In this paper, the authors study which compact \(3\)-manifold \(M\) with empty or toroidal boundary has the following property: any two connected finite covers of \(M\) of the same degree are homeomorphic to each other.
For a group \(G\), \(s_n(G)\) denotes the number of index-\(n\) subgroups of \(G\), and \(e_n(G)\) denotes the number of isomorphism types of index-\(n\) subgroups of \(G\). The study of \(s_n(G)\) has attracted the interest of many mathematicians, while studying \(e_n(G)\) is a more sensitive problem. An analogy of \(e_n(G)\) for manifolds is the following notation. For a connected manifold \(M\), \(e_n(M)\) denotes the number of homeomorphism types of connected \(n\)-sheeted covers of \(M\). In this paper, a manifold \(M\) is called exceptional if \(e_n(M)=0\) or \(1\) for all \(n\in \mathbb{N}\), thus any two connected \(n\)-sheeted covers of \(M\) are homeomorphic to each other.
It is straightforward to see that all closed orientable surfaces are exceptional. The main result in this paper (Theorem 1.2) classifies all compact exceptional \(3\)-manifolds with empty or tori boundary, which are: connected sums of \(S^1\times S^2\), \(S^1\tilde{\times} S^2\), \(S^1\times D^2\), \(T^2\times I\), \(T^3\), certain closed \(3\)-manifolds with finite fundamental groups.
It is not hard to see the above \(3\)-manifolds are exceptional, except for the last case, whose proof requires some argument on finite groups. To prove that all other compact \(3\)-manifolds are not exceptional, the authors did a case-by-case argument, and a nice flow chart can be found on Page 814. The proof consists of the following cases: hyperbolic \(3\)-manifolds (Section 4), Euclidean \(3\)-manifolds (Section 5), spherical \(3\)-manifolds (Section 6), Seifert \(3\)-manifolds not in Sections 5 or 6 (Section 7), Sol \(3\)-manifolds (Section 8), \(3\)-manifolds with nontrivial JSJ decomposition or nontrivial boundary (Section 9), nonorientable or non-prime \(3\)-manifolds (Section 10).
Besides the main result, the authors also prove several relevant results, including:

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An estimate of \(e_n(G)\) for Fuchsian groups that have finite covolume and contain torsion elements (Proposition 2.2).

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If \(G\) is an irreducible lattice in a semisimple linear Lie group not locally isomorphic to \(PGL_2(\mathbb{R})\), then \(G\) is not exceptional (Proposition 4.2).

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If \(G\) is a hyperbolic \(3\)-manifold group, then \(\limsup e_n(G)=\infty\), and these non-isomorphic subgroups of same index can be realized by normal subgroups (Proposition 4.5).

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If \(G\) is a lattice in the isometry group of an Euclidean space, then \(G\) is exceptional if and only if \(G\) is free abelian (Proposition 5.2).