The main result of this article is that if a $3$-manifold $M$ supports anAnosov flow, then the number of conjugacy classes in the fundamental group of$M$ grows exponentially fast with the length of the shortest orbitrepresentative, hereby answering a question raised by Plante and Thurston in1972. In fact we show that, when the flow is transitive, the exponential growthrate is exactly the topological entropy of the flow. We also show that takingonly the shortest orbit representatives in each conjugacy classes still yieldsBowen's version of the measure of maximal entropy. These results are achievedby obtaining counting results on the growth rate of the number of periodicorbits inside a free homotopy class. In the first part of the article, we alsoconstruct many examples of Anosov flows having some finite and some infinitefree homotopy classes of periodic orbits, and we also give a characterizationof algebraic Anosov flows as the only $\mathbb{R}$-covered Anosov flows up toorbit equivalence that do not admit at least one infinite free homotopy classof periodic orbits.
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机译：本文的主要结果是，如果$ 3 $-歧管$ M $支持anosov流，则$ M $基本组中的共轭类的数量将随着最短轨道代表的长度呈指数增长，从而回答一个问题由Plante和Thurston于1972年提出。实际上，我们证明了，当流为传递性时，指数增长率恰好是流的拓扑熵。我们还表明，在每个共轭类中仅取最短的轨道代表仍会产生最大熵度量的Bowen版本。这些结果是通过获得关于自由同态类内部的周期轨道数目的增长率的计数结果而获得的。在本文的第一部分中，我们还构造了许多Anosov流的例子，它们具有周期轨道的一些有限和一些无穷同伦类，并且我们还对代数Anosov流进行了表征，因为它是唯一覆盖$ \ mathbb {R} $的Anosov流不允许至少一个周期性轨道的无限自由同伦类的上升轨道当量。
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