Suppose that $g$ and $h$ are non-conjugate loxodromic isometries in a cusped hyperbolic $3$-manifold $M$ of finite volume. Fix a cusp $T$ of $M$. Can I choose a hyperbolic Dehn filling of $M$ along $T$ such that the images of $g$ and $h$ remain (a) loxodromic and (b) non-conjugate in the filled manifold?
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3$\begingroup$ It follows from geometric convergence of the filled manifolds to the original manifold. $\endgroup$– Moishe KohanCommented Dec 12 at 19:46
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It follows from the comment of Moisha Kohan above.
Another way to see it: take the closed oriented geodesics $\gamma, \eta$ in $M$ realizing the monodromy of $g$ and $h$ respectively. Since $g$ and $h$ are non-conjugate, they are distinct as oriented closed geodesics (note though that they could have multiplicity or intersect). Choose a horocusp $H$ disjoint from the images of $\gamma \cup \eta$. Choose a loop $\sigma$ in the boundary of $H$ which has length $> 2\pi$. Then by the proof of the $2\pi$ theorem, one may perform a filling along the slope $\sigma$ which removes $H$ and replaces it with a negatively curved metric on a solid torus so that the resulting manifold is negatively curved. Then the geodesic immersions $\gamma, \eta$ are still distinct in this metric, and hence represent non-conjugate elements in the fundamental group.
