Thompson's Group has two well known presentations:


$\langle x\_0,x\_1, ... $ | $ x\_k^{-1} x\_n x\_k = x\_{n+1}\forall k < n \rangle$


$\langle A,B $ | $ [AB^{-1}, A^{-1}BA], [AB^{-1}, A^{-2}BA^2] \rangle$


where $x\_0=A$ and $x\_n = A^{1-n}BA^{n-1}$


It is also known that every finite group exists as a subgroup of Thompson's Group. In particular, the Thompson Group has arbitrarily large torsion, as one can find any finite subgroup within it.


A simple construction like $G = \prod\_{n \in \mathbb{N}} G\_n$, where each $G\_n$ is a finite group yields another group with arbitrarily large torsion. 


What are good examples of other groups with arbitrarily large torsion? Can they be finitely generated or presented? 


Thank you =)



Groups With Arbitrarily Large Torsion
There exist many variations of Thompson's group $V$ which are finitely presented (or even of type $F\_\infty$) and which contain all the finite groups. For instance:


* Higman-Thompson groups $V\_{n,r}$ (same definition as $V$ but with $n$-adic subdivisions of $r$ disjoint copies of the Cantor set).
* Higher dimensional Thompson group $nV$.
* Rearrangement groups of fractals (as introduced [here](https://arxiv.org/abs/1510.03133)).


Another way to produce examples is to consider quasi-automorphism groups of trees. Given a tree $T$, a quasi-automorphism is a bijection $T^{(0)} \to T^{(0)}$ which preserves adjacency between all but finitely many pairs of vertices. 


* If $T$ is $n$ infinite rays gluing along a common vertex, then the corresponding group is (a finite extension of) the $n$th Houghton's group, which finitely presented as soon as $n \geq 3$.
* If $T$ is an infinite binary tree, you get the group $QV$. (See for instance [here](https://arxiv.org/abs/1705.01628).)


A last example: Using this [article](https://arxiv.org/abs/math/0509090), you can construct finitely presented wreath products containing $\bigoplus\limits\_{i=1}^\infty F$ for some (non-trivial) finite group $F$. 


