Subgroup separability in residually free groups

We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type $\mathrm{FP}_\infty$ are virtual retracts. We describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups.Comment: 8 pages, no figure

Finite and infinite quotients of discrete and indiscrete groups

These notes are devoted to lattices in products of trees and related topics. They provide an introduction to the construction, by M. Burger and S. Mozes, of examples of such lattices that are simple as abstract groups. Two features of that construction are emphasized: the relevance of non-discrete locally compact groups, and the two-step strategy in the proof of simplicity, addressing separately, and with completely different methods, the existence of finite and infinite quotients. A brief history of the quest for finitely generated and finitely presented infinite simple groups is also sketched. A comparison with Margulis' proof of Kneser's simplicity conjecture is discussed, and the relevance of the Classification of the Finite Simple Groups is pointed out. A final chapter is devoted to finite and infinite quotients of hyperbolic groups and their relation to the asymptotic properties of the finite simple groups. Numerous open problems are discussed along the way.Comment: Revised according to referee's report; definition of BMW-groups updated; more examples added in Section 4; new Proposition 5.1

Residual properties of free products

Let C be a class of groups. We give sufficient conditions ensuring that a free product of residually C groups is again residually C, and analogous conditions are given for locally embeddable into C groups. As a corollary, we obtain that the class of residually amenable groups and the one of LEA groups (or initially subamenable groups in the terminology of Gromov) are closed under taking free products. Moreover, we consider the pro-C topology and we characterize special HNN extensions and amalgamated free products that are residually C, where C is a suitable class of groups. In this way, we describe special HNN extensions and amalgamated free products that are residually amenable.Comment: 17 pages, no figures. revised and expanded versio

Residually free 3-manifolds

We classify those compact 3-manifolds with incompressible toral boundary whose fundamental groups are residually free. For example, if such a manifold $M$ is prime and orientable and the fundamental group of $M$ is non-trivial then $M \cong \Sigma\times S^1$, where $\Sigma$ is a surface.Comment: 19 pages, referee's comments incorporated, to appear in Algebraic & Geometric Topolog

Normalisers in Limit Groups

Let \G be a limit group, S\subset\G a subgroup, and $N$ the normaliser of $S$. If $H_1(S,\mathbb Q)$ has finite \Q-dimension, then $S$ is finitely generated and either $N/S$ is finite or $N$ is abelian. This result has applications to the study of subdirect products of limit groups.Comment: 10 pages, no figure

Volume gradients and homology in towers of residually-free groups

We study the asymptotic growth of homology groups and the cellular volume of classifying spaces as one passes to normal subgroups $G_n<G$ of increasing finite index in a fixed finitely generated group $G$, assuming $\bigcap_n G_n =1$. We focus in particular on finitely presented residually free groups, calculating their $\ell_2$ betti numbers, rank gradient and asymptotic deficiency. If $G$ is a limit group and $K$ is any field, then for all $j\ge 1$ the limit of $\dim H_j(G_n,K)/[G,G_n]$ as $n\to\infty$ exists and is zero except for $j=1$, where it equals $-\chi(G)$. We prove a homotopical version of this theorem in which the dimension of $\dim H_j(G_n,K)$ is replaced by the minimal number of $j$-cells in a $K(G_n,1)$; this includes a calculation of the rank gradient and the asymptotic deficiency of $G$. Both the homological and homotopical versions are special cases of general results about the fundamental groups of graphs of {\em{slow}} groups. We prove that if a residually free group $G$ is of type $\rm{FP}_m$ but not of type $\rm{FP}_{\infty}$, then there exists an exhausting filtration by normal subgroups of finite index $G_n$ so that $\lim_n \dim H_j (G_n, K) / [G : G_n] = 0 \hbox{for} j \leq m$. If $G$ is of type $\rm{FP}_{\infty}$, then the limit exists in all dimensions and we calculate it.Comment: Final accepted version. To appear in Math An