Degree-one maps between hyperbolic 3-manifolds with the same volume limit (Q2719047)

Generalizing Mostow rigidity, the Gromov-Thurston rigidity theorem implies that, if there is a degree-one map \(f:M \to N\) between closed hyperbolic manifolds, then the volume of \(N\) is not greater than the volume of \(M\); moreover, in dimensions \(\geq 4\), the volumes of \(M\) and \(N\) are equal if and only if \(f\) is homotopic to an isometry. The main result of the present paper is the following. Let \(f_n:M_n \to N_n\) be degree-one maps between closed hyperbolic 3-manifolds, \(n \in \mathbb N\), such that the volumes of \(M_n\) and of \(N_n\) converge to the same finite value. Then for all but finitely many \(n\), \(f_n\) is homotopic to an isometry. A special case of the argument gives a new proof (and a stronger version not assuming equality of volumes) of the Gromov-Thurston rigidity theorem, avoiding any ergodic theory which was applied in the original proof to the sphere at infinity of hyperbolic space, after radially extending a map. Another consequence is the fact that, for any ascending sequence of non-homotopy equivalence, degree-one maps between closed hyperbolic 3-manifolds, the volumes of the manifolds become arbitrarily large. Examples are given showing that the main result of the paper does not hold for maps of degree greater than one.