Generic length functions on countable groups

Abstract

Let L(G) denote the space of integer-valued length functions on a countable group G endowed with the topology of pointwise convergence. Assuming that G does not satisfy any non-trivial mixed identity, we prove that a generic (in the Baire category sense) length function on G is a word length and the associated Cayley graph is isomorphic to a certain universal graph U independent of G. On the other hand, we show that every comeager subset of L(G) contains 2ℵ0 “asymptotically incomparable” length functions. A combination of these results yields 2ℵ0 pairwise non-equivalent regular representations G→Aut(U). We also prove that generic length functions are virtually indistinguishable from the model-theoretic point of view. Topological transitivity of the action of G on L(G) by conjugation plays a crucial role in the proof of the latter result.