Infinite cyclic verbal subgroups of relatively free groups

We prove that there exist a relatively free group and a word in two variables such that the verbal subgroup of defined by is an infinite cyclic group whereas has only one nontrivial value in .

     

On the definability of verbal subgroups

We show that if G is a group of finite Morley rank, then the verbal subgroup <w(G)> is of finite width, where w is a concise word. As a byproduct, we show that if G is any abelian-by-finite group, then G ⁿ =<x ⁿ (G)> is definable. Received: 15 March 1996 / Published online: 18 July 2001     

On the intersection of subgroups of a free group

Summary The Hanna Neumann Conjecture says that the intersection of subgroups of rankn+1 andm+1 of a free group has rank at mostnm+1. This paper proves the conjecture for the casem=1. (See Theorem 1.) Our methods imply that the strengthened Hanna Neumann Conjecture is also true in this case (Theorem 2).Oblatum 31-V-1991 & 8-X-1991     

On free subgroups of units of rings

We prove that if for elements of a ring of characteristic zero and is not nilpotent, then there exists such that the group generated by and is free nonabelian. This is used to prove that a noncommutative positive-definite algebra with involution over an uncountable field contains a free nonabelian subsemigroup.

     

Invariant additive subgroups of a certain kind

LetR be a prime ring with a nonzero nil right ideal, and letM be the union of all nil right ideals ofR. IfW is an additive subgroup ofR which is invariant under conjugation by all special automorphisms 1+x forx ∈M, then eitherW is central orW contains a noncommutative Lie ideal ofR. Assuming thatW is invariant under only those 1+x forx ∈M andx ²=0, the same conclusion holds if the extended centroid ofR is not GF(2).     

On density properties of certain subgroups with boundedness conditions

LetG be a locally compact group acting on a topological space. Here we define some boundedness conditions for the action. For a nondiscrete locally compact vector spaceV andgG L (V), layering structures forV and the projective spaceP (V) ofV are obtained. From the layering structures, we derive then density properties of subgroups ofG with boundedness conditions. We generalize the Borel density theorem and Prasad's theorem on automorphisms of algebraic semi-simple groups. Some new results onp-adic groups are added.Partially supported by N. S. F. Grant 7702168.