Asymptotic bi-infinite geodesics in CAT($-1$) space coincides

Question

Suppose $\gamma_1,\gamma_2:(-\infty,\infty)\to (X,d)$ be two bi-infinite geodesic such that $d(\gamma_1(t),\gamma_2(t))<K$ for all $t$. Here $(X,d)$ is a CAT$(-1)$ space. Then Image($\gamma_1$)=Image($\gamma_2$).

Before posting I have seen that the solution to my problem would follow from this problem but there is no answer to it. I tried to use the hyperbolic law of cosines to estimate the distance $d(\gamma_1(t),\gamma_2(t))$ to make it arbitrarily small so that it finally tends to zero as $t$ goes to $\infty$.

Please help! Some hints would help me as well. Thank you.

Answer 1

This answer is the culmination of Moishe Kohan's comments and hints which he has provided in the comments. Thanks!

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Let us call the convex set generated by the image of $\gamma_1$ and $\gamma_2$ by $S$.

$S$ with subspace metric is CAT$(-1)$ as it is a convex subspace.

By the flat strip theorem $S$ is isometric to $[0, D]\times \mathbb{R}$, where $D$ is the Hausdorff distance between the images of two geodesics, hence $D\ge0$.

Note: For 'Flat Strip Theorem' refer to Metric Spaces of Non-Positive Curvature by André Haefliger and Martin Bridson page 182. Also note that a CAT($-1$) space is CAT($0$).

Suppose $D>0$ then we can consider a non-degenerate triangle $\Delta$ in $S$. Then by the conclusion of the flat strip theorem sum of the angles of $\Delta$ is $\pi$. But $S$ being CAT$(-1)$ has sum of the angles of $\Delta$ strictly less than $\pi$. This is not possible!

Hence we must have $D=0$ therefore Image$(\gamma_1)$=Image$(\gamma_2)$