Primitivity rank for random elements in free groups

Ilya Kapovich

doi:10.1515/jgth-2021-0150

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Primitivity rank for random elements in free groups

Ilya Kapovich

Published/Copyright: March 17, 2022

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From the journal Journal of Group Theory Volume 25 Issue 5

Abstract

For a free group F r of finite rank r ≥ 2 and a non-trivial element w ∈ F r , the primitivity rank π ⁢ ( w ) is the smallest rank of a subgroup H ≤ F r such that w ∈ H and 𝑤 is not primitive in 𝐻 (if no such 𝐻 exists, one puts π ⁢ ( w ) = ∞ ). The set of all subgroups of F r of rank π ⁢ ( w ) containing 𝑤 as a non-primitive element is denoted by Crit ⁡ ( w ) . These notions were introduced by Puder (2014). We prove that there exists an exponentially generic subset V ⊆ F r such that, for every w ∈ V , we have π ⁢ ( w ) = r and Crit ⁡ ( w ) = { F r } .

1 Introduction

Recall that an element 𝑔 of a free group 𝐹 is primitive in 𝐹 if 𝑔 belongs to some free basis of 𝐹. Let r ≥ 2 be an integer and F r = F ⁢ ( A ) , where A = { a 1 , … , a r } , the free group of rank 𝑟. In [20], Puder introduced the following natural measure of algebraic complexity for elements of F r . For an element 1 ≠ g ∈ F r , the primitivity rank π ⁢ ( g ) in F r is defined as the smallest rank of a subgroup H ≤ F r containing 𝑤 as a non-primitive element; if no such 𝐻 exists, put π ⁢ ( g ) = ∞ . Puder observed that, for 1 ≠ g ∈ F r , one has π ⁢ ( g ) = ∞ if and only if 𝑔 is primitive in F r . For any non-primitive, non-trivial g ∈ F r , one obviously has π ⁢ ( w ) ≤ r . For 1 ≠ g ∈ F r , the critical set of 𝑔, denoted Crit ⁡ ( g ) , is the set of all subgroups 𝐻 of F r containing 𝑔 as a non-primitive element and such that rank ⁡ ( H ) = π ⁢ ( g ) . It is known that the set Crit ⁡ ( g ) is always finite (see [22]) and that, given 𝑔, one can algorithmically compute π ⁢ ( g ) and the set Crit ⁡ ( g ) . Note that the primitivity rank is obviously automorphically invariant. For any 1 ≠ g ∈ F r and φ ∈ Aut ⁢ ( F r ) , we have π ⁢ ( g ) = π ⁢ ( φ ⁢ ( g ) ) . Moreover, in this case,

Crit ⁡ ( φ ⁢ ( g ) ) = φ ⁢ ( Crit ⁡ ( g ) ) = { φ ⁢ ( H ) ∣ H ∈ Crit ⁡ ( g ) } .

Puder [20] and later Puder–Parzanchevski [22] and Hanany–Puder [8] showed that the primitivity rank and the critical set are closely related to understanding the behavior of the word measures on finite symmetric groups S N corresponding to words in F r . Here, for a word w = w ⁢ ( a 1 , … , a r ) ∈ F r , the corresponding word measure on S N is the image under the map w : ( S N ) r → S r of the uniform probability distribution on ( S N ) r . This connection is illustrated in Corollary 1.2 to our main result below.

However, the primitivity rank and the critical set for an element of F r are natural algebraic notions that deserve to be studied in more detail in their own right. Indeed, Klimakov [14] extended the notion of primitivity rank to the context of free algebras of Schreier varieties and obtained some structural results there. Also, a recent series of papers of Louder and Wilton [15, 16] shows that, for w ∈ F r , there is a connection between the primitivity rank π ⁢ ( w ) and subgroup properties of the one-relator group G w = ⟨ a 1 , … , a r ∣ w = 1 ⟩ . In particular, they prove that if r ≥ 3 and π ⁢ ( w ) ≥ 3 , then the group G w is coherent, that is all of its finitely generated subgroups are finitely presented.

In the present paper, we study the primitivity rank and the critical set for “generic” or “random” freely reduced and cyclically reduced elements in F r . The idea of “genericity” in the context of infinite groups arose in the 1990s in the work of Gromov [6], Ol’shanskii [19], Arzhantseva [3, 1, 2] and others. The notion of genericity for subsets of free groups that we use in this paper was formalized [11] when defining generic-case complexity of group-theoretic algorithms. If Z ⊆ F r , a subset S ⊆ Z is exponentially generic in 𝑍 if

# ⁢ { w ∈ S : | w | A ≤ n } # ⁢ { w ∈ Z : | w | A ≤ n } → n → ∞ 1

with exponentially fast convergence. See Section 2.2 for more precise definitions and additional details. In this paper, we are particularly interested in the cases Z = F r and Z = U r , where U r is the set of all cyclically reduced words in F r = F ⁢ ( A ) .

Our main result is the following.

Theorem 1.1

Let r ≥ 2 be an integer, and let F r = F ⁢ ( A ) , where A = { a 1 , … , a r } , be the free group of rank 𝑟.

There exists a subset Y r ⊆ F r such that Y r is exponentially generic in F r and that, for every w ∈ Y r , we have π ⁢ ( w ) = r and Crit ⁡ ( w ) = { F r } .
There exists a subset Q r ⊆ U r such that Q r is exponentially generic in U r and that, for every w ∈ Q r , we have π ⁢ ( w ) = r and Crit ⁡ ( w ) = { F r } .
Let W n ∈ F ⁢ ( A ) be obtained by a simple non-backtracking random walk of length 𝑛. Then, with probability tending to 1 exponentially fast as n → ∞ , the word W n has the property that π ⁢ ( W n ) = r and Crit ⁡ ( W n ) = { F r } .

Note that the generic sets Q r and Y r in the conclusion of Theorem 1.1 can in fact be chosen to be algorithmically recognizable since the conditions P ′ ⁢ ( λ , μ , r ) and ( P ′ ⁢ ( λ , μ , r ) ) ∘ defining these sets in the proof of Theorem 1.1 are algorithmic if 𝜆 and 𝜇 are chosen to be rational numbers.

Puder [21, Corollary 8.3] also proved the existence of a generic subset of F r such that, for all elements 𝑤 of that subset, the primitivity rank is equal to 𝑟. His method of proof was very different from ours, and it did not imply Crit ⁡ ( w ) = { F r } for generic elements.

As an application of Theorem 1.1, combined with the result of Puder and Parzanchevski, we obtain precise asymptotics for the number of fixed points # ⁢ fix ⁡ ( σ ) of a random permutation 𝜎 in S N with respect to the “word measure” on the symmetric group S N defined by a generic word w ∈ F r .

Corollary 1.2

Let r ≥ 2 be an integer, and let F r = F ⁢ ( A ) , where A = { a 1 , … , a r } . Let Y r ⊆ F r , Q r ⊆ U r be the exponentially generic subsets provided by Theorem 1.1. Then, for every w ∈ Y r and every w ∈ Q r , we have

E w ⁢ [ # ⁢ fix ⁡ ( σ ) ] ⁢ = N → ∞ ⁢ 1 + 1 N r - 1 + O ⁢ ( 1 N r ) .

Here the expectation is taken with respect to the word measure on the symmetric group S N defined by the word w ⁢ ( a 1 , … , a r ) , that is σ = w ⁢ ( σ 1 , … , σ r ) , where σ 1 , … , σ r ∈ S N are chosen uniformly independently at random in S N .

Proof

Puder and Parzanchevski proved [22, Theorem 1.8] that one has, for every 1 ≠ w ∈ F r ,

E w ⁢ [ # ⁢ fix ⁡ ( σ ) ] ⁢ = N → ∞ ⁢ 1 + | Crit ⁡ ( w ) | N π ⁢ ( w ) - 1 + O ⁢ ( 1 N π ⁢ ( w ) ) .

The statement of the corollary now follows directly from Theorem 1.1. ∎

As noted above, the results of Louder and Wilton [15, 16] imply that if r ≥ 3 and w ∈ F r has π ⁢ ( w ) ≥ 3 , then the group G w = ⟨ a 1 , … , a r ∣ w = 1 ⟩ is coherent. Since generic elements w ∈ F r have π ⁢ ( w ) = r , it follows that, for r ≥ 3 and a generic w ∈ F r , the group G w is coherent. This fact was first proved by Sapir and Spukalova [23], and also observed by Louder and Wilton [16], with a reference to Puder’s result on the primitivity rank of generic elements [21].

Louder and Wilton conjectured [15] that if 1 ≠ w ∈ F r is not a proper power and has π ⁢ ( w ) > 2 , then the corresponding one-relator group G w is word-hyperbolic (note that, by [15, Theorem 1.4], all 2-generator subgroups in G w are free, and hence G w has no Baumslag–Solitar subgroups). Cashen and Hoffmann [5] obtained some experimental evidence for this conjecture, verifying it for all 𝑤 with | w | ≤ 17 and r ≤ 4 . For 1 ≠ w ∈ F r , which is not a proper power, Louder and Wilton define 𝑤-subgroups of F r as maximal with respect to inclusion elements of Crit ⁡ ( w ) . They show in [15] that, for 𝑤 as above, if π ⁢ ( w ) = 2 , then 𝑤 has a unique 𝑤-subgroup. In [5], they raise the question of whether or not the 𝑤-subgroup is always unique. Note that if π ⁢ ( w ) = r , then obviously F r is the unique maximal element in Crit ⁡ ( w ) , and thus, in particular, generic elements 𝑤 of F r have unique 𝑤-subgroups. Theorem 1.1 provides a maximally sharpened version of this statement for generic elements since, in that case, we conclude that the entire set Crit ⁡ ( w ) consists of a single element, namely F r itself.

As shown in [22], both π ⁢ ( w ) and the set Crit ⁡ ( w ) are algorithmically computable in terms of 𝑤. If w ∈ F r is a non-trivial cyclically reduced word (if we are given a word that is not cyclically reduced, we first cyclically reduce it), we write 𝑤 on a circle of length 𝑤 to get an 𝐴-graph C w . Then we consider all possible “quotients” of C w under surjective morphisms of folded 𝐴-graphs. That amounts to picking a partition of the vertex set of C w , collapsing each element of that partition and then folding. For each resulting 𝐴-graph Γ, we check, using Whitehead’s algorithm, if the closed path γ w labeled by 𝑤 at the base-vertex ∗ of Γ is a primitive element of π 1 ⁢ ( Γ , ∗ ) . We keep those Γ for which this path γ w is not primitive in π 1 ⁢ ( Γ , ∗ ) and take the minimum of the ranks of all such π 1 ⁢ ( Γ , ∗ ) . That minimum equals π ⁢ ( w ) , and the 𝐴-graphs Γ that realize this minimum give us elements of Crit ⁡ ( w ) . Note, however, that this algorithm sheds no light on the behavior of π ⁢ ( w ) and Crit ⁡ ( w ) for “random” cyclically reduced elements w ∈ F r . Therefore, we use rather different considerations in order to prove Theorem 1.1. The key tools there are provided by the “graph non-readability” conditions for generic elements of F r obtained by Arzhantseva and Ol’shanskii in [3, 1]; see Section 3 below. These genericity conditions also play a key role in the isomorphism rigidity results for “random” one-relator groups obtained in [12, 13].

The concept of the primitivity rank is dual to the notion of “primitivity index” for elements of free groups introduced by Gupta and Kapovich in [7]. For 1 ≠ g ∈ F r , the primitivity index d prim ⁢ ( g , F r ) of 𝑔 in F r is the smallest index of a subgroup H ≤ F r such that 𝐻 contains 𝑔 as a primitive element. It is shown in [7] that d prim ⁢ ( g , F r ) is always finite and moreover d prim ⁢ ( g , F r ) ≤ ∥ g ∥ A , where ∥ g ∥ A is the cyclically reduced length of 𝑔 in F ⁢ ( A ) . One of the main results of [7] shows that, for a “random” freely reduced w n ∈ F ⁢ ( A ) of length 𝑛, the primitivity index d prim ⁢ ( w n , F r ) is ≥ const ⁢ log 1 / 3 ⁡ n . Thus we see that the primitivity index and primitivity rank of generic elements in F r exhibit rather different quantitative behavior. However, a certain type of duality in these results is still preserved. They show that, for a long “generic” w ∈ F r , it is “hard” for 𝑤 to be non-primitive in a subgroup of small rank containing 𝑤 and it is “hard” for 𝑤 to be primitive in a subgroup of small index in F r containing 𝑤.

As part (3) of Theorem 1.1 shows, we can interpret genericity in the context of this theorem in terms of the element W n ∈ F r = F ⁢ ( A ) being produced by a simple non-backtracking random walk of length 𝑛 on F r = F ⁢ ( A ) . It would be interesting to understand if the conclusion of Theorem 1.1 holds for other types of random walks on F r . For example, let μ : F r → [ 0 , 1 ] be a discrete probability measure with finite support which generates F r . Let W n = s 1 ⁢ … ⁢ s n ∈ F r be obtained by a random walk of length 𝑛 on F r defined by 𝜇 (where the increments s 1 , s 2 , … are chosen using an i.i.d. sequence of random variables, each with distribution 𝜇). Is it then true that, with probability tending to 1 as n → ∞ , we have π ⁢ ( W n ) = r and Crit ⁡ ( W n ) = { F r } ? We suspect that the answer should be positive, but proving this fact would require establishing a suitable version of Proposition 3.4 below for W n , perhaps using the results of [18]. The original proof of Proposition 3.4 by Arzhantseva and Ol’shanskii deployed heavy duty counting arguments relying on certain entropy lowering considerations that were specific to the simple non-backtracking random walk context. Thus a new approach would be required for further generalizations.

We are grateful to Doron Puder for bringing to our attention his result from [21] that generic elements of F r have primitivity rank 𝑟 and for pointing out that the primitivity rank plays a key role in the work of Louder and Wilton [15, 16]. We also thank Henry Wilton for pointing us to the work of Cashen and Hoffmann [5].

2 Preliminaries

2.1 𝐴-graphs

For the remainder of this paper, unless specified otherwise, let r ≥ 2 be an integer, A = { a 1 , … , a r } , and let F r = F ⁢ ( A ) be the free group of rank 𝑟 with the free basis 𝐴.

As in [10], we adopt the same language and convention regarding Stallings subgroup graphs for F r and 𝐴-graphs more generally. Thus an 𝐴-graph is a labeled directed graph Γ where every oriented edge 𝑒 is given a label μ ⁢ ( e ) ∈ A ± 1 satisfying μ ⁢ ( e - 1 ) = ( μ ⁢ ( e ) ) - 1 . An edge-path 𝛾 in Γ is then naturally assigned a label μ ⁢ ( γ ) which is a word in the alphabet A ± 1 . An 𝐴-graph Γ is folded if, whenever e 1 , e 2 are two distinct oriented edges of Γ with the same initial vertex o ⁢ ( e 1 ) = o ⁢ ( e 2 ) , we have μ ⁢ ( e 1 ) ≠ μ ⁢ ( e 2 ) . Thus every vertex in a folded 𝐴-graph has at most degree 2 ⁢ r .

A connected 𝐴-graph Γ with a base-vertex ∗ is a core graph with respect to ∗ if Γ is equal to the union of all non-backtracking closed edge-paths from ∗ to ∗. A connected 𝐴-graph Γ is a core graph if it is a core graph with respect to each of its vertices.

For a finite 𝐴-graph Γ, we denote by vol ⁡ ( Γ ) the number of topological edges of Γ (where, for every oriented edge 𝑒 of Γ, the unordered pair e , e - 1 counts as a single topological edge).

Every subgroup H ≤ F r is uniquely represented by its Stallings subgroup graph Γ H , which is a connected folded 𝐴-graph with a base-vertex ∗ such that Γ H is a core graph with respect to ∗ and that a freely reduced word w ∈ F ⁢ ( A ) belongs to 𝐻 if and only if 𝑤 labels a closed path from ∗ to ∗ in Γ H . In this case, the labeling map 𝜇 provides a natural isomorphism between π 1 ⁢ ( Γ H , ∗ ) and 𝐻. A subgroup H ≤ F ⁢ ( A ) is finitely generated if and only if Γ H is finite. Moreover, 𝐻 has finite index in F ⁢ ( A ) if and only if every vertex of Γ H has degree 2 ⁢ r , and in this case, the index [ F ⁢ ( A ) : H ] is equal to the number of vertices in Γ H .

For H = F r , the corresponding Stallings subgroup graph is the 𝐴-rose R A , which is a wedge of 𝑟-loop edges, labeled a 1 , … , a r , wedged at a single vertex x 0 . For an arbitrary H ≤ F ⁢ ( A ) , one can obtain Γ H by first taking the covering space ( R ^ A , ∗ ) of ( R A , x 0 ) corresponding to H ≤ π 1 ⁢ ( R A , x 0 ) and then taking the topological core of this covering that is the smallest connected subgraph of R ^ A containing ∗ whose inclusion into R ^ A is a homotopy equivalence.

We refer the reader to [24, 10, 12] for additional background on 𝐴-graphs and Stallings subgroup graphs.

For an element g ∈ F r , we denote by | g | A the freely reduced length of 𝑔 with respect to 𝐴, and we denote by | g | A the cyclically reduced length of 𝑔 with respect to 𝐴. For a word 𝑤 over A ± 1 (not necessarily freely reduced), we denote by | w | the length of 𝑤. Similarly, for an edge-path 𝛾 in an 𝐴-graph Γ, we denote by | γ | the length of 𝛾.

Let Γ be a finite connected 𝐴-graph with a base-vertex ∗. The maximal arcs in Γ are closures of the connected components of the space obtained by removing from Γ all vertices of degree at least 3 and the vertex ∗. It is easy to check that if Γ is a finite connected 𝐴-graph with π 1 ⁢ ( Γ ) of rank 𝑚, then Γ has at most 3 ⁢ m maximal arcs.

2.2 Genericity

The definitions in this section follow [11, 12]. See [9] and [4, Chapter 1] for more general treatment of genericity.

For a sequence ( x n ) n ≥ n 0 of real numbers and x ∈ R , we say that

lim n → ∞ ⁡ x n = x with exponentially fast convergence

if there exist constants C > 0 and 0 < σ < 1 such that, for all n ≥ n 0 , we have

| x n - x | ≤ C ⁢ σ n .

Let S ⊆ F ⁢ ( A ) . For n ≥ 0 , we denote the number of elements w ∈ F ⁢ ( A ) with | w | A ≤ n by ρ n ⁢ ( S ) and the number of elements w ∈ F ⁢ ( A ) with | w | A = n by γ n ⁢ ( S ) .

Definition 2.1

Definition 2.1 (Exponentially generic subsets)

Let Z ⊆ F r be a subset such that there is n 0 ≥ 1 such that, for all n ≥ n 0 , we have ρ n ⁢ ( S ) ≠ 0 .

A subset S ⊆ Z is exponentially negligible in 𝑍 if
lim n → ∞ ⁡ ρ n ⁢ ( S ) ρ n ⁢ ( Z ) = 0 ,
and the convergence in this limit is exponentially fast.
A subset S ⊆ Z is exponentially generic in 𝑍 if Z ∖ S is exponentially negligible in 𝑍.

Recall that U r denotes the set of all cyclically reduced words in F r = F ⁢ ( A ) .

Both 𝑛-balls and 𝑛-spheres in F r and U r grow like ( 2 ⁢ r - 1 ) n . This fact has the following useful consequences; see [13, Lemma 6.1].

Proposition 2.2

The following hold.

Let S ⊆ F r . Then 𝑆 is exponentially negligible in F r if and only if
lim n → ∞ ⁡ γ n ⁢ ( S ) ( 2 ⁢ r - 1 ) n = 0
with exponentially fast convergence.
Let S ⊆ U r . Then 𝑆 is exponentially negligible in U r if and only if
lim n → ∞ ⁡ γ n ⁢ ( S ) ( 2 ⁢ r - 1 ) n = 0
with exponentially fast convergence.
Let S ⊆ F r . Then 𝑆 is exponentially generic in F r if and only if
lim n → ∞ ⁡ γ n ⁢ ( S ) γ n ⁢ ( F r ) = 1
with exponentially fast convergence.
Let S ⊆ U r . Then 𝑆 is exponentially generic in F r if and only if
lim n → ∞ ⁡ γ n ⁢ ( S ) γ n ⁢ ( U r ) = 1
with exponentially fast convergence.

We also need the following statement; see [13, Proposition 6.2].

Proposition-Definition 2.3

Let Z ⊆ U r . Let Z ∘ denote the set of all freely reduced words in F r whose cyclically reduced forms belong to 𝑍.

If 𝑍 is exponentially negligible in U r , then Z ∘ is exponentially negligible in F r .
If 𝑍 is exponentially generic in U r , then Z ∘ is exponentially generic in F r .

3 The genericity condition

Recall that r ≥ 2 is fixed, A = { a 1 , … , a r } and F r = F ⁢ ( A ) . We will need the following genericity condition introduced by Arzhantseva and Ol’shanskii in [3].

Definition 3.1

Definition 3.1 ([3])

Let 0 < μ ≤ 1 be a real number. A non-trivial freely reduced word 𝑤 in F ⁢ ( A ) = F ⁢ ( a 1 , … , a r ) is said to be 𝜇-readable if there exists a connected folded 𝐴-graph Γ such that

the number of edges in Γ is at most μ ⁢ | w | ,
the free group π 1 ⁢ ( Γ ) has rank at most r - 1 ,
there exists a reduced path in Γ with label 𝑤.

Definition 3.2

Definition 3.2 ([1])

Let 0 < μ ≤ 1 be a real number, and let L ≥ 2 be an integer. A non-trivial freely reduced word 𝑤 in F ⁢ ( A ) is said to be ( μ , L ) -readable if there exists a connected folded 𝐴-graph Γ such that

the number of edges in Γ is at most μ ⁢ | w | ,
the free group π 1 ⁢ ( Γ ) has rank at most 𝐿,
there is a path in Γ with label 𝑤,
the graph Γ has at least one vertex of degree less than 2 ⁢ r .

Definition 3.3

Let 0 < μ ≤ 1 be a real number, let L ≥ 2 be an integer, and let 0 < λ < 1 be a real number such that

λ ≤ μ 15 ⁢ L + 3 ⁢ μ ≤ μ 15 ⁢ r + 3 ⁢ μ < 1 6 .

We will say that a cyclically reduced word 𝑤 in F ⁢ ( A ) satisfies the ( λ , μ , L ) -condition if the following holds.

The (symmetrized closure) of the word 𝑤 satisfies the C ′ ⁢ ( λ ) small cancellation condition.
The word 𝑤 is not a proper power in F ⁢ ( A ) .
If w ′ is a subword of a cyclic permutation of 𝑤 and | w ′ | ≥ | w i | / 2 , then w ′ is not 𝜇-readable and not ( μ , L ) -readable.

We denote by P ⁢ ( λ , μ , L ) the set of all cyclically reduced words w ∈ F r satisfying the ( λ , μ , L ) -condition.

We refer the reader to [17] for the basic definitions and background information regarding small cancellation theory.

Note that (3) in Definition 3.3 implies that if w ∈ P ⁢ ( λ , μ , L ) , then 𝑤 is not 𝜇-readable and not ( μ , L ) -readable.

The results of Arzhantseva and Ol’shanskii [3, 1] imply the following result (cf. [12, Theorem 4.6]).

Proposition 3.4

Let r ≥ 2 and F r = F ⁢ ( a 1 , … , a r ) . Let λ , μ , L be as in Definition 3.3. Then the set P ⁢ ( λ , μ , L ) is exponentially generic in U r .

Remark 3.5

Note that it is fairly easy to see that, for any λ ∈ ( 0 , 1 ) , the set of all non-proper power words satisfying C ′ ⁢ ( λ ) is exponentially generic in U r . The main substance and power of Proposition 3.4 consists in verifying genericity of the non-readability conditions.

Note also that we do not actually need the full strength of the ( λ , μ , L ) -condition for the proofs in this paper. In particular, we do not use the “not a proper power” assumption on 𝑤. Also, in part (3) of Definition 3.3, we only really need non-readability of 𝑤 itself rather than of its sufficiently long subwords. Nevertheless, we state the ( λ , μ , L ) -condition in its original stronger form, as it was defined by Arzhantseva and Ol’shanskii, for which they established Proposition 3.4.

We also need the following slightly more restricted version of P ⁢ ( λ , μ , L ) .

Definition 3.6

Let r ≥ 2 and F r = F ⁢ ( a 1 , … , a r ) . Let λ , μ , L be as in Definition 3.3. Denote by P ′ ⁢ ( λ , μ , L ) the set of all words w ∈ P ⁢ ( λ , μ , L ) such that every freely reduced word of length 2 in F r occurs as a subword of 𝑤.

Proposition 3.7

Let r ≥ 2 and F r = F ⁢ ( a 1 , … , a r ) . Let λ , μ , L be as in Definition 3.3. Then the set P ′ ⁢ ( λ , μ , L ) is exponentially generic in U r .

Proof

For a given freely reduced word 𝑧 of length 2, the set of all words in U r containing 𝑧 as a subword is exponentially generic in U r ; see, for example [13, Proposition 6.3]. Since the intersection of finitely many exponentially generic subsets of U r is again exponentially generic in U r , the conclusion of the proposition now follows from Proposition 3.4. ∎

4 Primitivity rank of generic elements

Theorem 4.1

Let r ≥ 2 . Let 0 < λ , μ < 1 be such that

λ ≤ μ 15 ⁢ L + 3 ⁢ μ ≤ μ 15 ⁢ r + 3 ⁢ μ < 1 6 and λ < μ 3 ⁢ r .

Then, for every non-trivial cyclically reduced word w ∈ P ′ ⁢ ( λ , μ , r ) , we have

π ⁢ ( w ) = r and Crit ⁡ ( w ) = { F r } .

Proof

Let 1 ≠ w ∈ P ′ ⁢ ( λ , μ , r ) , where λ , μ are as in the assumptions of the theorem.

(a) Note first that 𝑤 is not primitive in F r because 𝑤 contains all freely reduced words in F r = F ⁢ ( A ) of length 2 as subwords, and hence the Whitehead graph of the cyclic word corresponding to 𝑤 is complete [25]. Hence π ⁢ ( w ) ≠ ∞ , and so π ⁢ ( w ) ≤ r .

(b) We claim that π ⁢ ( w ) = r . Indeed, suppose not. Thus π ⁢ ( w ) < r . Take a subgroup H ∈ Crit ⁡ ( w ) . Thus H ≤ F r is a subgroup of the smallest possible rank containing 𝑤 as a non-primitive element and rank ⁡ ( H ) = π ⁢ ( w ) < r . Let Γ H be the Stallings subgroup graph for 𝐻, and let ∗ be the base-vertex of Γ H . Then the cyclically reduced word 𝑤 is readable along a closed path γ w from ∗ to ∗ in Γ H . The minimality assumption on the rank of 𝐻 implies that γ w crosses every topological edge of Γ H . Moreover, since Γ H is folded and 𝑤 is cyclically reduced, the vertex ∗ has degree greater than 1 in Γ H , that is Γ H is a folded connected finite core 𝐴-graph. Put m = rank ⁡ ( H ) < r . The graph Γ H has at most 3 ⁢ m maximal arcs (recall that we view ∗ as an endpoint of maximal arcs even if ∗ has degree 2 in Γ H ). The case vol ⁡ ( Γ H ) ≤ μ ⁢ | w | is impossible since 𝑤 is readable in the graph Γ H of rank at most r - 1 and since, by assumption, 𝑤 satisfies the ( λ , μ , r ) -condition. Therefore, vol ⁡ ( Γ H ) ≥ μ ⁢ | w | . Let 𝛼 be the longest maximal arc of Γ H . Since Γ H has at most 3 ⁢ m maximal arcs, we have

| α | ≥ vol ⁡ ( Γ H ) 3 ⁢ m ≥ μ ⁢ | w | 3 ⁢ m ≥ μ ⁢ | w | 3 ⁢ r > λ ⁢ | w | .

The path γ w crosses over the arc 𝛼 (in some direction) at least once, and the C ′ ⁢ ( λ ) assumption on 𝑤 implies that it does so exactly once. Since γ w is a closed path at ∗ in Γ H , it follows that 𝛼 is a non-separating arc in Γ H and that γ w represents a primitive element in π 1 ⁢ ( Γ H , ∗ ) . Hence 𝑤 is primitive in 𝐻, yielding a contradiction with our assumption about 𝑤 and 𝐻. Thus, indeed, π ⁢ ( w ) = r , as claimed.

(c) We now claim that Crit ⁡ ( w ) = { F r } . We have already seen in (a) that 𝑤 is not primitive in F r , and we have just proved that π ⁢ ( w ) = r so that F r ∈ Crit ⁡ ( w ) . Suppose that Crit ⁡ ( w ) ≠ { F r } . Then there exists a subgroup H ∈ Crit ⁡ ( w ) such that H ≠ F r . Thus 𝐻 contains 𝑤 as a non-primitive element, and rank ⁡ ( H ) = r . Since H ≠ F r , it follows that 𝐻 has infinite index in F r . Again, let Γ H be the Stallings subgroup graph for 𝐻 with base-vertex ∗. Since w ∈ H , there is a closed path from ∗ to ∗ labeled by 𝑤. As in (b), we see that Γ H is a finite connected folded core 𝐴-graph. Also, the minimality assumption on the rank of 𝐻 implies that γ w crosses every edge of Γ H . Since [ F r : H ] = ∞ , the graph Γ H has a vertex of degree less than 2 ⁢ r . Since 𝑤 is readable in Γ H but 𝑤 is not ( μ , r ) -readable by our assumption that w ∈ P ′ ⁢ ( λ , μ , r ) , it follows that the case vol ⁡ ( Γ H ) ≤ μ ⁢ | w | is impossible. Hence vol ⁡ ( Γ H ) ≥ μ ⁢ | w | . Since π 1 ⁢ ( Γ H ) has rank 𝑟, the graph Γ H has at most 3 ⁢ r maximal arcs. Let 𝛼 be the longest maximal arc of Γ H . Then

| α | ≥ vol ⁡ ( Γ H ) 3 ⁢ r ≥ μ ⁢ | w | 3 ⁢ r > λ ⁢ | w | .

As in (b), the path γ w crosses the arc 𝛼 at least once in some direction, and the C ′ ⁢ ( λ ) condition implies that it crosses 𝛼 exactly once. Since γ w is a closed path, it again follows that 𝛼 is a non-separating arc in Γ H , and hence γ w is primitive in π 1 ⁢ ( Γ H , ∗ ) . Therefore, 𝑤 is primitive in 𝐻, yielding a contradiction with the choice of 𝐻. Thus | Crit ⁡ ( w ) | = 1 , as required. ∎

We can now prove Theorem 1.1 from the introduction.

Theorem 4.2

Let r ≥ 2 be an integer, and let F r = F ⁢ ( A ) , where A = { a 1 , … , a r } , be the free group of rank 𝑟.

There exists a subset Y r ⊆ F r such that Y r is exponentially generic in F r and that, for every w ∈ Y r , we have π ⁢ ( w ) = r and Crit ⁡ ( w ) = { F r } .
There exists a subset Q r ⊆ U r such that Q r is exponentially generic in U r and that, for every w ∈ Q r , we have π ⁢ ( w ) = r and Crit ⁡ ( w ) = { F r } .
Let W n ∈ F ⁢ ( A ) be obtained by a simple non-backtracking random walk of length 𝑛. Then, with probability tending to 1 exponentially fast as n → ∞ , the word W n has the property that π ⁢ ( W n ) = r and Crit ⁡ ( W n ) = { F r } .

Proof

Choose λ , μ as in Theorem 4.1, and put Q r = P ′ ⁢ ( λ , μ , r ) . Then Q r is exponentially generic in U r by Proposition 3.7. Then Q r satisfies the requirements of part (2) by Theorem 4.1.

Now put Y r = Q r ∘ . Then Y r is exponentially generic in F r by Proposition-Definition 2.3. Every element 𝑤 of Y r is conjugate in F r to some element w ′ of Q r in F r , that is w = g ⁢ w ′ ⁢ g - 1 for some g ∈ F r . Then π ⁢ ( w ) = π ⁢ ( w ′ ) and Crit ⁡ ( w ) = g ⁢ Crit ⁡ ( w ′ ) ⁢ g - 1 = { g ⁢ F r ⁢ g - 1 } = { F r } . Thus the requirements of (1) hold for Y r .

Note that the simple non-backtracking random walk W n of length 𝑛 on F ⁢ ( A ) induces the uniform probability distribution on the 𝑛-sphere in F ⁢ ( A ) . Therefore, (3) directly follows from (1) in view of part (3) of Proposition 2.2. ∎

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1905641

Funding statement: The author was supported by the individual NSF grant DMS-1905641.

Communicated by: Alexander Olshanskii

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Received: 2021-09-24

Revised: 2022-01-09

Published Online: 2022-03-17

Published in Print: 2022-09-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2021-0150