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 $\mathrm{tdc}_X(φ, G)$ 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 $n \rightarrow \infty$, where the twisting is done with respect to an endomorphism $φ$ of $G$. We compute $\mathrm{tdc}_X(φ, G)$ for several classes of groups, including virtually abelian groups, groups of subexponential growth, and free groups. We then study the twisted conjugacy ratio $\mathrm{tcr}_{X}(φ, G)$, which is the limit at infinity of the quotient of the twisted conjugacy and standard growth functions. We compute $\mathrm{tcr}_{X}(φ, G)$ for virtually abelian groups, and give examples of groups of exponential growth such that $\mathrm{tcr}_{X}(φ, G) = 0$.

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