Extended Glossary.
Abstract algebra.
Abstract algebra focuses on the structure of groups and related objects, such as rings and
modules. The founders of modern algebra used groups of isometries of a space to
investigate the space itself, that is, they studied groups associated with a space as a means
to understand that space. The modular group is a set of isometries of hyperbolic space.
By considering the actions taken by elements of the group and the structure of subgroups,
we provide examples of ideas that are central in an undergraduate abstract algebra course.
Conjugacy equivalence.
Conjugacy equivalence is a stronger condition than isomorphism. As abstract groups, the
infinite cyclic subgroups within the modular group are all isomorphic. However, for the
infinite cyclic groups generated by elements with trace values equal to two, there is just a
single conjugacy equivalence class, while for each trace value greater than two there are
more than one conjugacy equivalence class. While two subgroups may be isomorphic,
they can hold distinctly different positions in the overall subgroups structure of the
entire group.
It is easy to show that all order two elements in the modular group are conjugate
equivalent. This follows from how the modular group tessellation is constructed. Each
triangular shaped region is the image of a single triangular shaped fundamental
region that is acted on, in succession, by each modular group element.
Consequently, every triangular shaped region can be mapped onto every other triangular
shaped region. Since the right-angled vertex of each region is the fixed point of an order
two element, the order two action can be transferred, by conjugation, to any other order
two action. Thus, all order two actions are conjugate equivalent each and every other
order two action in the modular group. In a similar way, every 60 degree vertex of
triangular shaped regions can be mapped onto each other and are, thus, all conjugate
equivalent.
We say that the two finite order subgroups, order-two and order-three, form two distinct
conjugacy-equivalence classes. The traces of each are, respectively, zero and one. The
essential feature is that the trace value is less than two. This results in the action being
elliptic/rotational. The trace value for all parabolic/infinite cyclic elements is two. Each
such parabolic/infinite cyclic element has a single fixed point on the real axis. These are
all conjugate equivalent. In fact, the modular group tessellation provides a striking
visualization of the rationals. Full clusters (sometimes referred to as flowers) of
fundamental regions have their common point fixed at a rational number. Each rational
point on the x-axis serves as a common point for a cluster. It is as though each rational
point is announced and its location is marked by the emergence of a cluster from its
position on the x-axis.
The conjugacy classes of the hyperbolic elements, whose traces are all greater in absolute
value than two, are somewhat more complicated.
Conjugacy equivalence for hyperbolic elements.
Different trace values require different conjugacy classes. For elements with trace values
of zero, one, or two, there is a single conjugacy equivalence class for each respective
trace value. However, for traces with absolute values greater than two, there are
functions that have identical trace values, but that are not conjugate equivalent.
(example!) On the other hand, function with equal trace values and with equal
coefficients for z in the function denominator have fixed points that are mere integer
translations away from each other.
At issue is the mapping of fixed points of one element onto those of another. If there
exists an f in the modular group that can map the fixed points of one element onto those
of another, then these two are necessarily conjugate equivalent.
Conjugation.
We are interested in describing the conjugacy classes of maximal cyclic subgroups in the
modular group. Recall that the trace is a conjugacy invariant, that is, conjugation does
not affect the trace of an element. So, for any two elements, f and g, of the modular
group, the trace of f and the trace of g^-1ofog are the same.
Below are illustrations of functions that are conjugate equivalent to the first four
members of the family 
.
Generators.
Above are illustrated the actions of two generators of the modular group.
Combinations of these two actions will generate the action of each and every modular
group element. As examples, we illustrate how the action of the first four members of the
family of functions can be generated by compositions of the
two
generators, g_2(z)=-1/z and f(z)=z+1.
Groups.
A group is a set of elements together with an operation, which is associative and which is
closed under the operation and closed under inversion. We recall that integers are a
group under addition. However, integers are NOT a group
under multiplication, because the INVERSES under multiplication are fractions
and so,
requirement of closure under inversion is not met by the integers when multiplication is
the operation.
Another example of a group is all the translations of a square grid (think of a
checkerboard pattern sliding about on a flat plane) which bring the lines of the grid back
onto the same pattern. The translation group is ISOMORPHIC to the
CARTESIAN
PRODUCT of two infinite cyclic groups.
Think of the above as an example of two types of shifting translations which could be
along any of two directions, not necessarily perpendicular to each other. In this case, the
meaningful grid would be of parallelograms. However, no matter how we might change
the arrangement of the directions, we could never manage to find THREE directions for a
grid instead of two as in these examples.
Two dimensional hyperbolic space.
The leap into hyperbolic space gains us what? Well, for one, we get an infinite number
of directions that are independent of each other, not just the two we can find in the flat
plane.
The patterns of arcs and intersections can become much richer, mirroring the more
intricate subgroup structure compared to those possible in a tessellation group for a flat
surface. The modular group is one of the simplest types of tessellation groups for the
hyperbolic context, which makes it the ideal place to begin investigations with novices.
Direct contact with Teichmueller theory and other modern topics, make the modular
group a useful example to take into course in analysis, topology, algebra, and number
theory.
Open questions.
The questions here are currently unresoved for our group, including whether or not they
are open
open for the mathematical community as a whole.
What are the impacts of the coefficients, separately and in pairs, on the action and on the
conjugacy equivalence of modular group elements? For example, it is known that the
trace value
along with the value for the coefficient c determine conjugacy equivalence. Are there
other
groupings affected by the coefficient values?
Mathematica.
The computer algebra system (CAS) known as
Mathematica uses complex valued arithmetic, that is, it automatically treats
variables
as being complex valued with no special programming required of users such as us.
modular group.
The modular group is often manifested (not represented) as 2x2 matrices with
integer components and
such that the determinate of the matrix is equal to one. The group operation is the usual
matrix muotiplication. A remarkable feature is that these
same 2x2 matrices can be appropriated for an absolutely different manifestation of the
same group. In this different form, the group elements are LINEAR FRACTIONAL
TRANSFORMATIONS that act on the complex plane. The group operation in this
context is composition of functions. It is customary to note that composition of
funcitions is associative, which is ablsolutely required for a group operation. On the
other hand, commutivitiy is not required. Commutative groups have a special
label from the name of an adventurous mathematician, Abel. Commutative groups are
called abelian groups. It is of great interest to see the
differences and even get the connections between the two manifestations of the modualr
group, namely matrix
multiplications by a point (linear transformations) and the action of the linear fractional
transformations on a complex number.
Tessellation groups.
There are less than a few dozen different types of tessellations carried out with rigid
geometric figures in the plane. Tessellations in the hyperbolic plane have an unlimited
number of directions. This makes hyperbolic surfaces ideal for serving as
PARAMETER
SPACES for any sort of models with more than two or three independent variables.
All the groups that we consider here that act on a space will have a
FUNDAMENTAL
REGION, which completely tiles the space when the group elements, one after the
other,
act on that region. (Think of a checkerboard pattern being produced by a single square
being slide left and right and up and down.)
The additional directions make it a trickier business to return to a previous position. With
a square grid, motion in either horizontal or vertical directions can be taken in any order.
This is equivalent to saying that these translations commute with each other; the
order
in which the actions are taken does not matter at all.) The fact is that actions in different
hyperbolic directions are very noncommutative. The order matters very much.
The
result is a very rich subgroup structure within even very "simple" groups, such as the
modular group, generated by {-1/z,z+1}.
Trace
The trace of an element in SL[2,Z] is the sum of the two main diagonal entries, i.e., a+d.
This an extremely important item. In linear algebra, the trace is the sum of all
eigenvalues, that is,
the dilations in all "stable" directions for the transformation. For linear fractional
transormations,
the trace determines which of the three possible types of action will hold. The three
possible types
of actions are referred to as elliptic/rotations, as for the first two family members of
{f_n=-1/z+n}, parabolic/infinite cyclic, as for the third member of the family, and
hyperbolic/pushing forward, as
for all other family members.
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