2025/10/21 by Laura Ciobanu, Gemma Crowe, Ciobanu, Laura +4
Mathematics · #Geometric and Algebraic Topology #Finite Group Theory Research #Advanced Operator Algebra Research
paper · pdf · doi:10.48550/arxiv.2510.18980
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 tdcX(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 → ∞, where the twisting is done with respect to an endomorphism φ of G. We compute tdcX(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 tcrX(φ, G), which is the limit at infinity of the quotient of the twisted conjugacy and standard growth functions. We compute tcrX(φ, G) for virtually abelian groups, and give examples of groups of exponential growth such that tcrX(φ, G) = 0.