Cayley graphs and automatic sequences

TL;DR

This paper explores automatic sequences generated by Cayley graphs of finite groups, characterizing special classes like homogeneous and self-similar sequences, and establishing connections with algebraic and geometric objects such as dessins d'enfants.

Contribution

It introduces a characterization of 2-automatic sequences via homogeneity and self-similarity, and shows how to recover the underlying group and automaton from the sequence.

Findings

Self-similar 2-automatic sequences correspond to dessins d'enfants.

The automaton and group can be uniquely recovered from the sequence.

A rational function can be associated with each automatic sequence, computable from the Cayley graph.

Abstract

We study those automatic sequences which are produced by an automaton whose underlying graph is the Cayley graph of a finite group. For $2$-automatic sequences, we find a characterization in terms of what we call homogeneity, and among homogeneous sequences, we single out those enjoying what we call self-similarity. It turns out that self-similar $2$-automatic sequences (viewed up to a permutation of their alphabet) are in bijection with many interesting objects, for example dessins d'enfants (covers of the Riemann sphere with three points removed). For any $p$ we show that, in the case of an automatic sequence produced "by a Cayley graph", the group and indeed the automaton can be recovered canonically from the sequence. Further, we show that a rational fraction may be associated to any automatic sequence. To compute this fraction explicitly, knowledge of a certain graph is required. We prove that for the sequences studied in the first part, the graph is simply the Cayley graph that we start from, and so calculations are possible. We give applications to the study of the frequencies of letters.