Measure preserving words are primitive

Abstract

We establish new characterizations of primitive elements and free factors in free groups, which are based on the distributions they induce on finite groups. For every finite group G G , a word w w in the free group on k k generators induces a word map from G k G^{k} to G G . We say that w w is measure preserving with respect to G G if given uniform distribution on G k G^{k} , the image of this word map distributes uniformly on G G . It is easy to see that primitive words (words which belong to some basis of the free group) are measure preserving w.r.t. all finite groups, 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 Möbius inversions. Our methods yield the stronger result that a subgroup of F k \mathbf {F}_{k} is measure preserving if and only if 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 F k \mathbf {F}_{k} form a closed set in this topology.

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

“…In this section we address a subtler problem trying to distinguish words for which the fibres of the corresponding word map are of the same size, at least approximately. First note that for certain words all fibres of the word map : G → G are exactly of the same size for any finite group G. According to recent results of [104,105], this holds only for primitive words. Primitive words are asymptotically very rare (exponentially negligible, in the terminology of [65]): if we count them among all words of fixed length, their proportion tends to 0 exponentially fast (see, e.g., [95]).…”

“…Primitive words are asymptotically very rare (exponentially negligible, in the terminology of [65]): if we count them among all words of fixed length, their proportion tends to 0 exponentially fast (see, e.g., [95]). Another viewpoint at the set of primitive elements of F is that this set is closed in the profinite topology of F [105].…”

Growth of primitive elements in free groups

“…When G is finite and G k is given the uniform measure (or more generally when G is compact and G k is given the Haar measure), the push forward by w of this measure results in a new measure on G, which we denote by G w . It is an easy observation that if w 1 and w 2 are in the same Aut F k -orbit of F k , then they induce the same measure on every finite or compact group, namely G w1 = G w2 (see [9,Observation 1.2]). In particular, if w is primitive, then G w = G x1 , which is clearly the uniform (Haar) measure on G.…”

“…This conjecture is still wide open. However, the special case concerning primitives was settled in [9]. It is shown there that if G w is uniform for every finite group G, then w is primitive.…”

Системы Элементов, Сохраняющие Меру На Многообразиях Групп

“…Так, например, из [9] следует, что элемент v группы F примитивен тогда и только тогда, когда он унимодулярен. В [3] доказано, что элемент v ∈ F сохраняет меру (на многооб-разии всех групп) тогда и только тогда, когда он примитивен. Таким образом, для многообразия всех групп свойства элемента v ∈ F быть примитивным, сохранять меру и быть унимодулярным совпадают.…”