Primitive Words, Free Factors and Measure Preservation

TL;DR

This paper explores criteria for primitive words and free factors in free groups, introduces graph-theoretic and measure-preserving criteria, and proves their equivalence for rank two groups, advancing understanding of free group structure.

Contribution

It develops a graph-based procedure to identify free factors and primitive elements, and proves the equivalence of primitivity and measure preservation for F_2.

Findings

A simple graph-theoretic procedure to determine free factors.

Proof that primitivity implies measure preservation for F_2.

Primitive elements form a closed set in the profinite topology for F_2.

Abstract

Let F_k be the free group on k generators. A word w \in F_k is called primitive if it belongs to some basis of F_k. We investigate two criteria for primitivity, and consider more generally, subgroups of F_k which are free factors. The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H \le J \le F_k we present a simple procedure to determine whether H is a free factor of J. This yields, in particular, a procedure to determine whether a given element in F_k is primitive. Again let w \in F_k and consider the word map w:G x G x ... x G \to G (from the direct product of k copies of G to G), where G is an arbitrary finite group. We call w measure preserving if given uniform measure on G x G x ... x G, w induces uniform measure on G (for every finite G). This is the second criterion we investigate: it is not hard to see that primitivity implies measure preservation and it was conjectured that the two properties are equivalent. Our combinatorial approach to primitivity allows us to make progress on this problem and in particular prove the conjecture for k=2. It was asked whether the primitive elements of F_k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F_2.