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I am looking for a 3-manifold which is closed, aspherical, orientable, and atoroidal. And additionally I want to see an example that does not admit a fixed-point-free action on a simplicial tree. As a group theorist myself I am more interested in the fundamental group than the manifold and a full answer to this question would include a presentation of the group.

Since Thurston's geometrization program has been realised, such a manifold must be hyperbolic and finitely covered by a surface bundle over a circle. The Seifert--Weber dodecahedral space might be such an example but I do not know whether it is or not. Such an example would lay bare the need to pass to a finite cover when establishing the surface bundle over a circle feature of closed hyperbolic 3-manifolds and its fundamental group may have other interesting pathologies.

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  • $\begingroup$ In arxiv:2312.08913, it is mentioned that a theorem of Stallings shows that, given a closed, orientable, aspherical 3-manifold $M$, $\pi_1(M)$ has property FA iff $M$ is non-Haken. So it seems that your question amounts to finding explicit examples of non-Haken hyperbolic closed 3-manifolds. $\endgroup$
    – AGenevois
    Dec 16, 2023 at 16:13
  • $\begingroup$ @AGenevois yes that is right: an explicit example of a non-Haken hyprbolic closed 3-manifold would answer my question. So a natural question arises: is the Seifert--Weber dodecahedral space Haken. $\endgroup$ Dec 16, 2023 at 18:15
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    $\begingroup$ Can I suggest replacing "some pathology" in the title with "Property FA"? It's just as short and completely precise! $\endgroup$
    – HJRW
    Dec 22, 2023 at 13:47
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    $\begingroup$ A small remark. The word "atoroidal" has two different meanings in 3-manifold topology: no essential embedded torus and no essential "immersed" torus. Examples that are atoroidal in the first sense are easier to write down (although not necessarily hyperbolic): you can use Brieskorn spheres, Seifert-fibred spaces fibring over hyperbolic triangle orbifolds. $\endgroup$
    – HJRW
    Dec 22, 2023 at 13:54

1 Answer 1

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One answer to your question comes from the paper The Weber-Seifert dodecahedral space is non-Haken by Burton, Rubinstein, and Tillmann.

An earlier example is (say) the $(1, 2)$-Dehn filling of the figure-eight knot. This manifold is hyperbolic (4.22) and non-Haken (4.41). The references are page numbers in Thurston's lecture notes. We can use SnapPy to find presentations of the fundamental groups.

Before filling:

In[1]: M = Manifold("4_1")
In[2]: M.fundamental_group()
Out[2]: 
Generators:
   a,b
Relators:
   abbbaBAAB

After filling:

In[3]: M.dehn_fill((1, 2))
In[4]: M.fundamental_group()
Out[4]: 
Generators:
   a,b
Relators:
   abbbaBAAB
   abAbaBabAbaBAB

Edit: I remembered that Regina has the Weber-Seifert manifold as one of its examples. So, using the triangulation isomorphism signature, we can import this to SnapPy and find a presentation for its fundamental group. (We can also compute a presentation using Regina - but I prefer the notation for relations used by SnapPy).

In[5]: WS = Manifold("xvLvvvwMvQPPQQQQcehpjtqksntrtvoupwpsuwsvwcgacalvucahatbhapaggjgfk")
In[6]: WS.fundamental_group()
Out[6]: 
Generators:
   a,b,c,d
Relators:
   aDcbcdabdbbc
   aDabCADCbc
   abdbcdabdACBC
   aDBcdaDabbc
In[7]: WS.volume()
Out[7]: 11.1990647408
In[8]: WS.homology()
Out[8]: Z/5 + Z/5 + Z/5

The homology has rank three, so at least three generators are needed. It seems to be open (?) to compute the minimal number of generators.

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    $\begingroup$ Thanks Sam this is a very clear answer!! $\endgroup$ Dec 17, 2023 at 17:23
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    $\begingroup$ BTW, improvements in code and hardware allowed Ben Burton to verify that the Weber-Seifert space is non-Haken in 2016 running the code in 75 minutes. arxiv.org/abs/1812.11686 (see the end of Section 1). $\endgroup$
    – Ian Agol
    Dec 19, 2023 at 19:53

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