Twisted commutativity and conjugacy ratio in groups

In this paper we introduce and study the degree of twisted commutativity and the twisted conjugacy ratio of a finitely generated group $G$. The degree of twisted commutativity ${\mathrm{t}\mathrm{d}\mathrm{c}}_X(\phi ,G)$ generalises the degree of commutativity of $G$, by measuring the density of pairs of elements with trivial twisted commutators in the ball of radius $n$ of $G$, as $n\to \mathrm{\infty }$, where the twisting is done with respect to an endomorphism $\phi$ of $G$. We compute ${\mathrm{t}\mathrm{d}\mathrm{c}}_X(\phi ,G)$ for several classes of groups, including virtually abelian groups, groups of subexponential growth, and free groups. We then study the twisted conjugacy ratio ${\mathrm{t}\mathrm{c}\mathrm{r}}_X(\phi ,G)$, which is the limit at infinity of the quotient of the twisted conjugacy and standard growth functions. We compute ${\mathrm{t}\mathrm{c}\mathrm{r}}_X(\phi ,G)$ for virtually abelian groups, and give examples of groups of exponential growth such that ${\mathrm{t}\mathrm{c}\mathrm{r}}_X(\phi ,G)=0$.