The unrestricted degree of commutativity conjecture for finitely generated groups

Let $G$ be a finitely generated group, and let $X$ be a finite generating set. Define the degree of commutativity with respect to $X$ by

$$\operatorname{dc}_X(G)=\limsup_{n\to\infty}\frac{|\{(u,v)\in(\mathbb{B}_X(n))^2:uv=vu\}|}{|\mathbb{B}_X(n)|^2}.$$

Here $\mathbb{B}_X(n)$ is the ball of radius $n$ in the word metric associated with $X$. Unrestricted degree of commutativity conjecture. Then: (i)

$$\operatorname{dc}_X(G)>0\quad\Longleftrightarrow\quad G\text{ is virtually abelian};$$

and (ii)

$$\operatorname{dc}_X(G)>\frac{5}{8}\quad\Longleftrightarrow\quad G\text{ is abelian}.$$

The statement extends the paper’s theorem from finitely generated residually finite groups of subexponential growth to arbitrary finitely generated groups. It is open without the growth hypothesis; the corresponding assertion is proved in the paper for the stated residually finite subexponential-growth class.