Narrow Results

Here we characterize the residually finite groups G which are the fundamental groups of a finite graph of finitely generated torsion-free nilpotent groups. Namely we show that G is residually finite if and only if for each edge group of the graph of groups the two edge monomorphisms differ essentially by an isomorphism of certain subgroups of the Mal'cev completion of the corresponding vertex groups.

Recent work by T. Delzant and S. Hair shows that certain groups are unique product groups. In effect, they show that the groups have a locally invariant order, an idea introduced by D. Promislow in the early eighties. Having a locally invariant order implies the group is a unique product group, and a strict left (or right) ordering on a group is a locally invariant order. We study properties of the class of LIO groups, that is, groups having a locally invariant order. The main result gives conditions under which the fundamental group of a graph of LIO groups is LIO. In particular, the free product of two LIO groups is LIO. There is an analogous result for a graph of right orderable groups. We also study tree-free groups (those having a free action without inversions on a Λ-tree, for some ordered abelian group Λ). In particular, a detailed proof that tree-free groups are LIO is given. There is also a detailed proof of an observation made by Hair, that the fundamental group of a compact hyperbolic manifold is virtually LIO.

We give an example of a conjugacy separable, but not subgroup separable group. It is an adaptation of an example of Tavgen and the third author from [Closed orbits and finite approximability with respect to conjugacy of free amalgamated products, Math. Notes58 (1995) 1042–1048] to a theorem of Raptis [On finiteness conditions of certain graphs of groups, Internat. J. Algebra Comput.5 (1995) 719–724].

We show (using results of Wise and of Woodhouse) that a tubular group is always virtually special (meaning that it has a finite index subgroup embedding in a RAAG) if the underlying graph is a tree. We also adapt Gardam and Woodhouse’s argument on tubular groups which double cover 1-relator groups to show there exist 1-relator groups which are CAT(0) but not residually finite.

In this paper, we develop the theory of residually finite rationally $p$ (RFR$p$) groups, where $p$ is a prime. We first prove a series of results about the structure of finitely generated RFR$p$ groups (either for a single prime $p$, or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR$p$ groups, which we use to study the boundary manifolds of algebraic curves ${ℂ\mathbb{P}}^2$ and in ${ℂ}^2$. We show that boundary manifolds of a large class of curves in ${ℂ}^2$ (which includes all line arrangements) have RFR$p$ fundamental groups, whereas boundary manifolds of curves in ${ℂ\mathbb{P}}^2$ may fail to do so.

This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups, we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X, Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X, Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov's Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along π₁-injective boundary components with amenable fundamental group.