Infinite presentability of groups and condensation

TL;DR

This paper explores classes of infinitely presented groups that are condensation points in the space of marked groups, introducing a new class called infinitely independently presentable groups and providing criteria and examples.

Contribution

It introduces the class of infinitely independently presentable groups and establishes criteria for identifying such groups, expanding understanding of condensation points.

Findings

Infinitely independently presentable groups are condensation points.

Examples of finitely generated groups with no minimal presentation are constructed.

Every infinitely presented metabelian group is a condensation group.

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.