Doron Puder

Measure Preserving Words are Primitive

A word w 2 Fk, the free group on k generators, is called primitive if it belongs to some basis of Fk. Associated with w and a finite group G is the word map w : G×...×G ! G defined on the direct product of k copies of G. We call w measure preserving if given uniform measure on G×...×G, the image of this word map induces uniform measure on G (for every finite group G). It is easy to see that every primitive word is measure preserving, and several authors have conjectured that the two properties are, in fact, equivalent. Here we prove this conjecture. The main ingredients of the proof include random coverings of Stallings graphs, algebraic extensions of free groups and Mobius inversions. Our methods yield the stronger result that a subgroup of Fk is measure preserving iff it is a free factor. As an interesting corollary of this result we resolve a question on the profinite topology of free groups and show that the primitive elements of Fk form a closed set in this topology.

Primitive words, free factors and measure preservation

Let Fk be the free group on k generators. A word w ∈ Fk is called primitive if it belongs to some basis of Fk. We investigate two criteria for primitivity, and consider more generally subgroups of Fk which are free factors.The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H ≤ J ≤ Fk 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 Fk is primitive.Again let w ∈ Fk and consider the word map w: G × … × G → 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 × … × 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 Fk form a closed set in the profinite topology of free groups. Our results provide a positive answer for F2.

Growth of primitive elements in free groups

Let pk;N be the number of primitive words of length N in the free group Fk, k 3. We show that the exponential growth rate of pk;N is 2k 3, answering a question from the list ‘Open problems in combinatorial group theory’ [Baumslag-Myasnikov-Shpilrain 02’]. Moreover, we show that a generic conjugacy class of primitive elements has a representative containing some letter exactly once. Our proof also works for giving the exact growth rate of the larger class of elements belonging to a proper free factor.

Stallings Graphs, Algebraic Extensions and Primitive Elements in F 2

This paper studies the free group of rank two from the point of view of Stallings core graphs. The first half of the paper examines primitive elements in this group, giving new and self-contained proofs for various known results about them. In particular, this includes the classification of bases of this group. The second half of the paper is devoted to constructing a counterexample to a conjecture by Miasnikov, Ventura and Weil, which seeks to characterize algebraic extensions in free groups in terms of Stallings graphs.