Poster submission: Coset Enumeration and Geometric Ordering
D. Wright
Oklahoma State University
Limit sets of kleinian groups which contain a fuchsian subgroup of the
first kind are closures of unions of circles. One strategy for drawing
the limit set of such a group $G$ is to draw all the circles which are
equivalent under the group to the limit circle $C$ of the fuchsian
subgroup $H$. This is equivalent to enumerating all the left cosets
$aH$ of $H$ in $G$. In a wide class of examples, McShane, Parker and
Redfern created and used Finite Stata Automata to enumerate the right
cosets $Ha$, with transition rules of the form $Ha \mapsto Hab$ where
$b$ ranges over a set of generators of the group. Then the left cosets
are enumerated at the same time with representatives $a^{-1}$, and we
may plot the circles $a^{-1}(C)$, thus filling out the limit set. One
problem with this approach is that if $a_1$ and $a_2$ are coset
representatives which are ``close" in the lexicographical ordering
then $a_1^{-1}$ and $a_2^{-1}$ need not be. However, visually there is
often a clear geometric ordering to the plotting of the circles. In
some cases, we show how to preserve this geometric ordering in the
plot by using a specially constructed Finite State Automaton for the
whole group $G$, with generators listed in geometric order, together
with a plotting automaton or ``mask" that simply indicates whether or
not the currently enumerated element of $G$ should be accepted as a
coset representative or not. The first automaton indicates the tree
of words that should be followed, while the second automaton
identifies the accepted left cosets. The accepted states do not form
a connected subset of the tree of words in $G$. Finally, we shall show
how this method allows us to strikingly color the disks for certain
cusp groups, revealing certain interesting structures in the limit
set.

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