Sigma theory and twisted conjugacy classes

Daciberg Gonçalves and Dessislava Hristova Kochloukova

Vol. 247 (2010), No. 2, 335–352

DOI: 10.2140/pjm.2010.247.335

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 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 F_n,0 and their finite direct products, H = Aut(G).

Keywords

Reidemeister class, Thompson group, Sigma theory, automorphism of groups, R_∞ property, limit group

Mathematical Subject Classification 2000

Primary: 20F65, 20J05, 55M20

Secondary: 20E45

Milestones

Received: 22 January 2009

Revised: 13 December 2009

Accepted: 17 December 2009

Published: 11 August 2010