The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds

Mihajlo Cekić, Semyon Dyatlov, Benjamin Küster, Gabriel P. Paternain

Published 2020-09-17Version 1

We show that for a generic conformal metric perturbation of a hyperbolic 3-manifold $\Sigma$, the order of vanishing of the Ruelle zeta function at zero equals $4-b_1(\Sigma)$, contrary to the hyperbolic case where it is equal to $4-2b_1(\Sigma)$. The result is proved by developing a suitable perturbation theory that exploits the natural pairing between resonant and co-resonant differential forms. To obtain a metric conformal perturbation we need to establish the non-vanishing of the pushforward of a certain product of resonant and co-resonant states and we achieve this by a suitable regularisation argument. Along the way we describe geometrically all resonant differential forms (at zero) for a closed hyperbolic 3-manifold and study the semisimplicity of the Lie derivative.