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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.

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    $\begingroup$ You need the group to be finitely presented. Once you have such a group, take a presentation complex associated to some finite presentation. $\endgroup$
    – Tyrone
    Commented Aug 18, 2022 at 14:14
  • $\begingroup$ @Tyrone Thanks very much for the comment. You mean we use the fact that if $\mathcal{P}$ is a finite presentation of some group $G$, then there is a finite 2-dimensional CW-complex $K(\mathcal{P})$ with a single vertex in which the 1-cells correspond to the generators of $\mathcal{P}$ and the attaching maps of the 2-cells are given by the relations of $P$. Is that right? $\endgroup$
    – M.Ramana
    Commented Aug 18, 2022 at 15:48
  • $\begingroup$ Yes. See, for instance, pg.207 of Ferrario and Piccinini's Simplicial Structures in Topology for some details. $\endgroup$
    – Tyrone
    Commented Aug 18, 2022 at 20:38

2 Answers 2

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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 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 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.

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    $\begingroup$ Does the terminology "almost simple" for group with no nontrivial finite quotient really exist? Higman's group is so far from being simple... $\endgroup$
    – YCor
    Commented Aug 20, 2022 at 9:13
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    $\begingroup$ Caprace & Radu recently found a “smaller” example of the form $F_5 \ast_{F_{25}} F_5$ and even $F_3 \ast_{F_{11}} F_3$. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Aug 20, 2022 at 11:13
  • $\begingroup$ Thank you so much for your nice answer. $\endgroup$
    – M.Ramana
    Commented Aug 20, 2022 at 12:33
  • $\begingroup$ @YCor: You're right, I had misremembered. "Almost simple" means something different. (en.wikipedia.org/wiki/Almost_simple_group) In a 1993 paper, Bhattacharjee used the term "nearly simple" for a slightly stronger notion. I suspect that Higman's group may be "nearly simple" in Bhattacharjee's sense, but will edit for now. $\endgroup$
    – HJRW
    Commented Aug 20, 2022 at 22:48
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The tetrahemihexahedron’s fundamental group is isomorphic to $C_2$.

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    $\begingroup$ The body of the question specifies it asks for infinite groups. $\endgroup$
    – Wojowu
    Commented Aug 19, 2022 at 22:18

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