Twisted conjugacy classes and the R-infinity property for linear groups

The research project is devoted to the investigation of twisted conjugacy classes in linear groups from different points of view:

1) The R-infinity property. The question on describing groups which have an infinite number of twisted conjugacy classes was formulated by A.Felshtyn and R.Hill in 1991. This question is important for applications in different fields of mathematics, aspecially in algebra. There are connections between twisted conjugacy classes and isogredience classes of groups, representations of linear groups, invariants in Nielsen-Reidemeister fixed point theory. During the last years one obtained a lot of results on the R infinity property. Some results on this property in linear groups were obtained by the author (myself) and in the research project I am going to close this problem for classical linear groups answering the questions about the R-infinity property in linear groups over fields.

2) Properties of groups. Properties of twisted conjugacy classes reflect properties of group. For example Bardakov, Nasybullov (myself) and Neshchadim proved that every finite group for which the twisted conjugacy class of the unit element is a subgroup for every automorphism of this group is nilpotent. The question about an arbitrary group (not necessary finite) with condition above is formulated in the famous Kourovka notebook. I am going to study this problem for arbitrary groups and some other problems which connect properties of groups and twisted conjugacy classes.