Let $M$ be a complete hyperbolic manifold of dimension $n$, let $\varepsilon=\varepsilon_n$ be the Margulis constant. Let $M_{[\varepsilon,\infty)}$ be the thick part of $M$ with respect to $\varepsilon$. Suppose that $\pi_1(M)$ is infinite. Is it true that $\pi_1(M_{[\varepsilon,\infty)})$ is also infinite. Or, in case $M_{[\varepsilon,\infty)}$ is not connected, whether there exist a connected component of $M_{[\varepsilon,\infty)}$ such that its fundamental group is infinite.

PS: I am reading a paper, where this fact seems to be an important point in the proof. This is totally not my area, so it might as well be trivial for anyone familiar with these notions, however, I will be much obliged for an expalnation why this is (or is not) true.