Classifying nested 3-manifolds with fundamental group property

Question

Let $M_1\subseteq M_2\subseteq\mathbb R^3$ be closed connected subsets with smooth boundary. Suppose that every closed loop in $M_1$ is freely homotopic inside $M_2$ to a closed loop inside $M_2\setminus M_1^\circ$. Equivalently, every element of $\pi_1(M_1)$ is conjugate inside $\pi_1(M_2)$ to an element in the image of $\pi_1(M_2\setminus M_1)\to\pi_1(M_2)$. Also assume that the complements of $M_1$ and $M_2$ are connected (by Alexander's theorem this implies that $M_1$ and $M_2$ are both irreducible). What can be said about $M_1$ and $M_2$ in this situation?

My goal here is to force $\pi_1(M_1)\to\pi_1(M_2)$ to be zero. Unfortunately, this is not guaranteed in the situation above, since we could take $M_1=M_2$ to be a handlebody. Indeed, any loop inside a handlebody can be pushed to its boundary (use general position to make it disjoint from a graph $G\subseteq M_1$ of which $M_1$ is a regular neighborhood). But maybe this is "as bad as it gets"?

Without the assumption on the connectedness of $\mathbb R^3\setminus M_i$, there are more counterexamples such as $M_1=B(2)\setminus B(1)^\circ$ and $M_2=B(2)$, but I don't care much about this relaxation of the question.

The significance of the hypothesis that $M_2\subseteq\mathbb R^3$ (rather than just being any compact connected orientable irreducible three-manifold with boundary) is that I know it holds in the situation I am interested in. It may be end up being irrelevant as far as the question asked is concerned.

Answer 1

The only sort of examples that I can imagine are when there is a handlebody $H$ such that $M_1\subset H \subset M_2$. It holds more generally when $\pi_1(M_2-M_1)\to \pi_1(M_2)$ is onto. This might also be a necessary condition, but I don’t know how to prove it.

There may be an algorithm to determine when this condition holds, at least in the case that $\overset{\circ}{M_2}$ admits a complete hyperbolic metric. I’ll give a sketch of proof in this case (assuming the volume of $\overset{\circ}{M_2}$ is infinite).

Endow $\overset{\circ}{M_2}$ with a hyperbolic metric, then there are covering spaces $L\to \overset{\circ}{M_2}$ such that $\pi_1(L)=im\{\pi_1(M_2-M_1)\to \pi_1(M_2)\}$ and $N\to \overset{\circ}{M_2}$ such that $\pi_1(N)=im\{ \pi_1(M_1)\to \pi_1(M_2)\}$. If $\chi(M_2)<0$, then we may assume that $\overset{\circ}{M_2}, N, L$ are geometrically finite and have convex cores $C(M_2), C(N), C(L)$. The projections $C(N)\to \overset{\circ}{M_2}$ and $C(L)\to \overset{\circ}{M_2}$ will be finite volume locally convex immersions. Any loop $g$ in $M_1$ will lift to $N$ and hence be homotopic to a geodesic $\gamma$ immersed in $C(N)$ and hence projecting to an immersion geodesic in $C(M_2)$. Similarly any loop $g$ in $M_2-M_1$ will lift to $L$ and be homotopic to a closed immersed geodesic $\gamma$ in $C(L)$ which projects to an immersed geodesic curve in $C(M_2)$. If $g$ is homotopic into $M_2-M_1$, then we get a commutative diagram $\gamma\to C(L), C(N) \to C(M_2)$. Hence $\gamma$ lies in the pullback $P\subset C(L)\times C(N) \to C(M_2)$ of pairs of points with the same projection in $C(M_2)$. Such pullbacks are studied in the appendix of this paper (in the greater generality of hyperbolic groups and quasi-convex subgroups). In particular, $P$ has finitely many components with non-trivial convex core and fundamental group projecting to finitely generated subgroups of $\pi_1(N)$. A loop $g$ in $\pi_1(N)$ will be conjugate into $\pi_1(L)$ iff its geodesic realization $\gamma$ lifts to $P$. If all of these subgroups corresponding to components of $P$ are infinite index in $\pi_1(N)$, then I believe that not every geodesic in $C(N)$ will have a lift to $P$, and hence your condition will not hold. The point is that each component of $P$ will have convex core mapping to a proper subset of $C(N)$. Take a geodesic in $C(N)$ having points on it missing each of these convex cores, then this geodesic will not lift to $P$.

On the other hand, some component of $P$ might have image in $\pi_1(N)$ a finite-index subgroup. If this is the case, then there could be finitely many components which hit every conjugacy class. It seems hard to imagine an example where the index is not 1, in which case I think that $L=\overset{\circ}{M_2}$.

I think that there should be algorithms for computing $N, L, P$ and to determine if the condition holds. There are algorithms for computing subgroups of Kleinian groups. Using these, I think one should be able to compute (coarse) convex cores and $P$, and then determine if the conjugacy condition holds. However working out the details of such an algorithm may be involved and might be an appropriate project for a paper (it’s possible that such an algorithm, or various subroutines, exists in the literature, but I haven’t tried to do a search).