Twin-width VII: groups

TL;DR

This paper explores the twin-width parameter in groups, establishing finiteness for various classes, characterising it via pattern exclusion, and constructing examples of groups with infinite twin-width, thus advancing understanding of this graph invariant.

Contribution

It proves that many important classes of groups have finite twin-width, introduces uniform twin-width, and constructs examples of groups with infinite twin-width, addressing open questions.

Findings

Abelian, hyperbolic, ordered, solvable, and polynomial growth groups have finite twin-width.

Twin-width can be characterised by pattern exclusion in group actions.

Constructed a group with infinite twin-width, showing unbounded twin-width in some groups.

Abstract

Twin-width is a recently introduced graph parameter with applications in algorithmics, combinatorics, and finite model theory. For graphs of bounded degree, finiteness of twin-width is preserved by quasi-isometry. Thus, through Cayley graphs, it defines a group invariant. We prove that groups which are abelian, hyperbolic, ordered, solvable, or with polynomial growth, have finite twin-width. Twin-width can be characterised by excluding patterns in the self-action by product of the group elements. Based on this characterisation, we propose a strengthening called uniform twin-width, which is stable under constructions such as group extensions, direct products, and direct limits. The existence of finitely generated groups with infinite twin-width is not immediate. We construct one using a result of Osajda on embeddings of graphs into groups. This implies the existence of a class of finite graphs with unbounded twin-width but containing $2^{O(n)} \cdot n!$ graphs on vertex set $\{1,\dots,n\}$, settling a question asked in a previous work.