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Suppose S is a surface which admits a hyperbolic metric - by this I mean a complete Riemannian metric of constant negative curvature, with totally geodesic (possibly empty) boundary.

Fix some finite presentation of the fundamental group $$\pi_1(S) = \langle X\ |\ R \rangle.$$

For an element $g\in \pi_1(S)$, define the word length $|w|_X$, with respect to the generating set X, to be the minimum number of generators needed to express $w$ as a word in X.

Fact: Every homotopy class of curves in $S$ contains a unique geodesic.

Question: What's the relationship between the word length of $w$, and the hyperbolic length of the corresponding geodesic?

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    $\begingroup$ In group terms, define in any finitely generated group with given word metric, $c(g)$ as $\min_h|hgh^{-1}|$. Then in your setting the length of the geodesic defined by $w$ is comparable to $c(w)$. (All this is unrelated to the choice of relators. Only the choice of generators matters.) $\endgroup$ – YCor Jan 10 at 19:43
  • $\begingroup$ I'm not sure I understand the question. There are lots of hyperbolic metrics on your surface, that don't change the group, and lots of presentations, that don't change the metric. What kind of relationship are you looking for? $\endgroup$ – Hempelicious Jan 10 at 23:09
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    $\begingroup$ You are confusing/conflating based homotopy classes (aka elements of $G=\pi_1(S,x)$) and free homotopy classes (aka conjugacy classes in $G$). Which do you really mean? If you mean based homotopy classes then the relation of the word length and the hyperbolic length is given by Milnor-Schwarz lemma. $\endgroup$ – Moishe Kohan Jan 11 at 0:57
  • $\begingroup$ @MoisheCohen isn't it true that free homotopy classes have a one-to-one correspondence with conjugacy classes of $\pi_1(S,x)$? And could you expand on the significance of the Milnor-Schwartz lemma in this case? $\endgroup$ – 3891780 Jan 11 at 10:36
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    $\begingroup$ @3891780: 1. Yes, this is what I said in my comment. 2. I could, but it is unclear to me what your question really is. Are you talking about based or unbased (free) homotopy classes? Do you know what Milnor-Schwarz lemma is? Do you know about Morse lemma (stability of geodesics)? $\endgroup$ – Moishe Kohan Jan 11 at 18:06

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