Twisted conjugacy and commensurability invariance

A group 𝐺 is said to have property R ∞ R_{\infty} if, for every automorphism φ ∈ Aut ⁢ ( G ) \varphi\in\mathrm{Aut}(G) , the cardinality of the set of 𝜑-twisted conjugacy classes is infinite. Many classes of groups are known to have this property. However, very few examples are known for which R ∞ R_{\infty} is geometric, i.e., if 𝐺 has property R ∞ R_{\infty} , then any group quasi-isometric to 𝐺 also has property R ∞ R_{\infty} . In this paper, we give examples of groups and conditions under which R ∞ R_{\infty} is preserved under commensurability. The main tool is to employ the Bieri–Neumann–Strebel invariant.

Twisted conjugacy classes in twisted Chevalley groups

Let [Formula: see text] be a group and [Formula: see text] be an automorphism of [Formula: see text]. Two elements [Formula: see text] are said to be [Formula: see text]-twisted conjugate if [Formula: see text] for some [Formula: see text]. We say that a group [Formula: see text] has the [Formula: see text]-property if the number of [Formula: see text]-twisted conjugacy classes is infinite for every automorphism [Formula: see text] of [Formula: see text]. In this paper, we prove that twisted Chevalley groups over a field [Formula: see text] of characteristic zero have the [Formula: see text]-property as well as the [Formula: see text]-property if [Formula: see text] has finite transcendence degree over [Formula: see text] or [Formula: see text] is periodic.

The 𝑅∞-property for nilpotent quotients of Baumslag–Solitar groups

A group G has the {R_{\infty}}-property if the number {R(\varphi)} of twisted conjugacy classes is infinite for any automorphism φ of G. For such a group G, the {R_{\infty}}-nilpotency degree is the least integer c such that {G/\gamma_{c+1}(G)} still has the {R_{\infty}}-property. In this paper, we determine the {R_{\infty}}-nilpotency degree of all Baumslag–Solitar groups.

Twisted conjugacy classes in unitriangular groups

Let R be an integral domain of characteristic zero. In this note we study the Reidemeister spectrum of the group {{\rm UT}_{n}(R)} of unitriangular matrices over R. We prove that if {R^{+}} is finitely generated and {n>2|R^{*}|} , then {{\rm UT}_{n}(R)} possesses the {R_{\infty}} -property, i.e. the Reidemeister spectrum of {{\rm UT}_{n}(R)} contains only {\infty} , however, if {n\leq|R^{*}|} , then the Reidemeister spectrum of {{\rm UT}_{n}(R)} has nonempty intersection with {\mathbb{N}} . If R is a field and {n\geq 3} , then we prove that the Reidemeister spectrum of {{\rm UT}_{n}(R)} coincides with {\{1,\infty\}} , i.e. in this case {{\rm UT}_{n}(R)} does not possess the {R_{\infty}} -property.