When are two hyperbolic 3-manifolds homeomorphic?

A preprint of Lins and Lins appeared on the arXiv today, posing a challenge [LL].  In this post, I’m going to discuss that challenge, and describe a recent algorithm of Scott–Short [SS] which may point towards an answer.

The Lins–Lins challenge

The theory of 3-manifolds is now very advanced, and we can even say in a certain sense that we understand ‘all’ 3-manifolds (as I discussed in an earlier post).  But that understanding is very theoretical; the Lins–Lins challenge is to put this theory into practice.

They ask: ‘Are the two closed, hyperbolic 3-manifolds given by Dehn surgery on the following two framed links homeomorphic?’

(I’ve taken the liberty of copying the diagrams from their paper.)

The two manifolds are both hyperbolic, and calculations have been unable to distinguish their volumes or Witten–Reshetiken–Turaev invariants, up to many decimal places.  On the other hand, Lins–Lins’s techniques for finding homeomorphisms have also failed.  They (rather optimistically, in my opinion) bet that the failure of their techniques indicates that these manifolds really aren’t homeomorphic, and challenge the broader 3-manifold and combinatorial group theory communities to resolve the question.

They also note that, by Mostow Rigidity, we may equivalently try to determine whether or not the fundamental groups are isomorphic. They helpfully provide presentations.

(Again, I’ve copied this straight from their paper.)

There are some obvious things that one might attempt and which they don’t mention, such as computing the homology of the small-index subgroups of these groups.  Presumably they forgot to mention this, but it might be worth a try if anyone has any spare time.

(Added: This was a good idea! Nathan Dunfield did exactly this (see the comments below) and found that $G_1$ has a subgroup of index six with infinite abelianization, whereas $G_2$ does not. Readers are also referred to Nathan’s comments on the usefulness of SnapPea for this sort of problem.)

Regardless of the difficulty of this specific question, their broader complaint is valid.  Although we know that the homeomorphism problem for hyperbolic 3-manifolds (and, more generally, for all 3-manifolds) is solvable in theory by a remarkable algorithm of Sela, it has never been practical to even think about implementing that algorithm [S].

Aside: Sela’s algorithm

Sela’s algorithm is a tour de force; it solves the isomorphism problem for all ‘rigid’ torsion-free word-hyperbolic groups and, in fact, with rather more work, for all word-hyperbolic groups [DGu] and even all toral relatively hyperbolic groups [DGr].  But this very generality makes it daunting to think about implementing in practice.  One of the bounds is obtained from a classic Sela-style argument by contradiction on a limiting R-tree, so we get no control over this term. To add to the difficulty, the proof reduces the calculation to a set of equations and inequations over a non-abelian free group, which can be solved by Makanin’s algorithm [Mak]. Implementing Makanin’s algorithm is itself a highly non-trivial task.

The Scott–Short algorithm

I want to spend the rest of this post describing a remarkable recent observation of Scott–Short, which deserves to be much better known [SS].  They have a described a completely new solution to the homeomorphism problem for hyperbolic 3-manifolds, which relies solely on the geometry of hyperbolic 3-space. Their algorithm is technically much simpler than Sela’s, and huge credit must go to them for finding a better way of doing things.

Although I don’t for a moment think that their algorithm would be efficient as it stands, its simplicity and geometric nature bring the problem within reach.  Anyone with a working knowledge of hyperbolic geometry can understand it, which means that anyone can think about ways to improve it. Also, it has the huge advantage that it reduces the problem to calculations over the complex numbers,  rather than free groups.

Suppose we are given closed, orientable, hyperbolic manifolds $M_1$ and $M_2$ that we want to identify or distinguish. The starting point for Scott–Short is an algorithm of Manning (originally described by Casson) [Man], which constructs a fundamental domain $P_i$ in hyperbolic 3-space, together with side-pairing isometries $\{g_k^{(i)}\}$, for the manifold $M_i$. By Mostow Rigidity, we need to determine whether or not the subgroups

$G_i=\langle g_k^{(i)}\rangle \subseteq P⁢S⁢L_2⁢(\mathbb{C})$

are conjugate.

In practice, the fundamental domain and the side pairings are the sort of information that one usually already has about a given hyperbolic 3-manifold, so one shouldn’t need to implement Manning’s algorithm.  Perhaps an expert can tell me whether or not Snappea provides this information?  (It’s fine to go through non-exact computations like Snappea. If a candidate isomorphism is found, it’s easy to confirm whether or not it’s genuine; likewise, non-exact estimates are fine to compute the bounds we would need for a proof of non-isomorphism.)

Deciding whether or not two finite subsets of $PSL_2(\mathbb{C})$ are conjugate is just a matter of solving a system of equations over $\mathbb{C}$, which is relatively straightforward.  The difficulty arises because, even if $G_1$ and $G_2$ are isomorphic, the generating sets we have been given may be very different. Therefore, a priori, wenhave infinitely many different possible conjugacies to check.

Scott and Short use the geometry of the fundamental domains $P_i$ to reduce this to a finite number.  They define the n-star of $P_i$ to be the union of the translates of $P_i$ by the elements of word length at most n in $G_i$.  That is, in the tiling of hyperbolic 3-space by copies of $P_i$ it’s the set of tiles that are at most n steps away from $P_i$ itself.

They observe that, for any r, we can compute an n so that the metric ball of radius r around $P_1$ is contained in the n-star of $P_2$.  Since

$g_k^{(1)}⁢P_1\subseteq B⁢(P_1,\mathrm{diam}⁢P_1)$

for any k, it follows that we can compute an n so that the $g_k^{(1)}$ are all contained in the ball of diameter n in the word metric on $G_2$. We can then do the same thing the other way round, which reduces the problem to a finite computation as advertised, and we are done!

I recommend the Scott–Short paper, which is very readable.  It seems likely to me that an improvement of their algorithm, although it would no doubt be very slow on worst-case input, might well be effective enough to resolve questions like the Lins–Lins challenge. So I’ll finish with a challenge of my own.

Challenge:  Try to implement the Scott–Short algorithm!  Find ways to improve it!

Bibliography

[DGr] Dahmani, Groves, ‘The isomorphism problem for toral relatively hyperbolic groups’, Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 211–290.

[DGu] Dahmani, Guirardel, ‘The isomorphism problem for all hyperbolic groups’, Geom. Funct. Anal. 21 (2011), no. 2, 223–300.

[LL] Lins, Lins, ‘A challenge to 3-manifold topologists and group algebraists’, arXiv:1304.5964v1

[Mak] Makanin, ‘Decidability of universal and positive theories of a free group,’ (English translation) Mathematics of the USSR-Izvestiya 25 (1985), 75–88; original: Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya 48 (1984), 735–749.

[Man] Manning,  ‘Algorithmic detection and description of hyperbolic structures on closed 3-manifolds with solvable word problem’, Geom. Topol. 6 (2002), 1–25

[SS] Scott, Short, ‘The homeomorphism problem for closed 3-manifolds’, arXiv:1211.0264v1

[S] Sela, ‘The isomorphism problem for hyperbolic groups. I’, Ann. of Math. (2) 141 (1995), no. 2, 217–283.