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

Abstract:

Suppose that $f_n:M_n\longrightarrow N_n$ $(n\in {\mathbf N})$ are degree-one maps between closed hyperbolic 3-manifolds with \[ \lim _{n\rightarrow \infty } \operatorname {Vol} (M_n)=\lim _{n\rightarrow \infty }{\operatorname {Vol}}(N_n) <\infty . \] Then, our main theorem, Theorem 2, shows that, for all but finitely many $n\in {\mathbf N}$, $f_n$ is homotopic to an isometry. A special case of our argument gives a new proof of Gromov-Thurston’s rigidity theorem for hyperbolic 3-manifolds without invoking any ergodic theory. An example in §3 implies that, if the degree of these maps is greater than 1, the assertion corresponding to our theorem does not hold.