Densely related groups

TL;DR

This paper investigates densely related groups, which have relations at all scales, showing they are not simply connected at infinity, and explores their properties and examples within various classes of finitely generated groups.

Contribution

It introduces and studies the class of densely related groups, proving their invariance under quasi-isometry and providing examples like the Grigorchuk group.

Findings

Densely related groups are not simply connected at infinity.

The Grigorchuk group is densely related.

A law-satisfying, (infinite locally finite)-by-cyclic group is densely related.

Abstract

We study the class of densely related groups. These are finitely generated (or more generally, compactly generated locally compact) groups satisfying a strong negation of being finitely presented, in the sense that new relations appear at all scales. Here, new relations means relations that do not follow from relations of smaller size. Being densely related is a quasi-isometry invariant among finitely generated groups. We check that a densely related group has none of its asymptotic cones simply connected. In particular a lacunary hyperbolic group cannot be densely related. We prove that the Grigorchuk group is densely related. We also show that a finitely generated group that is (infinite locally finite)-by-cyclic and which satisfies a law must be densely related. Given a class $\mathcal{C}$ of finitely generated groups, we consider the following dichotomy: every group in $\mathcal{C}$ is either finitely presented or densely related. We show that this holds within the class of nilpotent-by-cyclic groups and the class of metabelian groups. In contrast, this dichotomy is no longer true for the class of $3$-step solvable groups.