Session 35. Topological fixed point theory and related topics

Twisted conjugacy classes in residually finite groups

Evgenij Troitsky, Moscow State University, Russia
The talk is based on the joint works with A. Fel'shtyn [1], [2], [3]
In this talk we will discuss the following statements:
  1. the number of twisted conjugacy classes (Reidemeister number) of an automorphism \(\phi\) of a finitely generated residually finite group is equal (if it is finite) to the number of finite dimensional irreducible unitary representations being invariant for the dual of \(\phi\) ;
  2. any finitely generated residually finite non-amenable group has the \(R_\infty\) property (i.e. any automorphism has infinitely many twisted conjugacy classes). This gives a lot of new examples and covers many known classes of such groups;
Some generalizations and related examples will be discussed, in particular, examples for non-finitely generated groups. Also we plan to discuss the state of the following our, two year old

Conjecture: a finitely generated, residually finite, non-\(R_{\infty}\)-group is solvable by-finite.

References
  1. A. Fel'shtyn and E. Troitsky, Twisted Burnside-Frobenius theory for discrete groups, J. reine Angew. Math., 613 (2007), 193-210.
  2. A. Fel'shtyn and E. Troitsky, Geometry of Reidemeister classes and twisted Burnside theorem, J. K-Theory, 2 (2008), 463-506.
  3. A. Fel’shtyn and E. Troitsky, Twisted conjugacy classes in residually finite groups, arXiv:1204.3175v2, 2012.
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