  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.
