Long, Reid, and Thistlethwaite have shown that some groups generated by representations of
the
triangle
group in
are thin, while the status of others is unknown. We take a new approach: For each
group we introduce a new graph that captures information about representations of
in the group.
We provide examples of our graph for a variety of groups, and we use information about the
graph for
to show that the chromatic number of the graph for
is at most 8. By generating a portion of the graph for
we
show its chromatic number is at least 4; we conjecture it is equal to 4.
Keywords
chromatic number of a graph, generators and relations,
special linear group, thin group, triangle group