The panted cobordism group of cusped hyperbolic 3-manifolds

For any oriented cusped hyperbolic $3$-manifold $M$, we study its $(R,\epsilon)$-panted cobordism group, which is the abelian group generated by $(R,\epsilon)$-good curves in $M$ modulo the oriented boundaries of $(R,\epsilon)$-good pants. In particular, we prove that for sufficiently small $\epsilon>0$ and sufficiently large $R>0$, some modified version of the $(R,\epsilon)$-panted cobordism group of $M$ is isomorphic to $H_1(\text{SO}(M);\mathbb{Z})$.