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I asked this question on math stackexchange (see here) but didn't get any answer so I thought I post it here too.

We have the following two well-known Theorems:

T1) For all $\delta > 0, \lambda \geq 1, \varepsilon \geq 0$ there exists a constant $R = R(\delta, \lambda, \varepsilon)$ with the following property: If $X$ is a $\delta$-hyperbolic geodesic space, $c$ is a $(\lambda, \varepsilon)$-quasi-geodesic in $X$ and $[p,q]$ is some geodesic segment joining the endpoints of $c$, then the Hausdorff distance between $[p,q]$ and the image of $c$ is less than $R$. (Hence there is some constant such that $[p,q]$ is contained in the neighbourhood of $c$ and vice versa)

T2) Let $X$ be a $\delta$-hyperbolic geodesic space and let $c: [a,b] \to X$ be a $k$-local geodesic, where $k > 8\delta$. Then:

(i) im(c) is contained in the $2 \delta$-neighbourhood of any geodesic segment connecting the endpoints of $c$.

(ii) $[c(a),c(b)]$ is contained the $3 \delta$-neighbourhood of im(c)

(iii) $c$ is a $(\lambda, \varepsilon)$-quasi-geodesic, where $\varepsilon = 2 \delta$ and $\lambda = (k + 4 \delta)/(k - 4 \delta)$.

I am wondering if Theorem 2 (T2) is also true (in an apropriate way) for $k$-local-$(\lambda, \varepsilon)$-quasi-geodesic, i.e. that such local quasi geodesics are actually quasi-geodesics. In the book of Bridson and Haefliger (Metric spaces of non-positive curvature) this should follow in the 'obvious' way of Theorem 1 (T1) and Theorem 2 (T2) above. However I have sme troubles writing this down explicitly.

More precisely in the above mentioned book the following is written:

"By combining (1.7) [our T1] and (1.13) [our T2] in the obvious way one gets a criterion for seeing that paths which are locally quasi-geodesic (with suitable parameters) are actually quasi-geodesics."

Question: This quote is exactly the content of my question, i.e. how to combine these two theorems and in particular what is the exact statement we get (e.g. for the constant $k$ in T2)?

My heuristic would be as follows: A local quasi geodesic is somehow close to a local geodesic by T1 which in turn is a quasi geodesic by T2. Thus the local quasi geodesic we were starting with is close to a quasi geodesic and hence one may deduce that it is itself a quasi geodesic.

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I don't have the book in front of me but in the book of Coornaert, Delzant and Papadopoulos you can find the fact that local quasi-geodesics are quasigeodics provided that the locality Parameter is sufficiently large.

PS: I checked, it is Théorème 1.4 of Chapitre 3

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  • $\begingroup$ I "know" the book (but my french skills tend rapidly to zero) and this statement, do you know an english translation? $\endgroup$ – M.U. Sep 16 '15 at 9:43
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    $\begingroup$ I don't think that the book has been translated. I did however write lecture notes a while ago that also treat this theorem. I can send you a copy if you email me. There might be other sources but I am not aware of any. $\endgroup$ – Richard Weidmann Sep 16 '15 at 9:52

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