Department of Mathematics

 Let G be a countable group. We prove that if G does not satisfy any non-trivial mixed identity, then a generic length function on G is a word length and the associated Cayley graph is isomorphic to a certain unbounded universal graph, which is independent of G. In particular, every such a group G admits a cobounded action on a metric space with unbounded orbits (whether every infinite countable group has this property is a well-known open problem).