Continuous deformation of the Bowen-Series map associated to a cocompact triangle group

Pub Date : 2024-04-02 DOI:10.1007/s10711-024-00887-2

引用次数: 0

Abstract

In 1979, for each signature for Fuchsian groups of the first kind, Bowen and Series constructed an explicit fundamental domain for one group of the signature, and from this a function on \({\mathbb {S}}^1\) tightly associated with this group. In general, their fundamental domain enjoys what has since been called both the ‘extension property’ and the ‘even corners property’. We determine the exact set of signatures for cocompact triangle groups for which this property can hold for any convex fundamental domain, and verify that for this restricted set, the Bowen-Series fundamental domain does have the property. To each Bowen-Series function in this corrected setting, we naturally associate four continuous deformation families of circle functions. We show that each of these functions is aperiodic if and only if it is surjective; and, is finite Markov if and only if its natural parameter is a hyperbolic fixed point of the triangle group at hand.

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与协整三角形群相关的鲍温系列图的连续变形

摘要 1979 年，对于第一类富奇异群的每个签名，鲍文和辑为签名中的一个群构造了一个明确的基域，并由此构造了一个与这个群紧密相关的 \({\mathbb {S}}^1\) 上的函数。一般来说，他们的基域具有后来被称为 "扩展性质 "和 "偶角性质 "的特征。我们确定了对于任何凸基域，这一性质都能成立的可紧密三角形群的精确签名集，并验证了对于这一受限集，鲍恩系列基域确实具有这一性质。对于这个修正设置中的每个鲍温系列函数，我们自然地关联了四个连续变形的圆函数族。我们证明，这些函数中的每个函数都是非周期性的，当且仅当它是可射的；并且，当且仅当它的自然参数是当前三角形群的双曲定点时，它是有限马尔可夫函数。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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