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Recall that the translation length $\tau(g)$ of an element $g \in G$ is the limit $d(1, g^n)/n$, where $d$ is the word metric on $G$ with resepct to some generating set.

It is a theorem of Gromov that the translation length of any hyperbolic element in a hyperbolic group is a rational number with denominator uniformly bounded.

This result is a bit surprising to me that the translation length is such a delicate thing. This also recalls me that the growth series of a hyperbolic group is a rational function (a theorem of Cannon).

Question 1: I'm wondering whether anyone could give some explaination to these such type of results? For example, what is the motivation of Gromov's theorem? any analogous results on Riemaninan manifolds?

Quesion 2: I would like to see whether there is certain relation over the spectums of translation lengths of all elements. An analogous result comes to my mind at this moment is the Mcshane identity over the lengths of geodesics on a punctured surfaces. See http://en.wikipedia.org/wiki/McShane%27s_identity

For the translation lengths, if we take the reciprocal of $\tau(g)^2$ and sum all of them, we see that it is a finite number. It is not clear how this finite number depends on the choices of generating sets. Can such type of an idea be further improved?

I saw some negative result in "On the absence of McShane-type identities for the outer space" by Ilya Kapovicha, Igor Rivinb. But I did not have a chance to loot at it very carefully, maybe their paper also gives a negative answer to my question here.

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Regarding question 1, this theorem of Gromov is a two-fold phenomenon related to the very special and very "discrete" nature of the word metric. First, that metric takes on only integer values. Second, the special effect of hyperbolicity of the Cayley graph is that the expression $d(1,g^n)/n$ achieves its limit at some finite value $n=N$ independent of $g$, and so the translation length equals an integer divided by $N$.

It is interesting to see what happens when you make some other assignment to the lengths of the generators. You could choose them to be rational numbers, for instance. But since there are only finitely many generators, then in computing any translation length $d(1,g^n)$ one is summing rational numbers with bounded denominator, and all such sums have the same denominator bound. The discrete nature of the Cayley graph still implies that ratio $d(1,g^n)/n$ achieves its limit at some finite $n=N$ independent of $g$, and so the translation lengths of all elements are still rational with bounded denominators. Or, you could choose the translation lengths of the generators to be in the field $\mathbb{Q}(\sqrt{2})$. Again the translation numbers of all elements of the group would be elements of the field $\mathbb{Q}(\sqrt{2})$ with bounded denominators.

You ask whether there are analogous results in Riemannian manifolds. If $M$ is a closed Riemannian manifold with word hyperbolic fundamental group, it seems possible to me that either of the two "discrete" phenomena mentioned earlier are candidates for failure. First, I am not sure whether it is still true that $d(1,g^n)/n$ achieves its limiting value at some finite $n=N$ independent of $g$. But even if it does, $d(1,g^n)$ is no longer computed by summing integers.

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    $\begingroup$ Both properties definitely fail in the realm of Riemannian manifold of negative curvature, even in the constant curvature case. $\endgroup$ – Misha Apr 26 '14 at 16:01
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    $\begingroup$ @lee-mosher Is it really true that "the expression d(1,g^n)/n achieves its limit at some finite value n=N independent of g"? In the free group of two generators a, b, if g=bab^{-1} then d(1,g^n)/n=(n+2)/n. Am I missing something stupid? Or is it necessary to replace g by an appropriate conjugate element? $\endgroup$ – Jairo Bochi Apr 9 '15 at 17:34

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