Symmetries of Euclidean 3-space
Two subsets, A and B say, of the plane or R3 are
said to be equivalently embedded if there is a topological
symmetry (ie a homeomorphism), h say, of Rn with
itself carrying one subset onto the other, h(A)=B. Thus A and B are
identically `positioned' in Rn. 
The first step in the project was to show that any two Cantor
subsets of the plane are equivalently embedded (a `folklore'
result). The second step was to explore the variety of inequivalent
embedings of a Cantor set in R3. The last
(incomplete) step was to link these so-called `wild' Cantor subsets of
Euclidean 3-space with fully invariant sets of smooth 3-d dynamical
systems.
The main result is that any metric profinite group is topologically
isomorphic to the group of self-equivalences of a Cantor subset of
R3. It follows that there are two groups of
self-equivalences of Cantor subsets of 3-space which are algebraically
isomorphic, and homeomorphic, but not topologically
isomorphic. Thus the algebraic structure of the group of
self-equivalences does not determine its topological
structure. Results of Kallman and others show that this is strikingly
different from the situation for groups of symmetries of manifolds.
Given a profinite group G, the aim is to find an embedding of a
Cantor set, C, into R3, such that the group of
self-equivalences of C is topologically isomorphic to G.
In the special case of G finite, we use a rigid Cantor subset of
3-space due to Schilepsky, and the fact that every finite group is
isomorphic to the group of color-preserving automorphisms of its
Cayley graph.
To extend to general G, we construct relevant Cantor
embeddings for all the finite homomorphic images of G, and carefully tie
these together...
Preprint (in postscript): preprint
1
Symmetries of the Plane
A separable space X
is countable dense homogeneous (CDH) if, for every pair A,B of countable
dense subsets of X, there is a homeomorphism of X onto itself that maps A
onto B. Thus, all countable dense subsets of X are `positioned' in X
in the same way. The plane is (CDH).
A space X is strongly locally homogeneous (SLH) if X has a basis of
open sets B such that, for any x and y in B, there is a homeomorphism h from
X onto X with h(x) = y and h(p) = p for all p not in B.
It had been shown
that an SLH Polish (ie separable completely metrizable) space is CDH,
but whether a metric CDH space is SLH remains unanswered (though a
number of papers - by Anderson, Curtis, van Mill,
Fitzpatrick, Zhou, Fletcher and McCoy among others - have appeared on the
subject).
The intent of the project was to study the topological properties
of the graphs of
discontinuous real-valued additive functions (Cauchy spaces), especially
discontinuous additive functions with connected graphs (Jones spaces), and
especially the homogeneity properties of such functions. It is easily
seen that every Jones function is not SLH, but it is
unknown whether some Jones function is CDH (Heath and Saltsman recently showed
that some Jones spaces are not CDH). Many papers and entire books (e.g.,
Kuczma, Kharazishvili) on Cauchy spaces have appeared but this and other
questions remain unanswered.
The Results: The question of whether there exists a CDH Jones function was not
answered, but a number of related results were obtained.
It was determined that a Cauchy space is connected (hence a Jones
space) if and only if it is not SLH. On the other hand the simplest
Cauchy space (a graph
of an additive function with only rational values) is not CDH. More generally
if f is any real-valued function on R with a dense, meagre graph, then that
graph is not CDH. Improving a consistency result by Fitzpatrick and Zhou, it
was shown that a CDH Cauchy function must have the Baire property. Moreover a
Jones space is Baire if and only if intersects every dense G-delta subset of
the plane. Some results concerning the embeddings of Jones spaces in the plane
were obtained.
Preprint (in postscript): preprint
2
