Primitive words, free factors and measure preservation

Abstract

Let F k be the free group on k generators. A word w ∈ 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 ≤ J ≤ 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 ∈ F k 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 F k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F 2 .

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]).…”

Growth of primitive elements in free groups

“…Stallings generalized the algorithm in order to detect words belonging to free factors of F k (see [12]). For other algorithms to detect primitives see, for example, [5,Chapter I.2] or [8].…”

Uniformly Distributed Systems of Elements on Metabelian Lie Rings

“…In this paper, we describe systems of elements that uniformly distributed on the variety of Lie rings (over the ring of integers Z). In [5,6], elements uniformly distributed on groups were studied. Such elements are called measure preserving.…”