Length spectra of the Hecke triangle groups
The principal results of this paper are the classification for all \(q\geq 3\) of \(G_q\)-equivalent classes of quadratic forms of \(\mathbb{Z}[\lambda_q]\) and the associated explicit construction of the geodesic length spectrum of \(G_q\backslash {\mathcal H}\), where \(G_q\) is the Hecke triangle group with \(\lambda_q= 2\cos \pi/q\), its translation length at \(\infty\). These results rely on techniques similar to earlier work of the first author concerning Rosen's continued fraction expansions of hyperbolic fixed points as well as on an extension to these Hecke groups of an algorithm of B. Fine for computing conjugacy class representatives. We give a variety of other results concerning the Hecke triangle groups. In the arithmetic cases, the set of fixed points is characterized and it is shown that for \(q= 4, 6\) Lagrange's Theorem on the periodicity of continued fraction expansion of quadratic irrationalities is false. The set of counterexamples is characterized. Periodic elements of the ground field exist in each of these cases, and their associated closed geodesics display conformal symmetry on a small finite cover of the corresponding \(G_q\backslash {\mathcal H}\). The matrices arising in the extension of Fine's method turn out to have axes which also project to fully symmetric geodesics on these finite covers. The length spectrum has substantial multiplicity, which can be easily effectively computed in these arithmetic cases.
explicit construction of the geodesic length spectrum
periodicity of continued fraction expansion of quadratic irrationalities
scientific article; zbMATH DE number 856571
scientific article