**Geometry Research**

My main research interest is in finite geometric spaces
with at least transitive symmetry groups. That is, the symmetries can move
any point to any other point of the space. I have several potential projects for
undergraduates, including two generalizing a recently published paper (see the
section on Highly Symmetric Spaces). By a geometric space I technically mean a complete edge colored graph:
the
graph is a set of *n* vertices with all *n(n-1)/2* edges between them
and the coloring is a function from the edges to their colors.

I often think intuitively of the colors as representing different distances between vertices: If two edges are the same color, then the distance between the endpoints of one edge equals the distance between the endpoints of the other edge. Thus the four vertices of a rectangle give rise to an edge colored graph with four vertices and three interesting colors, one for the "horizontal" length, one for the "vertical" length and one for the "diagonal" length. (One could add the "zero" color from a vertex to itself., which is sometimes useful to have.)

For certain colorings one can think of the colors as representing
lines (or parallel lines). That is, two edges of the same color belong to the same
line (or belong to parallel lines). For example, the Euclidean plane provides an
infinite example where the colors are real numbers *m*, corresponding to the slope of the
line determined by the two points and one more color, say, *V* for edges determining a
vertical line.

Alternatively, we can use more colors to have each
line of the Euclidean plane have its own color. We can "subdivide" the colors
of the previous paragraph with a
second parameter *b* to distinguish which line in a parallel class the edge
determines. So an edge would have color *(m,b)* if it is a subset of the line
*y = mx+b* and *(V,b)* if it is a subset of the vertical line *x = b*.

**Definitions.** An s*imilarity* (automorphism) of an edge colored graph is a bijection *f* of the vertices so that if edges *ab* and
*cd*
are the same color, written *C(ab) = C(cd)*, then *C(f(a)f(b)) =
C(f(c)f(d))*. A stronger condition, *f* is an *isometry*, requires that the
bijection *f* satisfy *
C(ab) = C(f(a)f(b))*, that is the edge

In the example of the rectangle above, any of the 24 bijections of the vertices is a similarity, but there are only 4 isometries, corresponding to the identity, vertical and horizontal mirror reflections and a 180 degree rotation. For both Euclidean examples, all affine transformations are similarities. For the first Euclidean example, the isometries are the Euclidean homotheties, that is, translations and rotations of 180 degrees around any point and dilations around any point. (Dilations are also called dilatations.) For the second example, only the identity is an isometry.

Each color of a space induces an ordinary graph on the vertices. The automorphisms of this graph preserve that color and so relate to the isometries of the space. Indeed, the intersection of the automorphism groups for each color is the group of isometries of the space.

**Generalizing Groups Geometrically**

A group has a natural geometric structure
similar to the Cayley directed graph of a group. We first generalize the
idea of absolute value on the reals for any group (G,*) by defining |a| = |a^{-1}|
= {a, a^{-1}}. We construct the geometric space whose vertices are
the elements of G and we "color" the edge ab with the value |a*b^{-1}|.
I showed that the space formed from a group is edge homogeneous. My 2011
thesis students, David Byrne and Matt Donner, classified the group of isometries
of any finite group. (Thesis links:
Donner
Byrne . Draft
of paper for publication:
paper.)
My student Kelsey Larson is working on classifying the similarity groups of
groups.

Further,
there are loops with identity and inverses that form 1 point homogeneous spaces
in the same way. Call these loops 1-point homogeneous. I proved
LaGrange's Theorem for these loops and the Sylow Theorems when the prime is odd.
The Sylow Theorems hold for p = 2 for loops with the stronger property of edge
homogeneity. Also such a space is 2-point homogeneous iff for all a, b in
the loop, |a*b| = |b*a|. These results and more can be found through the
links to the following
papers.

"Equidistance relations: A new bridge between geometric and
algebraic structures," Cuttington Research Journal, vol. 1 (1982), 19-25.

"Homomorphisms for equidistance relations," Cuttington
Research Journal, vol. 2 # 1 (1983) 1-11.

"Sylow-like theorems in geometry and algebra," Journal of
Geometry, vol. 30 (1987) pages 1-11.

**Highly Symmetric Spaces**

"On Classifying Finite Edge Colored Graphs with Doubly
Transitive Automorphism Groups" Journal of Combinatorial Theory, Series B
vol. 90/1 (Jan., 2004), 121-138.

Abstract: This paper classifies all finite edge colored graphs with
doubly transitive automorphism groups. This result generalizes the classification of
doubly transitive balanced incomplete block designs where distinct blocks have at most one
vertex in common as well as doubly transitive one-factorizations of complete graphs.
It also classifies all doubly transitive symmetric association schemes. The paper
uses the classification of finite doubly transitive groups, which depends on the
classification of finite simple groups.

Click here for a pdf file.

Click on Projects to view the file outlining projects generalizing this paper.

**Homogeneous Spaces**

During the summer of 2002 Eric Polley and I worked on two-point homogeneous spaces and the slightly more general edge homogeneous spaces. Our ultimate goal is to try to classify all two-point homogeneous spaces and all edge homogeneous spaces.

**Definitions.** A space is
*two-point homogeneous* provided for any four vertices *s, t, u, v*

if
*C(st) = C(uv)*,
then there is an isometry *f* with *f(s) = u* and *f(t) = v*. A space is *edge
homogeneous* provided for any four vertices *s, t, u, v* if *C(st) = C(uv)*, then there is
an isometry *f* with ( *f(s) = u* and *f(t) = v* ) or ( *f(s) = v* and
*f(t) = u* ).

Two-point homogeneous spaces are always edge homogeneous. The space derived above from a rectangle is two-point homogeneous. The first space derived from the Euclidean plane is also two-homogeneous. The second space is neither since it has only one isometry. The construction below turns any group into an edge homogeneous space.

Let (G,*) be any group with inverse '.
We first define an *absolute value* on G by |a| =
{a, a'}. The colors of G is the set of absolute values of G. An edge ab has color
C(ab) = |a*b'|. Then we can prove the function f_{a}: G --> G
given be f_{a}(x) = x*a is an isometry. These functions suffice to
show that this geometry is edge homogeneous. If * is commutative (that is,
G is abelian), then the space is two-point homogeneous. In fact, we can
weaken this condition to get the following result. For a group G, the
space is two-point homogeneous iff for all a and b in G,
|a*b| = |b*a|.

In graph theory there are two related topics, distance
transitive graphs and symmetric graphs. In a connected graph the *distance*
between two vertices is the length of the smallest path along the edges between
them. A graph is *distance transitive* provided for two pairs of points
*s, t*
and *u, v* if *d(s,t) = d(u,v)*,
then there is a graph automorphism of the (one color) graph taking *s* to
*u* and *t* to *v*.
By choosing the different distances as colors, each connected graph (on one color)
becomes a space and distance transitive implies two-point homogeneity.
However, there are many two-point homogeneous spaces that do not come from distance
transitive graphs. A *symmetric* graph has a group of auotomorphisms that is
transitive on the vertices and on the edges (of the one color) of the graph. This
comes close to matching the idea of edge homogeneous, although edge homogeneous requires
the transitivity of edges for every color, not just the one color.

Charlie O'Connor in his honors thesis in 2005 defined a semi-direct product of spaces similar to semi-direct products of groups. This method of constructing new spaces supplements direct products and wreath products, which were already investigated by previous students. Unfortunately, there are edge homogeneous spaces not built from groups through any of these constructions. The easiest such space to describe uses the 20 vertices of a regular dodecahedron with coloring corresponding to the distances between vertices. There is a homomorphism from this space to a space on ten vertices built from the Petersen graph, but I have not found a way to build the twenty vertex space from groups or smaller spaces using various constructions. There are other spaces that at present can't be built from groups and smaller spaces.

Click here for descriptions of other research areas.