Suffix-connected languages

TL;DR

This paper introduces suffix-connectedness, a new property of languages, and explores its implications for the structure of subgroups generated by return sets, including methods for explicit computation and examples.

Contribution

It defines suffix-connectedness and links it to subgroup conjugacy classes, providing computational methods and examples that extend prior work on recurrent languages.

Findings

Groups generated by return sets lie in a single conjugacy class

Rank of subgroups depends on extension graph components

Explicit computation of conjugacy class representatives

Abstract

Inspired by a series of papers initiated in 2015 by Berth\'e et al., we introduce a new condition called suffix-connectedness. We show that the groups generated by the return sets of a uniformly recurrent suffix-connected language lie in a single conjugacy class of subgroups of the free group. Moreover, the rank of the subgroups in this conjugacy class only depends on the number of connected components in the extension graph of the empty word. We also show how to explicitly compute a representative of this conjugacy class using the first order Rauzy graph. Finally, we provide an example of suffix-connected, uniformly recurrent language that contains infinitely many disconnected words.