By Margulis lemma, components of the $\varepsilon$-thin part are cusps or $\varepsilon$-tubes, so the interior of the $\varepsilon$-part is $M$ with cusps chopped off, and a finite (possibly empty) collection of embedded closed geodesics deleted. Thus the thick part is connected, and by general position the inclusion of the thick part into $M$ is $\pi_1$-surjective when $n=3$, and is a $\pi_1$-isomorphism when $n>3$. In particular, the fundamental group of the thick part is infinite, because it surjects onto an infinite group $\pi_1(M)$.

By Margulis lemma, components of the $\varepsilon$-thin part are cusps or $\varepsilon$-tubes, so the interior of the $\varepsilon$-part is $M$ with cusps chopped off, and a finite (possibly empty) collection of embedded closed geodesics deleted. Thus the thick part is connected, and by general position the inclusion of the thick part into $M$ is $\pi_1$-surjective when $n=3$, and is a $\pi_1$-isomorphism when $n>3$. In particular, the fundamental group of the thick part is infinite, because it surjects onto an infinite group $\pi_1(M)$.

EDIT: in the above I assumed that $n>2$.