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 ’ 2 H the Reidemeister number R.’/ is infinite. This includes all finitely generated nonpolycyclic groups G that fall into one of the following classes: nilpotent-by-abelian groups of type FP1; groups G= G 00 of finite Prufer rank; groups G of type FP2 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 Fn;0 and their finite direct products, H D Aut.G/.