On some infinite demensional linear groups
by
Leonid Kurdachenko
Dnipropetrovsk University, Ukraine
Coauthors: Igor Subbotin, National University, USA

The theory of finite dimensional linear groups is one of the best developed in Algebra. Because of the rich interplay between geometrical and algebraic ideas associated with finite dimensional linear groups they have played an important role in Group Theory. If the dimension is infinite, a situation is totally different. The study of the subgroups of infinite dimensional linear groups in this case is impossible without some essential additional restrictions. The series of the brilliant constructions developed by A. Yu. Olshanskii and his students is a very valuable argument supported this statement. The circumstances here are similar to those, which appeared in the early period of the development of Infinite Group Theory. One of the fruitful approaches there consisted from the application of finiteness conditions. The theory of finitary linear groups has been successfully developed on this way. The subgroup G of GL(F,A) (where F is a field, A is a vector space over F) is called finitary if for each element g from G the (subspace) centralizer of g in A has finite codimension in A. We consider another finiteness condition sporadically mentioned in some works but did not systematically studied yet. A subgroup G of GL(F,A) is said to be the linear group with finite orbits if the set aG = { ag : g from G } is finite for every element a of A. Similarly, we can consider not only G – orbits of the elements, but G – orbits of the subspaces as well. A subgroup G of GL(F,A) is said to be the linear group with finite orbits of subspaces if the set { Bg : g from G } is finite for every subspace B of A. In this case the index of the normalizer of B in G is finite for every subspace B of A. In particular, if the dimension of A in F is finite, then a linear group with finite orbits of subspaces is almost diagonal, thus it is abelian – by – finite. As the first step here we consider some classes of soluble linear groups with finite bounded orbits of elements (respectively finite bounded orbits of subspaces).

Date received: February 6, 2006