Infinite presentability of groups and condensation

We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor-Bendixson rank 1, and we prove that every infinitely presented metabelian group is a condensation group.Comment: 32 pages, no figure. 1->2 Major changes (the 13-page first version, authored by Y.C. and L.G., was entitled "On infinitely presented soluble groups") 2->3 some changes including cuts in Section

Invariable generation of prosoluble groups

A group G is invariably generated by a subset S of G if G = \u3008sg(s) | s 08 S\u3009 for each choice of g(s) 08 G, s 08 S. Answering two questions posed by Kantor, Lubotzky and Shalev in [8], we prove that the free prosoluble group of rank d 65 2 cannot be invariably generated by a finite set of elements, while the free solvable profinite group of rank d and derived length l is invariably generated by precisely l(d 12 1) + 1 elements. \ua9 2016, Hebrew University of Jerusalem

Quasi-isometric diversity of marked groups

We use basic tools of descriptive set theory to prove that a closed set $\mathcal S$ of marked groups has $2^{\aleph_0}$ quasi-isometry classes provided every non-empty open subset of $\mathcal S$ contains at least two non-quasi-isometric groups. It follows that every perfect set of marked groups having a dense subset of finitely presented groups contains $2^{\aleph_0}$ quasi-isometry classes. These results account for most known constructions of continuous families of non-quasi-isometric finitely generated groups. They can also be used to prove the existence of $2^{\aleph_0}$ quasi-isometry classes of finitely generated groups having interesting algebraic, geometric, or model-theoretic properties.Comment: Minor corrections. To appear in the Journal of Topolog