2
$\begingroup$

I am trying to verify the computations of the symmetry (isometry) groups of some closed hyperbolic 3-manifolds from the Hodgson-Weeks census with Regina or SnapPy. In particular (as one use-case) I want to verify that the isometry group of the Weeks manifold is dihedral of order 12.

SnapPy has the command symmetry_group(), but for some of the manifolds in the census it only gives a lower bound on the size, so I need something stronger in general.

How does one go about doing this?

Improve this question
$\endgroup$
2
  • $\begingroup$ I don't think SnapPy has the capability to compute these symmetry groups. It's only going to work when you have a cusped hyperbolic triangulation that agrees with the Epstein-Penner decomposition. If you knew how geodesics are acted-upon by the symmetry group of your manifold, you could start drilling out the group-orbits of some of the geodesics and hope you get a link in $S^3$, i.e. realize your manifold as a "symmetric filling". $\endgroup$ Commented yesterday
  • 1
    $\begingroup$ @RyanBudney - As I point out in my answer, Matthias has a new isometry signature for closed manifolds. With a bit of (not too much!) work, this can be used to give a verified symmetry group in the closed case. $\endgroup$ Commented yesterday

1 Answer 1

Reset to default
4
$\begingroup$

You should email Matthias Goerner and explain your use case. He has a new isometry signature for closed manifolds. For example:

sage: M = snappy.Manifold("m003(-3,1)")
sage: M.isometry_signature(verified=True)
'cPcbbbdxm(2,1)'

This will enable the rigorous computation of symmetry groups (and thus their sizes).

Written for previous version of the question: Matthias' new code for the length spectrum shows that the shortish geodesics in the Weeks manifold m003(-3, 1) have multiplicity three. So I very strongly suspect that three divides the order of the symmetry group of the Weeks manifold. In particular, the group probably isn't the dihedral group of order eight.

Improve this answer
$\endgroup$
6
  • $\begingroup$ Does the algorithm work roughly as I was suggesting? Write the manifold as a symmetric filling for a cusped manifold where you have the EP-decomposition. Snappy used to have a "hopeful" routine of that sort. $\endgroup$ Commented yesterday
  • 1
    $\begingroup$ But Matthias can get a canonical set anyway, by generating all closed geodesics in a certain interval. (In practice, the canonical set is just the set of systoles.) He drills all of these curves (one at a time) and computes isometry signatures (and filling slopes) of the surgery parents. The set of isometry signatures of the parents is an isometry signature for the child. $\endgroup$ Commented yesterday
  • 1
    $\begingroup$ Thanks, Sam. That's a helpful short summary. $\endgroup$ Commented yesterday
  • 1
    $\begingroup$ Sorry, 8 was a typo, the symmetry group is (known to be) dihedral of order 12. $\endgroup$ Commented yesterday
  • 1
    $\begingroup$ Thank you, this is helpful! $\endgroup$ Commented yesterday

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.