Sigma theory and twisted conjugacy classes

Abstract

Using Sigma theory we show that for large classes of groups G there is a subgroup H of finite index in Aut(G) such that for ϕ ∈ H the Reidemeister number R(ϕ) is infinite.This includes all finitely generated nonpolycyclic groups G that fall into one of the following classes: nilpotent-by-abelian groups of type FP ∞ ; groups G/ G of finite Prüfer rank; groups G of type FP 2 without free nonabelian subgroups and with nonpolycyclic maximal metabelian quotient; some direct products of groups; or the pure symmetric automorphism group.Using a different argument we show that the result also holds for 1-ended nonabelian nonsurface limit groups.In some cases, such as with the generalized Thompson's groups F n,0 and their finite direct products, H = Aut(G).