The Markoff spectrum is defined as the set of normalized values of arithmetic minima of indefinite quadratic forms. In the theory of the Markoff spectrum we observe various kinds of symmetry. Each of Conway’s topographs of quadratic forms which give values in the discrete part of the Markoff spectrum has a special infinite path consisting of edges. It has symmetry with respect to a translation along the path and countable central symmetries by which the path is invariant. We prove that these properties are obtained from the fact that the path is a discretization of a geodesic in the upper half-plane which corresponds to a value of the discrete part of the Markoff spectrum and projects to a simple closed geodesic on the once punctured torus with the highest degree of symmetry.

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Markoff 形式的地形图的对称性

马尔科夫谱被定义为不定二次形式的算术最小值的归一化值的集合。在马尔科夫谱的理论中，我们观察到各种对称性。在马尔科夫谱的离散部分给出值的康威的每一个二次形式的拓扑图都有一条由边组成的特殊无限路径。它具有关于沿路径平移的对称性和路径不变的可数中心对称性。我们证明了这些属性是从以下事实获得的：路径是上半平面中测地线的离散化，对应于马尔科夫谱的离散部分的值，并投影到曾经穿孔的圆环上的简单闭合测地线具有最高的对称性。