Admitting a coarse embedding is not preserved under group extensions

TL;DR

The paper constructs a finitely generated group that is an extension of two groups coarsely embeddable into Hilbert space but does not coarsely embed itself, challenging assumptions about preservation under extensions.

Contribution

It provides the first example of a finitely generated group that does not coarsely embed into Hilbert space despite being an extension of such groups.

Findings

Constructed a finitely generated non-embeddable group from embeddable groups

Introduced a new infinite monster group with unique embedding properties

Showed non-preservation of coarse embeddability under group extensions

Abstract

We construct a finitely generated group which is an extension of two finitely generated groups coarsely embeddable into Hilbert space but which itself does not coarsely embed into Hilbert space. Our construction also provides a new infinite monster group: the first example of a finitely generated group that does not coarsely embed into Hilbert space and yet does not contain a weakly embedded expander.