Example of three dimensional atoroidal Poincaré duality group with some pathology

Question

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.

Answer 1

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.