Groups With Arbitrarily Large Torsion [closed]

Question

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 =)

Answer 1

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).

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.)

A last example: Using this article, you can construct finitely presented wreath products containing $\bigoplus\limits_{i=1}^\infty F$ for some (non-trivial) finite group $F$.