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 $2^{\aleph_0}$ asymptotically incomparable length functions. A combination of these results yields $2^{\aleph_0}$ pairwise non-equivalent regular representations $G\to Aut(U)$. 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.