Short, highly imprimitive words yield hyperbolic one-relator groups

We give experimental support for a conjecture of Louder and Wilton saying that words of imprimitivity rank greater than two yield hyperbolic one-relator groups.


Introduction
An element in a free group is primitive if it is an element of some basis, or free generating set.Failure of primitivity can be quantified: define the imprimitivity rank of an element to be the minimal rank of a subgroup containing it as an imprimitive element, if such a subgroup exists, or infinite otherwise.An element has imprimitivity rank 0 if and only if it is trivial, 1 if and only if it is a proper power, and ∞ if and only if it is a primitive element.In these cases the quotient of the free group by the subgroup normally generated by the element is a hyperbolic group, either a free group, in the first and third cases, or a one-relator group with torsion, which is hyperbolic by the B. B. Newman spelling theorem [25], in the second case.Nonelementary, torsion-free two-generator one-relator groups have relators of imprimitivity rank 2. There are many nonhyperbolic groups of this form, such as Z 2 = a, b | aba −1 b −1 , the Baumslag-Solitar groups BS(m, n) = a, b | ab m a −1 b −n , and the groups considered by Gardam and Woodhouse [8].Louder and Wilton [22,Theorem 1.4] show that twogenerated subgroups of a higher imprimitivity rank one-relator group are free.Thus, they are of Type F and have no Baumslag-Solitar subgroups.It is a long-standing open question whether such groups must be hyperbolic.Louder and Wilton conjecture [22,Conjecture 1.6] a positive answer for one-relator groups.
We offer experimental support for their conjecture.Fix a basis for a free group, so that a group element can be uniquely represented as a freely reduced word, a product of basis elements and their inverses, of a well-defined length.
Theorem 1.1.Let w be a word in F r of length L and imprimivity rank not equal to 2. Then F r / w is hyperbolic if r ≤ 4 and L ≤ 17.
These results are achieved computationally, by a combination of efficient enumeration of representatives and brute force 1 .Details are in Section 4.
We also observe that a well-known result about hyperbolicity of one-relator groups is consistent with the conjecture.In these results |w| a denotes the total number of occurrences of a and a −1 in w, where w is freely reduced and a is an element of the chosen basis.
The proposition is proven in Section 3. As a consequence, we have: Let w be a cyclically reduced word in F r of imprimitivity rank not equal to 2 such that 0 < |w| a < 4 for some basis element a. Then F r / w is hyperbolic.
Corollary 1.4.Let w be cyclically reduced word in F r of length less than 4r and imprimitivity rank not equal to 2 such that every generator or its inverse occurs in w.Then F r / w is hyperbolic.
Combining these results with our experimental results, we have: Corollary 1.5.Let w be a word in F r of length at most 17 and imprimitivity rank not equal to 2. Then F r / w is hyperbolic.
Proof.F r / w is hyperbolic when the imprimitivity rank of w is 0, 1, or ∞, so suppose it is finite and at least 3. Up to replacing w by an element in the same automorphic orbit, we may, without increasing the length of w, assume that it is cyclically reduced and that there is s such that taking the first s basis elements and the last r − s basis elements gives a splitting F r = F s * F r−s where the F s factor is the smallest free factor containing w. Since w is imprimitive in F r , it is imprimitive in F s , so s is an upper bound on imprimitivity rank, which implies s ≥ 3. Furthermore, since F s is the smallest free factor containing w, all of the generators of F s or their inverses occur in w.Since F r / w ∼ = (F s / w ) * F r−s is hyperbolic if and only if F s / w is, we conclude by applying Theorem 1.1 or Corollary 1.4 to F s / w , according to whether s < 5 or s ≥ 5, respectively.
Additional conjectures.In checking the hyperbolicity conjecture, we enumerated the Aut(F 4 ) orbits of cyclic subgroups of F 4 that have a representative that can be generated by a word of at most length 16.We also computed imprimitivity ranks for these words.Armed with this data, we can test other questions involving imprimitivity rank.We check two additional conjectures and find that they are consistent with the data up to length 16.The first of these concerns uniqueness of the subgroup in the definition of imprimitivity rank, see Proposition 4.2.The second concerns the relationship between imprimitivity rank and stable commutator length, see Proposition 4.4.
with respect to X is denoted |f |, and the word length of the cyclic reduction of f with respect to X, the cylic length of f , is denoted ||f ||.
For our purposes, a finitely presented group is hyperbolic if there exists a linear function δ such that if w is a freely reduced word of length n in the generators or their inverses that represents the identity element of the group then it is possible to express w as the free reduction of a product of at most δ(n) conjugates of relators or their inverses.It turns out that while the precise function δ depends on the choice of finite presentation, its linearity does not, so being hyperbolic is a group property and not merely a property of a presentation.More on hyperbolic groups can be found in any textbook on Geometric Group Theory.
A Stallings graph is a based, directed, connected, X-labelled graph (Γ, o) that is folded and core with respect to o.The free group π 1 (Γ, o) is identified with a subgroup of F r via the labelling, and, in fact, Stallings graphs are in bijection with subgroups of F r .See Kapovich and Myasnikov [16] for details.
The group W I of Whitehead automorphisms of the first kind are automorphic extensions of maps defined on X by x i → x i σ(i) for 1 ≤ i ≤ r, where σ ∈ Sym(r) and i = ±1.
The set W II of Whitehead automorphisms of the second kind are automorphic extensions of maps defined on X ± as follows.Given an element x ∈ X ± and a subset Z ⊂ X ± \ {x, x −1 } take the map that fixes x and x −1 and for y ∈ X ± \ {x, x −1 } does: Together the Whitehead automorphisms generate Aut(F r ).Moreover, Whitehead [29] proves two stronger facts: • Call a word w ∈ F r Whitehead minimal if there does not exist a Whitehead automorphism α such that ||α(w)|| < |w|.An element has minimal length in its Aut(F r ) orbit if and only if it is Whitehead minimal.
• Define the Whitehead level-L graph to be the graph whose vertices are Whitehead minimal words of length L, where w and v are connected by an edge if there exists a Whitehead automorphism α such that v is the cyclic reduction of α(w).Then the partition of vertices by connected component in the Whitehead level-L graph is the same as the partition by Aut(F r ) orbits.Combining these two facts gives Whitehead's Algorithm for determining if two words are in the same Aut(F r ) orbit: they are if and only if their Whitehead minimal representatives have the same length, say L, and are contained in 2 Puder uses the term 'primitivity rank'.Louder and Wilton follow his terminology.We find it misleading.Compare, for instance, to the primitivity index of [10], which is the smallest index of a subgroup for which the element becomes primitive upon lifting to that subgroup.
the same component of the Whitehead level-L graph.In particular, a word represents a primitive element if and only if it Whitehead reduces to a word of length 1.

The Ivanov-Schupp criteria
Theorem 3.1 ([15, Theorem 3]).Let w be a freely and cyclically reduced word in F r and suppose that for some basis element a, the total number of occurrences of a and a −1 , |w| a , satisfies 0 < |w| a < 4. The group F r / w is not hyperbolic if and only if one of the following holds up to cyclic permutation and taking inverses: (1) |w| a = 2, w = auav and uv −1 is a proper power in F r .
( ).Let w = au 1 au 2 au 3 au 4 be a freely and cyclically reduced word in F r , such that |w| a = 4 and the subwords u i are pairwise different.Then the group F r / w is not hyperbolic if and only if for some i ∈ {1, . . ., 4} the following holds (with subscripts modulo 4): We check that nonhyperbolicity in these theorems implies imprimitivity rank 2: Proof of Proposition 1.2.Suppose w is of one of the forms in Theorem 3.1 and Theorem 3.2.For each case we exhibit a connected, based, rank 2 core graph with edges labelled by words in F r in which (a conjugate of) w labels a imprimitive element of the fundamental group.By subdividing edges we can arrange that edges are labelled by basis elements.The graphs are not necessarily folded, but from the hypothesis in Theorem 3.1 and Theorem 3.2 that the only occurrences of a ±1 are the explicit ones, it follows that in all of our examples folding will be a homotopy equivalence, so these graphs really do represent rank 2 subgroups.
In each of the figures the larger dot marks the base vertex, the triangular arrows mark a choice of edges in a maximal subtree, and the edges with the single and double arrows mark edges whose unique completion through the maximal subtree to a based loop represent generators α and β, respectively, of the fundamental group of the graph.
Let now |w| a = 3, w = atauav with ut −1 = z m and vt −1 = z n , where z is not a proper power and m, n satisfy one of the conditions (3a)-(3d) in Theorem 3.1.
Suppose in case (3a) we have m = 0 and n > 1, other variations of this case being similar.Then w ∼ = α 3 β n is imprimitive in Figure 2a.
In the subcase m = n of case (3b) that is not covered by case (3c), we may assume m > 1 by replacing z with z −1 , if necessary.Then w ∼ = α 2 β m αβ m in Figure 2a.This word admits a Whitehead reduction α −1 → βα −1 , which sends the w-loop to αβ −1 αβ 2(m−1) .Since m > 1, this word is Whitehead minimal, so the w-loop is imprimitive.
Next, consider the case |w| a = 3, w = ataua −1 v and t −1 ut = z m , v = z n where z is not a proper power.Again we can assume that |m|, |n| > 0 since otherwise |w| a would be less than 3. Then w = (at) 2 z m (at 2c by relabelling the β edge with z −n .
Finally, let w = au 1 au 2 au 3 au 4 be as in Theorem 3.2.We may assume that

The experiments
To prove Theorem 1.1, the idea is to enumerate words of each length in the given free group, compute their imprimitivity ranks, and for those of imprimitivity rank not equal to two, test to see if the resulting one-relator presentation is a hyperbolic group.4.1.Enumerating words/groups.For w ∈ F r an automorphism α ∈ Aut(F r ) induces an isomorphism between F r / w and F r / α(w) ±1 .Call these the 'obvious' isomorphisms between one-relator groups.To enumerate isomorphism types of one-relator groups it suffices to enumerate one generator of one representative of each automorphic orbit of cyclic subgroup.There is a canonical choice of such an element: we choose the one that is shortlex minimal with respect to the integer lexicographic order; that is, if (a 1 , . . ., a r ) is our fixed ordered basis for F r , we declare [23] that show that not all isomorphisms between one-relator groups are obvious, so our enumeration has some redundancies at the level of isomorphism type of one-relator groups.However, work of Kapovich and Schupp [17] and Kapovich, Schupp, and Shpilrain [18], says that there is a generic set of one-relator groups for which the only isomorphisms are the obvious ones, so the redundancies are rare, in a specific quantifiable sense.

extend to a shortlex ordering on reduced words. There are examples of McCool and Pietrowski
A naive algorithm for enumerating representatives of length L is to simply construct the Whitehead level-L graph.Additionally, since we are interested in cyclic subgroups and not just elements, we connect every vertex v to the vertex v −1 .Then choose the shortlex minimal word in each component.
We speed this algorithm up as follows.Permutation of generators and inversion of generators and conjugation by a generator are in Aut(F r ).Define the PCI class of a word to be those words that can be reached from it by a finite chain of P ermutation of generators, C yclic permutation, or I nversion of generators.Similarly, the PCI ± class is those words that can be reached by the above operations plus group inversion.Define a word to be SLPCI (±)  minimal if it is S hortLex minimal in its PCI (±) class.Notice that if we start with a cyclically reduced word then none of the above operations change the length of the word.Lemma 4.1.Aut(F r ) equivalence classes of cyclic subgroup such that the minimal generator length of a representative has length L are in bijection with connected components of the length-L SLPCI ± graph: the graph whose vertices are freely and cyclically reduced words of F r of length L that are both Whitehead and SLPCI ± minimal, and where two vertices u and v are connected by an edge if there exists an element α ∈ W II such that v is the SLPCI ± minimal representative of α(u).
Khan [19] used a similar construction, without inversion, to study the complexity of Whitehead's Algorithm in the special case r = 2.
Proof.Whitehead's result shows that the partition by components of the Whitehead level graph is the same as the partition by Aut(F r )-orbits.It is clear from the definitions that two words in the same component of the length-L SLPCI ± graph are in the same component of the Whitehead level-L graph.We show the opposite.The essential observation is that W I acts by conjugation on the set W II .
Elements that differ by an element of W I are in the same PCI class, so suppose α ∈ W II and α(u 1 ) = u 2 where u i = a i v i a −1 i with v i cyclically reduced, and suppose Thus there is an element of W II that takes w 1 to α (w 1 ) ∼ σ 1 (u 1  2 ), which is in the same PCI ± class as u 2 , so w 2 is the SLPCI ± minimal representative of α (w 1 ).
The lemma says we can run the naive algorithm but instead of enumerating all words of a fixed length, it's enough to enumerate SLPCI ± minimal ones.This is a benefit because SLPCI ± minimality is falsifiable by a subword: if w is a word that contains a prefix p and a subword v of equal length such that there is a W I automorphism that takes v or v −1 to a word that lexicographically precedes p, then w is not SLPCI ± minimal.We enumerate words of a fixed length by an odometer and check for such subwords v.If we find such a subword then we increment the odometer at the rightmost position of v.This potentially allows us to skip over large ranges of words that do not contain any SLPCI ± minimal words.
As the wordlength grows and exponential growth in the free group builds up steam, it impractical to hold the entire SLPCI ± graph in memory.Instead, for each SLPCI ± and Whitehead minimal word w we start constructing its graph component as described in Lemma 4.1.If in this construction we encounter a shortlex predecessor then we throw w away and proceed to the next candidate.If no such element occurs then w is minimal in its component.This procedure would be most effective if the SLPCI ± graph consists of many small components, and if in each component it is easy to verify whether or not a given word is the shortlex minimal one.Unfortunately for the latter case, there do exist examples of components with shortlex local minima.For example, here is a component of the graph in rank 2 at length 9 (Capitalization indicates inversion, and the base ordering is B < A < a < b.) that contains a word w := BBABBAAbA that is a shortlex local minimum but not the global minimum: BBABBAAbA − BBABAbAbA − BBBABBAAA So, to verify that w is not the global minimum in its component we have to construct the entire component.That is easy in this example because the component is small.It turns out that most components are small.Figure 4 shows the observed number of components of each size in rank 3 at length 15.In this example 99% of the components have size at most 14.
For all 3 11 ≤ L ≤ 15 the component frequency plot looks much like Figure 4, with most values clustered left and one prominent spectrum at multiples of 1  2 ((L − 7) 2 + 11(L − 7) + 30), with a unique largest component of size L−7 2 ((L − 7) 2 + 11(L − 7) + 30) represented by C L−8 BCACaBAA.Myasnikov and Shpilrain [24] proved that components of the Whitehead level-L graph in rank r = 2 have size bounded by a polynomial of degree 2r − 2 in L, see also [19,5], and conjectured that this should be true in higher ranks (see the conjecture and discussion following [24, Corollary 1.2]).The conjecture has been proven in some cases with additional technical hypotheses [20,21].Myasnikov and Shpilrain also, citing experimental evidence, give a specific quartic polynomial for rank 3 bounding the size of the largest component, and a representative of that component.Their representative is in the same Aut(F 3 )-orbit as C L−8 BCACaBAA.We enumerated Aut(F r ) equivalence classes of cyclic subgroup up to length 16 for r ≤ 4. Table 1 shows the resulting number of representatives of each length.Lists of these representatives can be found at: https://www.mat.univie.ac.at/ ~cashen/orgcensus/ Our tools for working with free groups and enumerating equivalence classes are extensions of those developed with Manning for [3].4.2.Computing imprimitivity rank.We compute imprimitivity rank by inductively building Stallings graphs Γ representing finite rank subgroups H of F r containing w as an imprimitive word.Since we are interested in minimal rank subgroups containing w, we may assume that the loop labelled by w traverses every edge of Γ.Furthermore, since we are interested in subgroups  1.The number of length L Aut(F r ) equivalence classes of cyclic subgroup not contained in a proper free factor of F r , for L ≤ 16. containing w as an imprimitive element, we may assume w traverses every edge at least twice.In particular, Γ can contain at most |w| a /2 edges labelled a for each basis element a.These constraints cut down on the number of possible graphs Γ.
Table 2 shows the observed number of equivalence classes of cyclic subgroup of given imprimitivity rank at word lengths 14-16.2. The number of length L Aut(F r ) equivalence classes of cyclic subgroup not contained in a proper free factor, by rank and imprimitivity rank, for 14 ≤ L ≤ 16.
In item (5) we used the function IsHyperbolic (with parameter ε = 1/100) of the GAP [7] package walrus [27] which is based on an algorithm of Holt, et al. [14].The function tries to verify that the RSym curvature distribution scheme defined in [14] succeeds on every van Kampen diagram over the presentation defined by w.This step is crucial, since although small cancellation words are generic, there are still far too many words that evade checks (1)-( 4) to feasibly check with kbmag.Step ( 5) is based on the second author's investigation of the application of RSym and its variants to hyperbolicity of one-relator groups [12].(Another recent application of RSym, to a different class of groups, was conducted by Chalk [4].) The implementation of IsHyperbolic in the version of walrus we used does not capture the full power of the algorithms described in [14]: • IsHyperbolic quits and answers inconclusively if it encounters certain potential bad van Kampen diagrams, but sometimes it can be checked by hand that such a diagram does not really exist.We considered implementing an enhanced RSym algorithm, but it turned out in our experiments that Checks (1)-( 5) caught enough words that kbmag could finish off the rest in a reasonable amount of time.4.4.Word length 17 and beyond.We have described the experiments up to length 16.To extend Theorem 1.1 to length 17 we altered the algorithm.It turns out that hyperbolicity checks (1)-( 5) are fast compared to computing equivalence classes and imprimitivity ranks.Also, the imprimitivity rank computation can be short-circuited to give a faster decision of whether the imprimitivity rank is greater than 2. For length 17 we enumerated SLPCI ± and Whitehead minimal words and checked for hyperbolicity using checks (1)-( 5) first.If some check answered 'hyperbolic' we moved on to the next candidate.Otherwise, we checked if the imprimitivity rank was equal to 2. If so, we moved on to the next candidate.In the remaining cases where hyperbolicity was inconclusive and imprimitivity rank was greater than 2, then we proceeded to check if the word was the the shortlex minimal generator of a cyclic subgroup in its Aut(F r ) equivalence class, and if so verified hyperbolicity with kbmag.
This still took ∼ 4 years of CPU time.The problem is completely parallelizable over the words of fixed length in a free group, so conceivably our programs could be run on a larger cluster to extend the results to length 18 or 19, if there were any particular reason to expect that a counterexample would be revealed at these lengths.We did have a reason to push as far as length 17: in rank 3 at length at most 12, kbmag is not necessary-checks (1)-( 5) always succeed in verifying hyperbolicity.We conjectured, and verified, that the same phenomenon would repeat in rank 4-checks (1)-( 5) suffice up to length 16, but beginning with length 17 additional complexity appears that requires kbmag.This leaves us with the question of whether in rank r all highly imprimitive words of length at most 4r can be verified hyperbolic using only checks (1)-( 5), or, similarly to Theorem 4.3, using some enhancement of RSym?If so, this would improve Corollary 1.4.4.5.Stable commutator length.The commutator length (cl) of an element in the commutator subgroup of a group is the minimal number of factors in the expression of that element as a product of commutators.The stable commutator length (scl) is scl(w) := lim n→∞ cl(w n )/n.Heuer [11,Conjecture 6.3.2]conjectures a generalization of the Duncan-Howie scl-gap theorem [6] saying that scl ≥ (irank − 1)/2.We confirm Heuer's conjecture on our dataset: |w| a = 2, w = aua −1 v and either u and v are conjugate to powers of the same word in F r or u and v are both proper powers in F r .(3) |w| a = 3, w = atauav and ut −1 = z m , vt −1 = z n where z is not a proper power, such that one of the following holds: (a) min(|m|, |n|) = 0 and max(|m|, |n|) > 1.(b) min(|m|, |n|) > 0 and |m| = |n| = 1.(c) min(|m|, |n|) > 0 and m = −n.(d) min(|m|, |n|) > 0 and m = 2n (or n = 2m).(4) |w| a = 3, w = ataua −1 v and t −1 ut = z m , v = z n where z is not a proper power and either |m| = |n| or m = −2n (or n = −2m).Theorem 3.2 ([15, Theorem 4 (3)]

u 1 u 3 a a u 2 Figure 3 .
Figure 3.A graph for |w| a = 4 in the proof of Proposition 1.2.

Figure 4 .
Figure 4. Number of connected components of the SLPCI ± graph by size in rank 3 at length 15.

Figure 5 .
Figure 5.The number of Aut(F 4 ) equivalence classes of cyclic subgroups of length at most 16 by stable commutator length and imprimitivity rank.

•
[14]RSym algorithm in[14]takes a depth parameter d.Success for any d implies hyperbolicity.IsHyperbolic only implements d = 1.•[14]also defines an enhanced version of RSym called RSym + that is not implemented.The second author showed by hand that the enhanced version of RSym often succeeds when IsHyperbolic is inconclusive.For example: Theorem 4.3 ([12, Theorem 5.6]).If w ∈ F 3 has imprimitivity rank 3 and length at most 12 then RSym + succeeds at depth 2.