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A consequence of the famous Jørgensen inequality is that there is a lower bound for the distance between closed geodesics in hyperbolic three-manifolds: for any $R>0$ there is a c>0 such that for any such manifold $N$ and any two distinct closed geodesics loops $c_1,c_2$ on $M$ both of length less than $R$ we have $$ d_H(c_1,c_2) \ge c \quad (\ast) $$ where $d_H$ is Hausdorff distance. This is false for general locally symmetric spaces, since the higher-rank ones can contain flat tori. My question is the following: if we fix a real-rank one symmetric space $X$ (i.e. $X$ is a real, complex or quaternionic hyperbolic space, or octonionic hyperbolic plane) is there a c>0 (depending on the space) such that $(\ast)$ holds for all quotients of $X$ and all pairs of distinct closed geodesics of bounded length on such? (More generally one could ask the same question for Riemannian $n$-manifolds with sectional curvature between $-1$ and $\kappa<0$, with a $c$ depending on $\kappa$ and $n$ and the bound on the length).

Edit: As noted in Anton's comments the opening remark is false if one does not bound the length of the geodesics; I edited to add that.

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  • $\begingroup$ One can not fix such constant $c$ for all metrics with curvature $-1$ on the surface of genus 2. But probably you can choose $c$ for each negatively pinched manifold. $\endgroup$
    – Anton Petrunin
    Commented Aug 19, 2013 at 16:10
  • $\begingroup$ By the way, I think you wanted to say Fréchet distance; the Hausdorff distance between the images can be made arbitrary small. $\endgroup$
    – Anton Petrunin
    Commented Aug 19, 2013 at 16:14
  • $\begingroup$ @AntonPetrunin: I may have misphrased the question, but I am looking for a bound which depends only on the local geometry for the minimal distance between two closed (nontrivial) geodesic loops, for the distance defined between two closed sets $X,Y$ by the smallest $d$ such that both $X$ and $Y$ are contained in the $d$-neighbourhood of the other. I think this is universally bounded below for hyperbolic surfaces because you cannot have two noncommuting hyperbolic elements too close to each other in a discrete subgroup of ${\rm PSL}_2(\mathbb{R})$. $\endgroup$
    – Jean Raimbault
    Commented Aug 19, 2013 at 17:24
  • $\begingroup$ If you measure the Hausdorff distance between the images. Choose two generators $a$ and $b$ and note that the distance from $a^n\cdot b^n$ to $a^{n+1}\cdot b^{n+1}$ goes to $0$ as $n\to \infty$. $\endgroup$
    – Anton Petrunin
    Commented Aug 19, 2013 at 17:34
  • $\begingroup$ @AntonPetrunin: OK, I think I see that now: there may be pairs of geodesics which are very close but one "makes more turns" than the other, so the corresponding elements in the fundamental group are away from each other in ${\rm PSL}_2(\mathbb{R})$ and this does not contradict J&oslash;rgensen's inequality. I will change the question accordingly. $\endgroup$
    – Jean Raimbault
    Commented Aug 19, 2013 at 17:54

2 Answers 2

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I think such a bound exists (depending only on pinched curvature constant $\kappa$, dimension $n$, and $R$). Suppose one has an infinite sequence of pinched negatively curved manifolds where geodesics of length $\leq R$ have Hausdorff distance approaching $0$. By the generalized Margulis Lemma (see Ballmann-Schroeder), there is an $\epsilon$ so that in a $\kappa$-pinched $n$-dim. manifold, $\epsilon$-thin regions are tubular neighborhoods about short geodesics of radius $>R$. Thus, we may assume that the curves pass through the thick part of the manifold, otherwise they must be homotopic to short geodesics in the thin part, in which case it follows. One may take a pointed Cheeeger-Gromov limit (with basepoint on either curve in the thick part), to get a limit in which the Hausdorff distance between the two curves is zero. This gives a contradiction, since these curves are homotopic in the limit, but this implies they will be homotopic in the approximates.

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  • $\begingroup$ Thanks for the answer; this is good enough for me although I did not check all lemmas involved in this proof. The only place I am not completely comfortable with is that something could happen in the thin part if the injectivity radius goes to 0, although the following considerations (if true) rule this out: Magulis' lemma implies that closed goedesics of length less than $R$ stay out of the $\delta$-thin part for a $\delta>0$ depending on $R$ (which would imply, if I'm not mistaken, that the Gromov-Hausdorff limit you use is a manifold near the geodesics we are interested in). $\endgroup$
    – Jean Raimbault
    Commented Aug 23, 2013 at 16:22
  • $\begingroup$ Yes, that's right - I didn't really explain this part of the argument. By choosing $\epsilon$ small enough, the $\epsilon$-thin has an embedded a tubular neighborhood of radius $R$ about it. If a geodesic of length $\leq R$ entered this tube (but wasn't a multiple of the core geodesic), then it couldn't get back out again of the tubular neighborhood. The way I phrased it wasn't quite right. $\endgroup$
    – Ian Agol
    Commented Aug 23, 2013 at 16:54
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This is really a question about the fundamental group, since two such geodesics necessarily belong to distinct free homotopy classes of loops. One immediate answer is given by the injectivity radius of the space. The distance between the loops must be at least the injectivity radius, since otherwise one can be homotoped into the other.

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  • $\begingroup$ Thanks for the remark; I agree, but I am looking for a bound which does not depend on the global geometry but only on the universal cover. $\endgroup$
    – Jean Raimbault
    Commented Aug 19, 2013 at 17:25
  • $\begingroup$ @JeanRaimbault, you may have such bounds only if you can estimate the injectivity radius. $\endgroup$
    – Anton Petrunin
    Commented Aug 19, 2013 at 17:39

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