Infinite presentability of groups and condensation

Abstract

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.

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 4

References in corpus (3)

• Structure and finiteness properties of subdirect products of groups
• A sofic group away from amenable groups
• The space of subgroups of an abelian group

Cited by in corpus (4)

• Locally compact wreath products
• On the finiteness length of some soluble linear groups
• Counting submodules of a module over a noetherian commutative ring
• Densely related groups