Algebraic characterization of dendricity

TL;DR

This paper establishes an algebraic characterization of dendric shift spaces, showing they are precisely those where return words form a basis of the free group, thus connecting combinatorial and algebraic properties.

Contribution

It proves the converse of the Return Theorem for dendric shift spaces, providing an algebraic criterion involving free group bases.

Findings

Dendric shift spaces are characterized by their return words forming a free group basis.

The paper proves the equivalence between dendricity and an algebraic property of return words.

It offers a new algebraic perspective on the structure of dendric shift spaces.

Abstract

Dendric shift spaces simultaneously generalize codings of regular interval exchanges and episturmian shift spaces, themselves both generalizations of Sturmian words. One of the key properties enforced by dendricity is the Return Theorem. In this paper, we prove its converse, providing the following natural algebraic perspective on dendricity: A minimal shift space is dendric if and only if every set of return words is a basis of the free group over the alphabet.