One answer to your question comes from the paper *[The Weber-Seifert dodecahedral space is non-Haken][1]* 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][2]. We can use [SnapPy][3] 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][4] 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][5] (?) to compute the minimal number of generators. [1]: https://arxiv.org/abs/0909.4625 [2]: https://archive.org/embed/ThurstonTheGeometryAndTopologyOfThreeManifolds [3]: https://snappy.computop.org/ [4]: https://regina-normal.github.io/ [5]: https://mathoverflow.net/questions/146529/heegaard-genus-of-the-hyperbolic-dodecahedral-space-is-it-3-or-4