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.