Logan, Alan D. (2014) The outer automorphism groups of three classes of groups. PhD thesis, University of Glasgow.
Full text available as:

PDF
Download (1MB)  Preview 
Abstract
The theory of outer automorphism groups allows us to better understand groups through their symmetries, and in this thesis we approach outer automorphism groups from two directions. In the first direction we start with a class of groups and then classify their outer automorphism groups. In the other direction we start with a broad class of groups, for example finitely generated groups, and for each group Q in this class we construct a group G_Q such that Q is related, in a suitable sense, to the outer automorphism group of G_Q.
We give a list of 14 groups which precisely classifies the outer automorphism groups of oneended twogenerator, onerelator groups with torsion. We also describe the outer automorphism groups of such groups which have more than one end. Combined with recent algorithmic results of Dahmani–Guirardel, this work yields an algorithm to compute the outer automorphism group of a twogenerator, onerelator group with torsion.
We prove a technical theorem which, in a certain sense, writes down a specific subgroup of the outer automorphism group of a particular kind of HNNextension. We apply this to prove two main results. These results demonstrate a universal property of triangle groups and are as follows. Fix an arbitrary hyperbolic triangle group H. If Q is a finitely generated group then there exists an HNNextension G_Q of H such that Q embeds with finite index into the outer automorphism group of G_Q. Moreover, if Q is residually finite then G_Q can be taken to be residually finite. Secondly, fix an equilateral triangle group H = ⟨a, b; a^i, bi, (ab)^i⟩ with i > 9 arbitrary. If Q is a countable group then there exists an HNNextension G_Q of H such that Q is isomorphic to the outer automorphism group of G_Q. The proof of this second main result applies a theory of Wise underlying his recent work leading to the resolution of the virtually fibering and virtually Haken conjectures.
We prove a technical theorem which, in a certain sense, writes down a specific subgroup of the outer automorphism group of a semidirect product H
Item Type:  Thesis (PhD) 

Qualification Level:  Doctoral 
Keywords:  Group theory, outer automorphism groups, residual finiteness, onerelator groups 
Subjects:  Q Science > QA Mathematics 
Colleges/Schools:  College of Science and Engineering > School of Mathematics and Statistics > Mathematics 
Supervisor's Name:  Pride, Professor Stephen J. and Brendle, Dr. Tara 
Date of Award:  2014 
Depositing User:  Mr Alan D. Logan 
Unique ID:  glathesis:20145210 
Copyright:  Copyright of this thesis is held by the author. 
Date Deposited:  27 Jun 2014 10:34 
Last Modified:  27 Jun 2014 10:35 
URI:  http://theses.gla.ac.uk/id/eprint/5210 
Actions (login required)
View Item 