The outer automorphism groups of three classes of groups

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 one-ended two-generator, one-relator 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 two-generator, one-relator 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 HNN-extension. 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 HNN-extension 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 HNN-extension 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 semi-direct product