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I am looking for an example of $M$ and $N$ two orientable hyperbolic complete without boundary 3-manifolds ( with infinite hyperbolic volume) such that $\pi_{1}M\cong \pi_{1}N$ but $M$ is not isometric to $N$.

Is there a concrete such example ?

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    $\begingroup$ @Wojowu afaik S^2 x R cannot have an hyperbolic complete structure. $\endgroup$
    – GSM
    Commented Nov 7, 2021 at 14:48
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    $\begingroup$ I am ignorant of the subject, so perhaps this is silly. But a loxodromic transformation $\varphi$ of hyperbolic 3-space acts without fixed points, as does a parabolic transformation $\psi$. These isometries are not conjugate. This should imply that the resulting quotients $\Bbb H^3/\langle \varphi\rangle$ and $\Bbb H^3/\langle \psi\rangle$ should be both diffeomorphic to $S^1 \times \Bbb R^2$ but non-isometric. Have I made an error here? $\endgroup$
    – mme
    Commented Nov 7, 2021 at 15:14
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    $\begingroup$ @mme: You are correct. $\endgroup$
    – Moishe Kohan
    Commented Nov 7, 2021 at 15:18
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    $\begingroup$ (Obviously in the comment above I meant for both manifolds to have the same fundamental group, or it has a tautological answer.) $\endgroup$
    – mme
    Commented Nov 7, 2021 at 16:38
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    $\begingroup$ @mme: It's easy already in the 2d case: one-hole torus and 3-holed sphere. In 3d this is also not hard, say, trivial and nontrivial interval bundles over genus 2 surface. $\endgroup$
    – Moishe Kohan
    Commented Nov 7, 2021 at 18:13

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Yes, I believe there are many.

For example, if you think of hyperbolic $2$-space as a geodesic subspace of hyperbolic $3$-space, any group of hyperbolic isometries of hyperbolic $2$-space extends naturally to a group of hyperbolic isometries of hyperbolic $3$-space.

So if you take the fundamental group of a hyperbolic $2$-manifold, and embed it in this manner in the group of isometries of hyperbolic $3$-space, and take the quotient of hyperbolic $3$-space, you get a hyperbolic $3$-manifold of infinite volume such that the hyperbolic $2$-manifold is a totally geodesic submanifold. Topologically, it has the form $\Sigma \times \mathbb R$ but the metric is flared out away from $\Sigma \times \{0\}$ so that the manifold has infinite volume.

This is pretty much the tautological example.

For different hyperbolic metrics on $\Sigma$ you get different hyperbolic $3$-manifolds.

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