  The main result of this article is that if a $3$-manifold $M$ supports an
Anosov flow, then the number of conjugacy classes in the fundamental group of
$M$ grows exponentially fast with the length of the shortest orbit
representative, hereby answering a question raised by Plante and Thurston in
1972. In fact we show that, when the flow is transitive, the exponential growth
rate is exactly the topological entropy of the flow. We also show that taking
only the shortest orbit representatives in each conjugacy classes still yields
Bowen's version of the measure of maximal entropy. These results are achieved
by obtaining counting results on the growth rate of the number of periodic
orbits inside a free homotopy class. In the first part of the article, we also
construct many examples of Anosov flows having some finite and some infinite
free homotopy classes of periodic orbits, and we also give a characterization
of algebraic Anosov flows as the only $\mathbb{R}$-covered Anosov flows up to
orbit equivalence that do not admit at least one infinite free homotopy class
of periodic orbits.
