Reduction theory over quadratic imaginary fields (Q1767656)

This paper concerns the explicit reduction theory of elements of a symmetric space such as hyperbolic \(n\)-space \(H^{n}\) modulo a discrete arithmetic group \(\Gamma\). \ The first section considers a geometric approach to Gauss reduction of real binary quadratic forms using the method of Farey triangles for \(n=2\) and \(\Gamma=\text{GL}(2,\mathbb{Z})\). \ The second section concerns Hurwitz's least remainder continued fractions and the connection with reduction theory for \(\text{SL}(2,\mathbb{Z})\). \ The third section uses Ford disc packing [see \textit{J. H. Conway}, The sensual (quadratic) form, Math. Assoc. America (1997; Zbl 0885.11002)] to get a geometric interpretation of Hurwitz reduction. It is shown that the Farey tiling and the Ford discs are dual geometric configurations that underlie Gauss reduction and Hurwitz reduction, respectively. Section 4 describes reduction theory for compact quotients of hyperbolic \(n\)-space \(H^{n}\) -- as a special case of the theory of Markov partitions for the geodesic flow on \(H^{n}/\Gamma\) due to \textit{R. Bowen} [Symbolic dynamics for hyperbolic space, Am. J. Math. 95, 429--460 (1973; Zbl 0282.58009)] and \textit{M. Ratner} [Markov partitions for Anosov flows on \(n\)-dimensional manifolds, Isr. J. Math., 15, 92--114 (1973; Zbl 0269.58010)]. Sections 5 and 6 show how Hurwitz reduction extends to a discrete group \(\Gamma\) of isometries of hyperbolic \(n\)-space \(H^{n}\) such that the quotient \(H^{n}/\Gamma\) has finite volume. The main case investigated is \(n=3\), with the Bianchi group \(\Gamma=\text{PSL}(2,O_{K}),\) where \(O_{K}\) is the ring of integers in an imaginary quadratic number field such as \(K=\mathbb{Q} (\sqrt{-3}).\) When \(K\) has a Euclidean algorithm, the theory is based on least-remainder continued fractions. The paper concludes with a section on examples and open problems.