Twisted conjugacy and separability
Sam Tertooy
TL;DR
This work investigates how twisted conjugacy separability interacts with subgroups, quotients, and finite extensions, and identifies when complete twisted conjugacy separability (CTCS) holds. It develops a nest-based framework and proves that CTCS is equivalent to residual finiteness with respect to nests (NRF) in general, with stronger equivalences for broader group classes. Specifically, for nilpotent-by-finite and polycyclic-by-nilpotent-by-finite groups, it establishes the equivalence $SRF \iff ERF \iff NRF \iff CTCS$, extending known results about residual finiteness and separability. The findings provide a unified perspective linking classical separability notions to twisted conjugacy phenomena, with implications for profinite methods and structural group theory; they also offer explicit CTCS constructions outside traditional finite-type classes. Overall, the paper clarifies when twisted conjugacy conditions are preserved under common group constructions and how nests govern these properties.
Abstract
A group $G$ is twisted conjugacy separable if for every automorphism $\varphi$, distinct $\varphi$-twisted conjugacy classes can be separated in a finite quotient. Likewise, $G$ is completely twisted conjugacy separable if for any group $H$ and any two homomorphisms $\varphi,ψ$ from $H$ to $G$, distinct $(\varphi,ψ)$-twisted conjugacy classes can be separated in a finite quotient. We study how these properties behave with respect to taking subgroups, quotients and finite extensions, and compare them to other notions of separability in groups. Finally, we show that for polycyclic-by-nilpotent-by-finite groups, being completely twisted conjugacy separable is equivalent to all quotients being residually finite.