*On
automorphisms of structures in logic and orderability of groups in
topology*

** **

**by
Ataollah Togha**

Ph.D. dissertation, The George Washington University, 2004, 76 pages.

**Abstract**

We
investigate properties of non-standard models of set theory, in
particular, the
*countable recursively saturated* models while having the
automorphisms of
such models in mind. The set of automorphisms of a model forms a
group that in
certain circumstances can give information about the model and even
recover the
model. We develop results on conditions for the existence of
automorphisms
that fix a given initial segment of a countable recursively saturated
model of
ZF.

In
certain cases an additional axiom **V** = **OD** will be needed
in order
to establish analogues of some results for models of Peano Arithmetic.
This
axiom will provide us with a definable global well-ordering of the
model. Models
of set theory do not automatically possess such a well-ordering, but a
definable well-ordering is already in place for models of Peano
Arithmetic,
that is, the natural order of the model.

We
investigate some finitely presentable groups that arise from topology.
These
groups are the *fundamental groups* of certain manifolds and
their orderability
properties have implications for the manifolds they come from. A
group (*G*, ◦) is called left-orderable if there is a total
order relation < on *G*
that preserves the group operation ◦ from the left.

Finitely presentable groups constitute an important class of finitely generated groups and we establish criteria for a finitely presented group to be non-left-orderable. We also investigate the orderability properties for Fibonacci groups and their generalizations.