For $n\geq 2$, $P\mathbb{R}^n$ is a simple example of finite polyhedron with finitely generated simple fundamental group. I was wondering if someone could give me an example of a finite polyhedron with infinite finitely generated simple fundamental group. Thanks in advance.


Here by a simple group I mean in the group theoretical sense.



Examples of finite polyhedra with finitely generated simple fundamental group
As suggested in the comments, what you are asking for is essentially the presentation complex of a finitely presented, infinite, simple group. Thus it suffices to exhibit a presentation for such a group. Some are known, but not many.


Probably the easiest examples are Thompson's groups $T$ and $V$. Google gives me a link to an explicit finite presentation for $T$ in §11 of some [notes](http://www.macs.hw.ac.uk/%7Eaal7/docs/Introduction%20to%20Thompson%27s%20Groups%20F,%20T%20and%20V.pdf) of Levine, based on the classic notes of Cannon, Floyd and Parry.


Even more remarkable examples were constructed by Burger and Mozes. Their examples are CAT(0) amalgams of free groups, and in particular their presentation complex is aspherical. [This survey](https://arxiv.org/abs/1709.05949) of Caprace is a good place to start learning about these. It looks like the smallest known example is an amalgam of free groups of the form $F\_7\*\_{F\_{49}}F\_7$ (where the subscripts indicate the ranks of the free groups). [UPDATE: Carl-Fredrik Nyberg Brodda points out in comments that there is now an example of the form $F\_3\*\_{F\_{11}}F\_3$.]


Finally, if you would be satisfied with a group without non-trivial *finite* quotients, then *Higman's group*


$\langle a,b,c,d\mid bab^{-1}a^{-2}, cbc^{-2}b^{-2}, dcd^{-1}c^{-2},ada^{-1}d^{-2} \rangle$


provides a fairly digestible example.


