Generic length functions on countable groups

TL;DR

This paper studies the typical properties of length functions on countable groups, showing that generically they are word lengths with universal Cayley graphs, and explores the diversity and model-theoretic indistinguishability of these functions.

Contribution

It proves that a generic length function on a countable group is a word length with a universal Cayley graph, and demonstrates the abundance of non-equivalent regular representations.

Findings

Generic length functions are word lengths with universal Cayley graphs.

There are continuum many asymptotically incomparable length functions in each comeager set.

There exist continuum many non-equivalent regular representations of the group.

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.