Linear Groups and Group Rings

This paper consists of two parts. The first is concerned with free products in linear groups and uses the usual “ping pong” lemma and attractors to prove the results. What is new here is that we allow certain subspaces of V associated with the semisimple and generalized transvection operators to have dimensions larger than 1. The second part is concerned with applications of this machinery to integral groups rings Z[G] of finite groups. We show, for example, that if G is nonabelian of order prime to 6, then Z[G] contains two Bass cyclic units that generate a nonabelian free group. 1. Linear operators and attractors Let F be a field and let | | : F → R = {r ∈ R : r ≥ 0} be an absolute value defined on F . Here R is the field of real numbers and, by definition, we have |ab| = |a|·|b| |a+ b| ≤ |a|+ |b| |a| = 0 ⇐⇒ a = 0 for all a, b ∈ F . In particular, |1| = 1 = | − 1|. Indeed, |a| = 1 if a is any root of unity in F . See [J, Chapter 9] or [B, Chapter VI] for additional basic properties. If we define δ : F × F → R by δ(a, b) = |a − b|, then F clearly becomes a metric space using δ as a metric. We assume throughout that F is locally compact in this topology, so that each a ∈ F has a neighborhood with compact closure. As a consequence of the product formula |ab| = |a|·|b|, it follows that every closed, bounded subset of F is compact. Furthermore, | | : F → R is a continuous function and F is complete, in that every Cauchy sequence has a limit. If | | is archimedean, then Ostrowski’s Theorem [J, page 552] implies that F = R or C, the field of complex numbers, and that | | is the ordinary absolute value defined on C. In particular, | | is the identity function on |F | = R ⊆ F , where |F | is the set of absolute values taken on by elements of F . Thus, if 0 6= a ∈ F , then a/|a| is an element of F having absolute value 1. On the other hand, if | | is non-archimedean, then we know that |a+ b| ≤ max{|a|, |b|} for all a, b ∈ F . Indeed, if |a| 6 = |b|, then |a + b| = max{|a|, |b|}. In this situation, O = {a ∈ F : |a| ≤ 1} is a subring of F , the valuation ring associated with | |, and O has the unique maximal ideal m = {a ∈ F : |a| < 1}. Since F is locally compact, O is compact and hence the residue class field O/m is finite, since the 2000 Mathematics Subject Classification. 16S34, 16U60, 20E06, 20H20. The first author’s reasearch was supported in part by CNPq grant 303.756/82-5 and FapespBrazil, Proj. Tematico 00/07.291-0. The second author’s research was supported in part by NSA