Example of unstable group

Question

What are examples of unstable groups in a purely group theoretic language? It is easy with a ring theoretic language (e.g. Given $(\mathbb{Q}, +, .)$ we can define $a<b$ using a formula which allows us to prove that Th$(\mathbb{Q}, +, .)$ is unstable).

Answer 1

For $G$ a group and $g, h\in G$, write $g\le_c h$ if $$\forall x\in G, gx=xg\implies hx=xh.$$ That is, $g\le_ch$ if $g$ commutes with "fewer" elements than $h$ does.

(For example, if $g\in Z(G)$ then $g\le_ch$ for every $h\in G$.)

I believe that we can construct groups whose associated $\le_c$-preorder has infinite chains; such a group will be unstable.

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Another example: consider the group $G$ on generators $g_i$ ($i\in\mathbb{N}$), and relations $g_ig_j^2g_i^{-3}$ for $i<j$. Unless I miss something, this group is unstable: the set of standard generators is totally ordered by the formula "$xy^2x^{-3}=e$".

Answer 2

Here is another example of an unstable group: The braid group with infinitely many generators. The proof on this page shows that this group is not NIP $\implies$ unstable.