Generalizations of Conway’s Topograph arising from Arithmetic Coxeter Groups
Conway’s topograph can be used in the study of binary quadratic forms (BQFs)
to replace tedious algebraic computations with straightforward geometric arguments.
The crux of his method is the isomorphism between the arithmetic group PGL2(Z) and
the Coxeter group (3, infinity). We introduce the arithmetic groups called dilinear groups
and construct generalizations of Conway’s topograph called dilinear topographs. Then
we use them to study variants of BQFs, called binary quadratic diforms (BQDs). The
payoff can be seen in the last chapter in our investigation of minimum value bounds for
diforms and pairs of BQFs.