Residual finiteness of hyperbolic 3-manifold groups

Question

So the consequence of the geometrization (according to 3-manifold group note) is that any finite-volumed hyperbolic 3-manifold is residually finite. So the question is:

Q1. If $M$ is an infinite-volumed hyperbolic 3-manifold, in particular, a compact 3-manifold whose interior admits a hyperbolic structure such that it contains at least one boundary component with genus $\geq 2$, then $\pi_1(M)$ is residually finite?

Since any subgroup of a residually finite group is again residually finite, I can also ask:

Q2. Any infinite-volumed hyperbolic 3-manifold is a covering of a finite-volumed hyperbolic 3-manifold? i.e., if $M = \Bbb H^3/\Gamma$ is infinite-volumed, then I can find $\tilde{\Gamma}>\Gamma$ such that $\Bbb H^3/\tilde{\Gamma}$ is finite-volumed?

I think the first question is positive but couldn't find a reason.

Edit: I think this answer answers Q1 but I'm not sure if it's dealing with my case (infinite volumed case) because I think the main reference of that is 3-manifold groups note and I think that note mainly deals with finite volumed ones.

Answer 1

Here's another negative answer for Q2. I'm assuming (as in Sam Nead's answer) that the covering should be locally isometric. By Ahlfors-Bers, a tame infinite volume hyperbolic manifold with ends of the form $\Sigma \times [0,\infty)$ where $\Sigma$ is a higher-genus surface will have uncountably many hyperbolic structures (coming from the Teichmüller space of $\Sigma$). On the other hand, there are only countably many finite-volume hyperbolic manifolds. So most of those uncountably many hyperbolic manifolds don't cover a finite volume one.

The proof of the latter statement is that there are only finitely many homeomorphism types, since each finite volume manifold is the interior of a compact manifold with boundary. But now Mostow rigidity says that the hyperbolic structure is unique.

Answer 2

Sam Nead's answer does it, but perhaps I can offer a slightly different perspective on Question 1. No complicated hyperbolic gluing results are needed.

I assume we are satisfied with the characterisation of a hyperbolic 3-manifold as a quotient $M=\Gamma\backslash\mathbb{H}^3$, where $\Gamma$ is a torsion-free discrete subgroup of $SL_2(\mathbb{C})$. If $M$ is tame then, by definition, $M$ is homotopy equivalent to a compact $3$-manifold with boundary, and in particular $\Gamma=\pi_1M$ is finitely generated. Therefore, $\Gamma$ is residually finite, by Malcev's theorem that finitely generated linear groups are residually finite [1]. (Note that this is also the theorem one uses in the finite-volume case, so there's no avoiding it.)

By the tameness theorem, $M$ being tame is the same as $\pi_1M$ being finitely generated. But I have no idea what happens outside the finitely generated case. Which is to say, I don't know if every infinitely generated torsion-free Kleinian group is residually finite. Presumably this is well known to someone... [In comments, Misha links to a nice construction of a non-residually finite Kleinian group, but it has torsion.]

[1] A.I. Malcev: On isomorphic matrix representations of infinite groups of matrices (Russian), Mat. Sb. 8 (1940), 405–422 & Amer. Math. Soc. Transl. (2) 45 (1965), 1–18

Answer 3

The answer to the first question is "yes" and the answer to the second is "no", assuming that you are looking for a covering which is a locally isometric. If you do not require a locally isometric covering then the answer to the second is "yes" and it follows from the first.

Claim: Tame hyperbolic three-manifolds have residually finite fundamental groups.

Proof: In fact, such manifolds embed nicely into finite volume ones. Here are the details.

Suppose that $M$ is a compact three-manifold whose interior admits a hyperbolic structure. Let $\partial_0 M$ be the boundary components of $M$ which are tori. If $\partial_0 M = \partial M$ then we are done. Suppose that $\partial_+ M = \partial M - \partial_0 M$ is non-empty - so $M$ has boundary components of higher genus. For each component $S$ of $\partial_+ M$ we choose a "sufficiently complicated" mapping class $f_S$. Let $F_+$ be the union of the $f_S$. We glue two copies of $M$ using $F_+$ to obtain $D(M, F_+)$. The manifold $D(M, F_+)$ is a "double" of $M$; it is hyperbolic and of finite volume. Also, the inclusion of $M$ into $D(M, F_+)$ induces an injection of fundamental groups. So $\pi_1(M)$ is residually finite. $\Box$

Claim: There are infinite-volume hyperbolic three-manifolds which do not cover (locally isometrically) any finite volume hyperbolic three-manifold.

Proof: Suppose that $N$ is hyperbolic, of finite volume, and without cusps. Then there is a constant $\epsilon_N > 0$ so that, at all points of $x$ of $N$ the injectivity radius of $N$ is bounded below by $\epsilon_N$. Thus the same is true for all covers (finite or infinite), using the same constant $\epsilon_N$.

On the other hand, there are infinite volume three-manifolds $M$, homeomorphic to $S \times \mathbb{R}$ ($S$ closed of genus two), which have a sequence of geodesics $\gamma_n$ (exiting one end of $M$) so that the length of $\gamma_n$ tends to zero as $n$ tends to infinity. $\Box$

Answer 4

The answer to Q1 is negative in general (allowing infinitely generated fundamental group). See Example 2 which is a discrete torsion-free subgroup $G< PSL_2(\mathbb{C})$, hence $\mathbb{H}^3/G$ is a hyperbolic manifold with non-residually finite fundamental group:

Cremaschi, Tommaso; Souto, Juan, Discrete groups without finite quotients, Topology Appl. 248, 138-142 (2018). ZBL1404.20023.

Of course this also gives a negative answer to Q2, even as a covering space in the topological category.

As observed in some other answers, if $M$ is a hyperbolic 3-manifold and $\pi_1(M)$ is finitely generated, then $M$ admits a Scott-Shalen core and hence has residually-finite fundamental group by the compact case.

Q2 is almost true in the finitely generated case, in the sense that a finitely generated discrete torsion-free group $ G <PSL_2(\mathbb{C})$ is an algebraic limit of geometrically finite groups. In turn, these can be perturbed to be covers of finite-volume 3-manifolds.

Brooks, Robert, Circle packings and co-compact extensions of Kleinian groups, Invent. Math. 86, 461-469 (1986). ZBL0578.30037. MR0860677