ell;. Let <em>r</em><sub>1</sub>, <em>r</em><sub>2</sub> ... <em>r</em><sub>n</sub> be the words (in terms of the generators and their inverses which correspond to the faces of the boundary) which are the shortest words which fix &ell;<sub>1</sub>, &ell;<sub>2</sub>, ... &ell;<sub>n</sub> respectively (These are defined to within inversion.). If <em>r</em><sub>i</sub><sup>n<sub>i</sub></sup> = 1 are relations of the group, then the condition is for all X</p> <p><table align="center" border="0" cellpadding="0" cellspacing="0"> <tr><td align="center">n<br><span style="font-size: 350%;">∑</span><br align="center">i=1</br></br></td> <td> </td> <td align="center">1<hr noshade="" size="1">n<sub>i</sub></hr></td> <td> <span style="font-size: 350%;">><span></span></span></td> <td> (n - 1).</td></tr></table></p> <p>This theorem is proved by a detailed analysis of the neighbourhood of points X in the space.</p> <p>Another theorem which is a corollary of the methods used in proving the first theorem is:-</p> <p><em>THEOREM</em> Given a group acting on a three-dimensional space with a fundamental region satisfying the conditions above then the group has a soluble word problem.</p></p>
|
|---|