No abstract

Following Serre's original description of groups having the fixed point property for actions on trees, Bass has introduced the notion of a group of type FA'. Groups of type FA' can not be nontrivial free products with amalgamation. We show that a locally compact (hausdorff) topological group with a compact set of connected components is of type FA'. Furthermore, any locally compact group which is a nontrivial free product with amalgamation has an open amalgamated subgroup. l A group G is called an amalgam if it is a free product with amalgamation of subgroups A and B along C, i.e., G = A*B, so that CΦ A,CΦB.

Integral sets of finite groups are discussed and related to the integral Cayley graphs. The Boolean algebra of integral sets are determined for dihedral group and finite abelian groups. We characterize the finite abelian groups as those finite groups where the Boolean algebra generated by  integral sets equals  the Boolean algebra generated by its subgroups.

Following Lyndon's axiomatic treatment of Nielsen's cancellation arguments for free products [5], I. M. Chiswell showed the equivalence of the Bass-Serre theory of group actions (without inversions) on a tree and integer-valued length functions on a group [2]. In the process Chiswell defined, for a group with a real-valued length function, a contractible metric space X on which there is an action of the group. For integer-valued length functions, X\s a tree in the ordinary simplicial sense. The results of our first section show that X, the metric completion of X, has the following treelike properties:(i) any two points of X are the endpoints of the image of an isometric embedding of a closed interval s u,

Our main result provides necessary and sufficient conditions for a finitelygenerated subgroup of GL n (C), n > 0, to have finite virtual cohomological dimension. A group has finite virtual cohomological dimension (VCD) if it has a subgroup of finite index which has finite cohomological dimension; this dimension is, in fact, the same for all torsion-free subgroups of finite index. It is, of course, necessary for a group r with VCD(T) < °° to have torsion-free subgroups of finite index; this is guaranteed in the case of finitely-generated linear groups by a well-known result of Selberg which extends ideas of Minkowski.A subgroup of GL n (C) is called unipotent if it is contained in a conjugate of the group of upper triangular matrices with all diagonal entries equal to one. Any unipotent subgroup is nilpotent; hence, a finitely-generated unipotent subgroup is poly cyclic and torsion-free. It is well known that a poly cyclic group has finite cohomological dimension if and only if it is torsion-free; moreover, the cohomological dimension is the same as the Hirsch rank. For a solvable group T with solvable series, 1 = T n < F n _ l < • • • < I\ = T, the Hirsch rank, h(T) = Sj^Tj 1 dimQ(ryr /+1 <8> Q), is independent of the choice of solvable series; thus, for a polycyclic group r, h(T) is the number of infinite factors in a normal series with cyclic quotients. We announce our main result. THEOREM. Let A be a finitely-generated integral domain of characteristic zero. A group T C GL n (A), n > 0, has finite VCD if and only if there is a finite upper bound on the Hirsch ranks of its finitely-generated unipotent subgroups.We obtain easily the following curious corollary. COROLLARY 1. Every finitely-generated subgroup of the unitary group U n (C), n > 0, has finite virtual cohomological dimension.