The Markov spectra for triangle groups
Let \(V\) be a three-dimensional vector space over \(\mathbb{R}\) and \(Q:V \to\mathbb{R}\) an indefinite ternary quadratic form of signature (2,1) with associated symmetric bilinear form \((x,y)= (Q(x+y) -Q(x-y))/4\). We may regard \(D^2= \{x\in P^2 |Q(x) <0\}\), \(P^2\) the projective plane, as a model for the hyperbolic plane and a Fuchsian group \(G\) as a discrete subgroup of the group of isometries of \(D^2\). If \(G\) is a Fuchsian group of the first kind and \(w\in D^2\) is not a fixed point then, for an indefinite point \(x\in P^2\), we denote \[ v_w(x) = \inf \bigl |(x,gw) \bigr|Q^{-{1\over 2}} (x), \] where the infimum is taken over all \(g\in G\). The set \[ M_w(G) =\bigl\{ v_w(x) \mid x\in P^2,\;Q(x) >0 \bigr\} \] is called the Markoff spectrum of \(G\) with respect of \(w\). The complete description of the discrete part of the Markoff spectrum for any Hecke group \(G(\lambda)\) is given. Here \(\lambda=2\cos {\pi \over q}\), \(q \geq 3\), and \(G(\lambda)\) is generated by the transformations \(\left( \begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix} \right)\) and \(\left( \begin{smallmatrix} 1 & \lambda \\ 0 & 1 \end{smallmatrix} \right)\).
scientific article; zbMATH DE number 1097546
scientific article