The $R_\infty$ property for nilpotent quotients of generalized solvable Baumslag-Solitar groups

Abstract

We say a group G has property R ∞ if the number R(ϕ) of twisted conjugacy classes is infinite for every automorphism ϕ of G. For such groups, the R ∞ -nilpotency degree is the least integer c such that G/γ c+1 (G) has property R ∞ . In this work, we compute the R ∞ -nilpotency degree of all Generalized Solvable Baumslag-Solitar groups Γ n . Moreover, we compute the lower central series of Γ n , write the nilpotent quotients Γ n,c = Γ n /γ c+1 (Γ n ) as semidirect products of finitely generated abelian groups and classify which integer invertible matrices can be extended to automorphisms of Γ n,c .

Valuations, completions, and hyperbolic actions of metabelian groups

“…It is thus a natural, but typically very difficult, problem to attempt to classify all of the hyperbolic actions of a group. Classifying the hyperbolic actions of a group can be used to compute Bieri-Neumann-Strebel invariants (see Section 1.2 and [2,20]) and thereby may also be useful for studying related notions such as property 𝑅 ∞ (see [37]).…”