CONJUGATOR LENGTH IN FINITELY PRESENTED GROUPSpi.math.cornell.edu/.../Conjugator_Length.pdf · CONJUGATOR LENGTH IN FINITELY PRESENTED GROUPS M. R. BRIDSON, T. R. RILEY AND A. W

CONJUGATOR LENGTH IN FINITELY PRESENTED GROUPS

M. R. BRIDSON, T. R. RILEY AND A. W. SALE

Abstract. The conjugator length function of a finitely generated group G maps n to theminimal N such that if u and v are words representing conjugate elements of G with the sumof their lengths at most n, then there is a word w of length at most N such that uw = wv in G.We explain how a group-subgroup pair can be used to construct a finitely presented groupwhose conjugator length function reflects the pair’s subgroup distortion. As a consequence,for all α ∈ N, we exhibit an example of a finitely presented group for which the conjugatorlength function grows like nα. And we give an example for which it grows like 2n. Wesurvey what is known about the conjugator length function for different classes of groupsand we state some open problems.2000 Mathematics Subject Classification: 20F65, 20F10, 20F06Key words and phrases: conjugacy problem, conjugator length

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. A survey of conjugator length bounds and open problems . . . . . . . . . . . . . . . . . . . 5

3. Diagrams and corridors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4. From distortion to conjugator length: proof of Theorem 2. . . . . . . . . . . . . . . . . . 12

5. Conjugator length of BS(1,m): proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . 14

6. Conjugator length of extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7. Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8. Alternative approaches and future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1. Introduction

1.1. Background. Suppose G is a group with a finite generating set A. Write u∼ v whenwords u and v on A±1 represent conjugate elements of G. For such u and v,

CLG(u, v) := min `(w) | uw = wv in G

Date: February 28, 2020.We gratefully acknowledge the financial support of the Royal Society (MRB) and the Simons Foundation

(TRR–Simons Collaboration Grant 318301).1

CONJUGATOR LENGTH IN FINITELY PRESENTED GROUPS 2

where `(w) denotes the number of letters in the word w. The conjugator length functionCLG : N → N is defined so that CLG(n) is the minimum N such that for all words u and vwith `(u)+`(v) ≤ n and u∼ v in G, we have CLG(u, v) ≤ N. The conjugator length functionis also known in the literature as the conjugacy length function. Groups for which it is atmost linear may be said to satisfy the ‘linearly bounded conjugator property.’ We drop thesubscript G, writing CL, when the group is clear from context.

Many estimates on conjugator length functions for particular families of groups appear inthe literature. We will survey these in Section 2 and highlight some open problems.

The conjugator length function is natural and important from a number of points-of-view.

Firstly, it is a means to quantify the difficulty of the conjugacy problem. Given a finitelygenerated group G, the conjugacy problem of Dehn [Deh12] asks for an algorithm whichwith input words u and v on the generators, declares whether or not u ∼ v in G. Theconjugacy search problem asks for an algorithm which on input a pair of words u and v suchthat u∼v outputs a word w with uw = wv in G. The conjugacy problem is a generalisationof the word problem. The word problem asks for an algorithm which on input a wordon A±1 declares whether or not it represents the identity in G. (It is the restriction of theconjugacy problem to the case where v is the empty word.) If there is an algorithm solvingthe word problem for G, then the conjugacy problem is solvable if and only if CL(n) is arecursive function (equivalently, is bounded from above by a recursive function). There arefinitely presented groups which have decidable word problem but undecidable conjugacyproblem [Bok68, Col69, Mil71, OS], and therefore have non-recursive conjugator lengthfunctions.

Cryptographic schemes based on the conjugacy search problem for certain groups G havebeen proposed; a fast growing conjugator length function for G, such as those we willexhibit here, represents an obstruction to naıve (e.g. length-based) attacks on such schemes.See, for example, Section 11.2 of the survey [MSU08].

The conjugator length function is also geometric. Rich geometry arises from the wordproblem for finitely presented groups in the form of isoperimetry: the study of the ge-ometry of discs spanning loops of bounded length in spaces associated to the group. Ina combinatorial guise, these discs are van Kampen diagrams—they portray schemes forreducing words that represent the identity to the empty word by applying defining rela-tions in the group; see [Bri02, BRS07, Ger93] for surveys. For the conjugacy problem forfinitely presented groups G annuli provide the natural analogue to discs—a direction firststudied systematically in Schupp’s thesis [Sch66] and further described in [LS01, Sch68].Whenever words u and v represent conjugate non-identity elements of G, there is an an-nular diagram such that u labels the outer boundary and v the inner boundary. A word wsuch that uw = wv in G labels a path from the inner boundary to the outer boundary. (Moredetails will be given in Section 3.)

The conjugator length function is a group invariant: different finite generating sets for thesame group give '-equivalent CL functions in the following sense. For f , g : N→ N writef g when there exists C > 0 such that f (n) ≤ Cg(Cn + C) + C for all n ≥ 0. Thenwrite f ' g when f g and g f . However, conjugator length is not a quasi-isometryinvariant. Indeed, Collins & Miller [CM77] exhibited groups for which the decidability ofthe conjugacy problem does not pass to or from index-2 subgroups. (Their examples are


finitely presented and have decidable word problem.) So there are quasi-isometric finitelypresented groups whose conjugator length functions are not '-equivalent.

The conjugator length function is also natural in the study of equations over groups. Dio-phantine equations are, of course, extensively studied in number theory both in terms ofalgorithmic decidability results and of bounding the sizes of solutions in terms of the sizesof the coefficients. Bounding the conjugator length function is an analogue of the latter ofthese two directions.

The conjugacy problem is a basic example of an equation over a group: viewing u and vas coefficients, we seek to solve ux = xv for x. Indeed, the conjugacy problem is a fun-damental example of a quadratic equation. These are perhaps the most important class ofequations over finitely presented groups as their solutions can be represented by diagramson surfaces [Sch80] (in the case of ux = xv, the annular diagrams we mentioned earlier).Results quantifying solutions of equations over groups have begun to emerge. Sela [Sel05]provides a quantitative analysis of Makanin–Razborov diagrams which encode solutionsets of systems of equations over limit groups. Kharlampovich and Vdovina [KV12] givelinear estimates for the lengths of solutions of quadratic equations over free groups in termsof the sums of the lengths of the coefficients. And in [KMTV17], working with Mohajeriand Taam, they generalize to torsion-free hyperbolic groups and to toral relatively hyper-bolic groups.

Conjugator length complements conjugacy growth—the number of conjugacy classes in-tersecting the ball of radius n—whose systematic study was initiated by Guba & Sapir in[GS10]. The permutation conjugator length function PCL is a variant on conjugator length.It records the length of a minimal length word that conjugates some cyclic conjugate of uto some cyclic conjugate of v. Proposition 2.2 of [AS16] gives an inequality relating PCLto conjugacy growth. A related geometric invariant arising from annular diagrams is theannular Dehn function introduced by Brick & Corson [BC98]. It gives the minimal upperbound on the area of an annular diagram for u ∼ v as a function of `(u) + `(v).

1.2. Our results. For an element g of a group G with generating set S , we write |g|S (or|g| or |g|G when S is clear from the context) for the length of the shortest word on S ±1

representing g. We abide by the notational conventions [x, y] := x−1y−1xy and xy := y−1xy.

Our main result is the following.

Theorem 1. For all integers m ≥ 1, there is a finitely presented group with CL(n) ' nm.Also, there is a finitely presented group with CL(n) ' 2n.

To put this into perspective, as we will see in our survey in Section 2, for most groups wherea non-trivial estimate on conjugator length appears in the literature, either the estimate islinear or quadratic, or else one has an upper bounds unaccompanied by a matching lowerbound (or, less commonly, vice versa). But ultimately, we see no reason why conjugatorlength functions of finitely presented groups should be any more constrained than Dehnfunctions are [BORS02, SBR02].

Our basic strategy will be to promote subgroup distortion functions to conjugator lengthfunctions. For a subgroup H of a group G with finite generating sets T and S , respectively,the distortion function DistGH : N→ N is

DistGH(n) := max|g|T | g ∈ H with |g|S ≤ n

.


Up to ', DistGH does not depend on the choices of finite generating sets.

Many functions are distortion functions. Indeed, Olshanskii [Ol′97] has shown that ev-ery computable, subadditive function N → N that is bounded above by an exponentialfunction is '-equivalent to a distortion function of an infinite cyclic subgroup of a finitelypresentable group.

After preliminaries concerning van Kampen diagrams, annular diagrams, and their cor-ridors in Section 3, we will explain in Section 4 how distortion functions for central Z-subgroups lead to groups with the corresponding conjugator length functions:

Theorem 2. Let Λ be the group given by the finite presentation 〈A, λ | R〉. SupposeCLΛ DistΛ〈λ〉 for the distortion function DistΛ〈λ〉 of a central infinite cyclic subgroup 〈λ〉 inΛ. Then the conjugator length function of the iterated HNN extension

Σ := 〈 A ∪ λ, p, q, s | R, [p, s], [q, λs], [λ, s] 〉

satisfies CLΣ ' DistΛ〈λ〉.

Theorem 2 should lead to a wide range of examples of conjugator length functions forfinitely presented groups since one expects the inequality CLΛ DistΛ〈λ〉 holds in greatgenerality. However, it appears less than straight-forward to prove this bound in specificexamples.

To obtain Theorem 1 from Theorem 2 we will need suitable Λ and bounds on CLΛ. Ourconstruction of these Λ uses an input group H, an extension G of H, and then obtainsΛ as an extension of G. We begin with Theorems 3 and 4 below, giving the conjugatorlength bounds for our input groups H. For the exponential examples we start with theBaumslag–Solitar group. For the polynomial examples we start with a well-known familyof free-by-cyclic groups in which the defining free-group automorphism is polynomiallygrowing.

Theorem 3 ([Sal16a]). For all m ≥ 2, the conjugator length function of BS(1,m) = 〈a, s |s−1as = am〉 satisfies CL(n) ' n.

Theorem 4 ([BRS]). Let m ≥ 2 and F = F(a1, . . . , am) be a rank-m free group. Defineϕ ∈ Aut(F) by ϕ(ai) = aiai−1 for 2 ≤ i ≤ m and ϕ(a1) = a1. The conjugator length functionof F oϕ Z satisfies CL(n) ' n.

The conjugator length functions of Theorem 1 will originate in

DistBS(1,2)〈a〉 (n) ' 2n and DistFoϕZ

F (n) ' nm.

Theorem 3 is a special case of results of the third author in [Sal16a]. We will give ashort proof in Section 5 to make our treatment self-contained and because it serves asan introduction to more elaborate arguments in Section 6.3. Theorem 4 is proved in acompanion paper to this one, [BRS].

The remaining steps towards building our group Λ for Theorem 2 are in Sections 6.2 and6.3, where we extend H to a group G, and find a bound on the conjugator length functionof G, and in Section 7, where it is explained how to promote subgroup distortion in H tothe distortion of a central Z-subgroup in Λ, which is an extension of G, and how conjugatorlength is affected by taking these extensions.


This article also includes an estimate of the conjugator length of Stallings’ group. Stallings’group is the kernel of any map F(a, b) × F(c, d) × F(e, f ) →→ Z from the product of threerank-2 free groups sending each direct factor onto Z.

Theorem 5. The conjugator length function of Stallings’ group grows quadratically.

Theorem 5 contrasts with the examples we construct to prove Theorem 1. The non-linearbehaviour of the conjugator length functions of Stallings’ group stems from an issue thatseems more inherent to the conjugacy problem—specifically, the sizes of solutions to Dio-phantine equations—as opposed to the “imported” phenomenon of subgroup distortion inTheorems 1 and 2.

We conclude this article with a discussion in Section 8 of possible alternative approaches tothe construction of finitely presented groups with prescribed conjugator length functions.

2. A survey of conjugator length bounds and open problems

We begin with a survey, apologising for any omissions. All the statements of growth ratesof conjugator length functions should be understood to be up to , , or ' as defined inSection 1.

2.1. Hyperbolic groups. The conjugator length function of a non-elementary hyperbolicgroup is linear. Indeed, for a hyperbolic group G and a finite generating set, there existsC > 0 such that if words u and v represent conjugate groups elements, then there are cyclicpermutations u and v of u and v, respectively, and a word w of length at most C such thatuw = wv in G. See, for example, [Lys89] or [BH99, pp. 451–454] . A similar result for thelist conjugacy problem for hyperbolic groups can be found in [BH05]. The linear lowerbound can be seen by exploiting a non-abelian free subgroup.

2.2. Relatively hyperbolic groups. Suppose G is hyperbolic relative to parabolic subgroupsHω, for ω ∈ Ω. Antolın and the third author [AS16] prove that CLG(n) ' maxCLHω

(n) :ω ∈ Ω + n. This extends prior results of Ji, Ogle and Ramsey [JOR10] and complementsBumagin’s study of the complexity of the conjugacy problem [Bum04, Bum15].

2.3. Biautomatic groups, CAT(0) groups, and semihyperbolic groups. The general solu-tion to the conjugacy problems in biautomatic, CAT(0), or more generally semihyperbolicgroups [AB95, BH99, ECH+92, GS04] lead to an exponential upper bound on conjuga-tor length. We are not aware of an example where this cannot be improved to linear. Inparticular, we ask what the optimal upper bound is for biautomatic, and CAT(0) groups.(Kokarev [Kok13] gives bounds, but they depend on conjugacy class.)

There are synchronously combable groups for which the conjugacy problem is undecidable[Bri03]. These examples have combings with quadratic length functions and hence admitcubic upper bounds on their Dehn functions. They do not resolve the following outstandingproblem: do synchronously automatic groups have solvable conjugacy problem [ECH+92,Question 2.5.8]?

The key property of a synchronous (bi)combing is that if we move a pair of beads at unitspeed along the two adjacent combing paths in the Cayley graph then the beads are nevermore than a uniformly bounded distance apart. We can relax this condition by allowing


the speed of travel of one bead along its path to vary, giving us the asynchronous-fellow-travelling property. This leads to analogous definitions of asynchronously (bi)combablegroups and asynchronous (bi)automatic groups—see [ECH+92, Page 153]. Miller [Mil92,Section 7] explains how it follows from results in [BGSS91, Mil71] that there are asyn-chronously automatic groups with unsolvable conjugacy problem; these are of the formFr o F5 and have cubic Dehn functions.

2.4. Groups with quadratic Dehn functions. Hyperbolic groups are precisely the finitelypresentable groups that have the slowest possible growing Dehn functions: they are linear(or equivalently strictly subquadratically). A wide assortment of finitely presented groupshave the next slowest growing Dehn functions—namely, quadratically growing. Many ofthese have conjugator length functions growing at most linearly—for example, the free-by-cyclic groups of Theorem 4. Stallings’ group, the (2n+1)-dimensional integral Heisenberggroups for n > 1, and Thompson’s group F (more on which below), all of which havequadratic Dehn functions (see [DERY09], [All98, OS99], and [Gub06], respectively), haveconjugator length functions all grow at least quadratically (see Theorem 5, Theorem ??,and Section 2.12, respectively). Recently, Olshanskii and Sapir [OS], solving a problemof Rips, constructed the first example with undecidable conjugacy problem, and so non-recursive conjugator length function.

2.5. Bestivina–Brady groups. These and more general kernels have decidable conjugacyproblem by [Bri01, Theorem 3.1]. As for conjugator length functions, in Section 6.1 wewill prove that of Stallings’ group grows quadratically.

2.6. Mapping class groups, 3-manifold groups, right-angled Artin groups (RAAGs), graphproducts, virtually special groups, and hierarchically hyperbolic groups. Hemion gavethe first solution to the conjugacy problem for mapping class groups [Hem79], Preauxfor 3-manifold groups [Pre06, Pre16], and Servatius for RAAGs [Ser89]. Some of theother groups in the list above are covered by the solution to the conjugacy problem forsemihyperbolic groups [AB95] discussed above, including fundamental groups of specialcube complexes, Coxter groups, and certain graph products.

Mapping class groups have linear conjugator length function: Masur & Minsky [MM00]obtained a linear upper bound for pairs of conjugate pseudo-Anosov elements; J. Tao[Tao13] completed the remaining cases. Behrstock & Drutu [BD14] gave new proofs ofthese bounds along with quadratic upper bounds on the conjugator length functions of fun-damental groups of prime 3-manifolds. From Servatius’ solution to the conjugacy problemfor RAAGs [Ser89] it follows that the conjugator length function is at most linear. (Firstcyclically reduce both input words. Then a sequence of cyclic permutations and applica-tions of commutator relations will take one cyclically reduced word to the other, assumingthe input words were conjugate.) More generally, Genevois recently gave bounds on theconjugator length functions of graph products [Gen19].

Generalising [MM00, BD14], Abbott & Behrstock [AB18] established upper bounds onthe conjugator length for certain pairs of elements in hierarchically hyperbolic groups. Inthe special case of conjugate pairs of infinite order elements of virtually compact specialcubical groups, they obtain a linear upper bound.

2.7. Nilpotent groups. Macdonald, Myasnikov, Nikolaev, & Vassileva obtain a polyno-mial upper bound for the conjugator length functions of finitely generated nilpotent groups[MMNV, Thm. 4.6]. Their estimate on the polynomial’s degree is at least cc2

, where c is


the nilpotency class, and is likely far from optimal. Earlier, Ji, Ogle, and Ramsey showedthat the conjugator length functions of 2-step nilpotent groups G grow at most quadrati-cally [JOR10]. They were motivated by the fact that having a polynomial upper boundon conjugator length appears on a list of conditions they had found which together sufficeto imply that a group satisfies the `1(G) Idempotent Conjecture or the `1-Stronger-BassConjecture.

2.8. Polycyclic groups. Remeslennikov [Rem69] proved that finitely generated polycyclicgroups are conjugacy separable, from which it follows that they have decidable conjugacyproblems and therefore recursive conjugator length functions. We are not aware of anyfurther general bounds on the conjugator length functions of polycyclic groups.

2.9. Solvable groups. Kharlampovich constructed three-step solvable groups of derivedlength three that have undecidable word problem (and so undecidable conjugacy prob-lem) [Kha81]. On the other hand, a number of classes of solvable groups have decidableconjugacy problem, including finitely generated metabelian groups [Nos82], and amongthese there are some in which the conjugator length function is known. Examples includethe solvable Baumslag–Solitar groups, lamplighter groups (both linear), and free solvablegroups (at most cubic) [Sal15, Sal16a, Sal16b]. We are not aware of any general boundson conjugator length.

2.10. Lattices in Lie groups. Decidability of the conjugacy problem was proved for arith-metic groups by Grunewald & Segal [GS79], and hence also for higher rank lattices byMargulis Arithmeticity. The behaviour of the conjugator length function for such latticesis almost wholly unknown. This is even true for SLn(Z). See [Sal14] for linear bounds inthe case of real hyperbolic and unipotent elements in semisimple Lie groups.

2.11. Free-by-cyclic groups. In [BRS] we prove the groups of Theorem 4 have linearconjugator length functions and polynomial-time solutions to their conjugacy problems,and we discuss conjugacy for free-by-cyclic groups more widely.

2.12. Thompson’s group F and other diagram groups. The conjugacy problem for F haslong been solved (see [Hig74, Theorem 9.8]). More recently a solution due to Guba &Sapir [GS97] for a family of diagram groups which includes F and has been implementedin [BM14] and [BHMM]. Belk and Matucci [BM] tell us that they can exhibit a familyof words establishing that the conjugator length of F grows at least quadratically and thatthey expect the conjugator length function to be quadratic for F, and indeed for all diagramgroups.

2.13. Permanence results. It is easy to see that the conjugator length function of a directproduct is the maximum of the conjugator length functions of the factors. The same upperbound holds for free products—this is a special case of the result for relatively hyperbolicgroups [AS16].

There are upper bounds on the conjugator length functions of wreath products in [Sal15].For more general group extensions, conjugacy can be complicated. Estimates for upperbounds on conjugator length functions for some types of extension can be found usinggeneral methods in [Sal16a]. Collins & Miller have examples of groups where the solv-ability of the conjugacy problem does not pass to or from index-2 subgroups [CM77]. The


infinite dihedral group D∞ = 〈a, b | a2, b2〉 is an elementary example where passing to a fi-nite index subgroup, namely Z, disrupts conjugator length. The conjugator length functionof D∞ is linear but unbounded, while CLZ(n) = 0.

2.14. Groups with undecidable conjugacy problem. Per our discussion in Section 1, thereare finitely presentable groups with undecidable conjugacy problem and, among these,there are examples which have decidable word problem. What conjugator length functionscan occur for finitely presentable groups with undecidable word problem?

2.15. Relationship to K-theory. The property of having a polynomial bound on the conju-gator length function of a group has been shown to be useful in the study of the K-theoryof discrete groups [JOR10, CY19].

3. Diagrams and corridors

3.1. Van Kampen and annular diagrams. For background beyond the following essen-tials, see [LS01, Sch66, Sch68] or, for a more recent account, [BH99, page 454].

Suppose G is a group with finite presentation P = 〈A | R〉. A van Kampen diagram for aword w on A±1 is a finite planar contractible 2-complex with its edges directed and labelledby elements of A in such a way that around each face one reads a word in R±1 and aroundthe boundary one reads w. Van Kampen’s Lemma states that a word w represents 1 in G ifand only if it admits a van Kampen diagram. A diagram displays how w representing 1 isa consequence of the defining relations R. Let Area(w) the least integer N such that thereis a van Kampen diagram for w with N 2-cells. The Dehn function Area : N→ N of P is

Area(n) := max Area(w) | w = 1 in Γ and `(w) ≤ n .

Equivalently, Area(n) is the minimal N such that if w = 1 in Γ and `(w) ≤ n, then w freelyequals a product of N or fewer conjugates of defining relations or their inverses.

An annular diagram over P for a pair of words u, v is a finite planar combinatorial 2-complex Ω homotopy equivalent to a circle or, in the degenerate case (where u or v is theempty word), a point; its edges are directed and are labelled by elements of A in such away that around each face one reads some word in R±1, and anticlockwise around the twoboundary cycles from vertices ?u and ?v one reads words u and v, respectively.

A van Kampen diagram or annular diagram is reduced when it does not contain an adja-cent pair of ‘cancelling’ 2-cells—that is, a pair of 2-cells with a common edge such thataround the attaching 2-cycles of each cell, starting with that common edge, one reads thesame word, clockwise for one and anticlockwise for the other. A van Kampen diagram orannular diagram that is of minimal area among all diagrams for its boundary word(s) willbe reduced.

Our next lemma is the conjugacy analogue of van Kampen’s Lemma. It asserts that annulardiagrams exist in all meaningful situations in which u ∼ v in G. In many situations thisis straight-forward. Suppose u ∼ v in G. Then w−1uwv−1 = 1 for some word w, and sothere is a van Kampen diagram ∆ for w−1uwv−1. One might hope to obtain an annulardiagram by identifying the boundary arcs of ∆ labelled w and w−1, and to go from anannular diagram to a van Kampen diagram by reversing the process. However such anidentification is problematic when, for instance, the boundary arcs are not disjoint simplepaths. The treatment below is more careful.


Lemma 3.1 (Cf. Lemma 5.2 of [LS01], also [Sch66, Sch68]). If u, v admit an annular dia-gram, then they represent conjugate elements of G. Conversely, if u, v represent conjugateelements of G and do not represent 1 in G, then they admit an annular diagram.

Proof. Suppose Ω, as above, is an annular diagram for u, v. Let ρ be a simple path in the1-skeleton of Ω from ?u to ?v. Let w be the word one reads along ρ. Following ρ−1 from?v to ?u, then following u around one boundary cycle, then returning from ?u to ?v alongρ, and then following v around the other boundary cycle gives a cycle around which onereads w−1uwv−1. This cycle is contractible in Ω, so u ∼ v in Γ. Indeed, roughly speaking,cutting Ω along ρ yields a van Kampen diagram for w−1uwv−1.

For the converse, we expand on the proof of Lemma 5.2 given in [LS01]. They assume uand v do not represent 1 in G, are cyclically reduced in the free group F(A), and are notconjugate in F(A). Then u freely equals a conjugate of v times a product of (one or more)conjugates of elements of R±1. One can then form a van Kampen diagram for u by foldingup a ‘lollipop diagram’ in the manner of a proof of van Kampen’s lemma (as explained in[Bri02], for example). A technicality is that not every term in the product of conjugatesneed give rise to a 2-cell in this van Kampen diagram. Nevertheless the 2-cells will be asubset of those in the lollipop diagram (see Remark 4.2.5 in [Bri02]). So, as u , 1 in G,these faces must include one (and no more than one) with boundary word v. Removingthis face gives an annular diagram for u, v.

Lyndon and Schupp’s hypothesis that u and v are cyclically reduced in F(A) can be re-moved. After all, if u and v are cyclically reduced words that are conjugate to u and v,respectively, then an annular diagram for u, v can be obtained from one for u, v by addingappropriate 1-cells to the inner and outer boundary cycles.

Similarly, in the case where u and v are words that represent conjugate elements in thefree group F(A), they are both conjugate to the same cyclically reduced word z, and soan annular diagram for u and v (with no 2-cells) can be constructed by adding appropriate1-cells to a polygonal circuit labelled z.

The case where u and v represent the identity, excluded in Lemma 3.1, is not significant tothe study of conjugator length functions—after all, then u = v in G, and so CL(u, v) = 0.

In contrast to van Kampen diagrams, there is no label-preserving combinatorial map froman annular diagram for u, v to the Cayley 2-complex of the presentation, unless u and vrepresent the identity.

In terms of annular diagrams, for words u, v such that u∼v, but neither of which representthe identity, CL(u, v) is the minimal L such that there is an annular diagram Ω for u, v asabove with ?u joined to ?v by a path of length at most L in the 1-skeleton of Ω.

3.2. Corridors. Suppose some t ∈ A has the property that every defining relation in whicht±1 occurs contains exactly one t and one t−1. A t-corridor in a van Kampen or annulardiagram is either

• a union of distinct faces (that is, 2-cells) σ1, . . . , σm such that σi shares a t-edgewith σi+1 for i = 1, . . . ,m − 1, or


• a single t-edge in the 1-dimensional part of the diagram; that is, a t-edge not in theboundary of any 2-cell.

The length of a t-corridor is the number of 2-cells it contains. It is a t-annulus (or annulart-corridor) when there is an additional t-edge shared by σm and σ1. A t-corridor in anannular diagram is radial when it connects a t-edge on the inner boundary component toa t-edge on the outer boundary component. A t-annulus is circling when it encloses theinner boundary.

A setting in which t-corridors arise is when t is the stable letter of an HNN-extension. Hereis an instance of this which we will encounter a number of times. Suppose H = 〈A | R〉 isa finitely presented group. Let W be a (finite) set of words on A±1, and define the group Gby

G = H∗〈W〉 = 〈A, t | R, [w, t] ∀w ∈ W〉,which is the HNN-extension of H in which the stable letter commutes with all elements ofthe subgroup 〈W〉. We will be concerned with estimating the conjugator length functionsof such G. Here are some general remarks towards simplifying the task.

The following lemma reduces the task of estimating conjugator length to pairs of wordswhich admit annular diagrams in which all the t-corridors are radial and there are no cir-cling t-annuli.

Lemma 3.2. Let G = H∗〈W〉, as above. Suppose u, v are words on (A∪t)±1 that representnon-identity conjugate elements of G.

(1) There exist a pair of words u′, v′ that admits an annular diagram where all t–corridors are radial and• u ∼ u′ and v ∼ v′ in G,• `(u′) ≤ `(u) and `(v′) ≤ `(v),• CLG(u, v) ≤ CLG(u′, v′) + `(u) + `(v).

(2) If u, v contain no letters t±1, then they are conjugate in H and CLG(u, v) = CLH(u, v).

Proof. If u can be expressed (as a word) as αt±1βt∓1γ where t commutes in G with β, thenremove the t±1 and t∓1 to give αβγ. Repeat until there are no more such t±1, t∓1-pairs toremove. Let u be the resulting word. Obtain v from v likewise.

Now u ∼ v and so they admit an annular diagram Ω by Lemma 3.1. Suppose this annulardiagram admits a t-corridor C connecting two edges on the same boundary component.Suppose this component is the component labelled by u. As above, let ?u denote the vertexon this boundary component such that along the boundary path starting (and ending) at ?u

we read u. Then, by the construction of u, we must have that ΩrC′, where C′ is the union ofthe interiors of the t-edges in C and the faces in C, contains a connected component that isa disk and has ?u in its boundary. Denote this component by ΩD, and the other componentby ΩA (which is an annular diagram). Since t-corridors do not cross, we can choose Cso that ΩD is contains all other t-corridors connecting edges in the boundary componentlabelled u. Let ?u′ be a vertex of ΩA that is also a vertex of the boundary component of Ω

labelled by u (such a vertex exists by the nature of this construction), and let u be the wordread around the boundary component of Ω starting at ?u′ . Then u is a cyclic conjugate ofu. Let u′ be the word obtained from u by the process outlined in the opening paragraphof this proof—remove pairs t, t−1 where the subword between these letters represents anelement of 〈W〉. In particular, u′ = u ∼ u = u in G, and `(u′) ≤ `(u) = `(u) ≤ `(u).


We can construct an annular diagram for u′ ∼ v from Ω by excising the corridors like C.Indeed, the word read along both sides of the corridor C are the same, so we can glue ΩD toΩA by identifying these two segments of each boundary. (There is a technicality here thatoccurs when the boundary of the t-corridor is not simple: one may excise the t-corridorby ‘zipping it up’ as follows. First identify two vertices at opposite ends of a t-edge in thecorridor, and then, making one’s way along the corridor in either direction, identify onepair of edges at a time. The resulting diagram fails to be planar if and only if we reach apair of edges being zipped together that have the same end vertices and bound a region of Ω

not in C. The initial vertices of these edges have already been glued together so the edgeseither form a bigon, or they are both single edged loops. In order to preserve planarity,the subdiagram(s) that this pair of edges encloses should be removed (this is Case 3 ofLemma 8.1 in [BR09]).)

Finally, we can repeat the process above to obtain v′ from v in a similar way, so that u′ ∼ v′

admits an annular diagram in which there are no t-corridors connecting edges on the sameboundary component. The claim that CLG(u, v) ≤ CL(u′, v′) + `(u) + `(v) follows since u′

and v′ are obtained from u and v, respectively, by reduction and cyclic permutations.

Let Ω be of minimal area among all annular diagrams for u′, v′ with no t-corridors connect-ing edges on the same boundary component. Then it must also have no t-annuli (circling ornot), and so have only radial t-corridors as claimed in (1). This is because the words readaround each boundary component of the annulus are the same: we can delete the annulusand glue the two new boundary segments together (bearing in mind the same technicalityas mentioned above).

Under the additional hypothesis that u and v contain no letters t±1 we get u′ = u and v′ = vas words, and there can be no radial t-corridors in Ω, so Ω demonstrates that u ∼ v in Has claimed in (2). Now CLG(u, v) ≤ CLH(u, v) is immediate. For the reverse inequality,suppose w is a minimal length conjugator in G—that is, a minimal length word such thatuwv−1w−1 = 1 in G. Then w can have no subword t±1βt∓1 where t commutes in G with β(the t±1 and t∓1 could be removed, shortening w). Suppose then there was t±1 in w. Then(by Britton’s Lemma) the rightmost t±1 in w and the leftmost t∓1 in w−1 would bookenda subword of uwv−1w−1 with which t commutes. So that t±1 could be removed from w togive a shorter conjugator. We conclude that w is a word on A±1 and so is a conjugator foru ∼ v in H, and therefore CLH(u, v) ≤ CLG(u, v).

Remark 3.3. The same proof works with multiple stable letters that do not interact witheach other. Specifically it works for groups with presentation of the form

〈A, t1, . . . , tk | R, [wi, ti] ∀wi ∈ Wi, i = 1, . . . , k〉

where W1, . . . ,Wk are (finite) sets of words on A±1. The proof of Lemma 3.2 gives usa method for refining an annular diagram, excising corridors until we ultimately have anannular diagram where all ti-corridors are radial. We note that ?u′ (and similarly ?v′ )can be defined by choosing the corridor that determines ΩA and ΩD in the proof aboveamong corridors for any stable letter. The bound on CLG(u, v) in (1) may be replaced byCLG(u′, v′) + `(u) + `(v).


4. From distortion to conjugator length: proof of Theorem 2.

We have a central Z-subgroup 〈λ〉 in a finitely presented group Λ = 〈A, λ | R〉 and wedefine a group Σ by adding further generators and defining relations:

Σ := 〈 A, λ, p, q, s | R, [p, s], [q, λs], [λ, s] 〉 .

We will prove Theorem 2: if CLΛ DistΛ〈λ〉, then CLΣ ' DistΛ〈λ〉. We begin with:

Lemma 4.1. Let Λ′ = 〈 A, λ, s | R, [λ, s] 〉. Then CLΛ′ (n) ≤ CLΛ(n) + 2n for all n ∈ N.

Proof. In light of Lemma 3.2, at a cost of n to conjugator length, it suffices to analyzethe conjugator lengths of pairs of words u, v on (A ∪ λ, s)±1 that represent non-identityconjugate elements of Λ′ and admit an annular diagram Ω in which there is at least ones-corridor and all the s-corridors are radial.

Cutting such an Ω along one side of an s-corridor gives a van Kampen diagram for λmuλ−mv−1

for some cyclic conjugates u and v of u and v (respectively) and some m ∈ Z. But λ is cen-tral in Λ′, so u = v in Λ′, and therefore u and v have cyclic conjugates in common, soCLΛ′ (u, v) ≤ n.

Proof that DistΛ〈λ〉 CLΣ. Suppose w is a word on (A∪λ)±1 equaling λα in Λ for some α ∈Z. Then pq−1 ∼ pwq−1w−1 in Σ via an annular diagram shown schematically in Figure 1.We claim that this and indeed any annular diagram Ω demonstrating pq−1 ∼ pwq−1w−1 inΣ contains circling s-annuli nested to a depth |α|. Then DistΛ〈λ〉(n) ≤ CLΣ(n) will follow.

There is one radial p-corridor and one radial q-corridor in Ω. The word along the sidesof this p-corridor may or may not be freely reduced, but must freely equal sβ for someβ ∈ Z. Likewise, the word along the sides of the q-corridor must freely equal (λs)γ forsome γ ∈ Z. (It may be that the p- or q-corridor has zero area, in which case β = 0 orγ = 0.) But then s−β(λs)γ = w in Σ, so (λs)γ = sβλα. (See Figure 1.) By killing p, q, ands, we see that (λs)γ = sβλα in Σ induces an equality in 〈λ〉 = Z which tells us that γ = α.We can then map onto 〈s〉 = Z by killing all other generators, giving β = γ. No s-corridorsterminate on the boundary of Ω, so all form s-annuli. If an s-corridor emanating fromthe side of the radial p-corridor intersects the radial p-corridor twice, then there would bea sub-disc-diagram whose boundary word is xy−1 for some word x on s±1 read along aportion of the side of that radial p-corridor, and a word y along a portion of the side of thats-corridor (as in the left annular diagram of Figure 2). But y contains no s and so x freelyequals the empty word on account of the retraction Σ →→ 〈s〉 = Z. It follows that there mustbe circling s-corridors nested to a depth |α|.

Proof that CLΣ DistΛ〈λ〉. Suppose Ω is an annular diagram of minimal area for a pair ofwords u and v with u ∼ v in Σ. Let n = `(u) + `(v).

Given Lemma 3.2 and Remark 3.3, we may assume that at least one p or q occurs in u or vand that every p and q that occurs in u or v is part of a radial corridor in Ω. We will provethat then

(1) CLΣ(u, v) ≤ DistΛ〈λ〉(n) + CLΛ′ (n) + 4n.

Since distortion functions are by definition at least linear, the result will then follow byLemma 4.1.


p

p

q

q

w wλα λα

(λs)γ (λs)γ

sβ sβ

Figure 1. An annular van Kampen diagram over the group Σ.

We will consider three cases in turn.

Firstly, suppose Ω contains both a p- and a q-corridor. As p- and q-corridors cannot cross(as no defining relation involves both p and q), such a pair of radial corridors together withportions of the inner and outer boundary of Ω bound a disc sub-diagram Ω0 of Ω withthe following properties. Along the sides of these p- and q-corridors we read words sβ

and (λs)γ, respectively. (Since Ω is of minimal area these words will be reduced and soare powers or s and λs, respectively.) The word one reads around the boundary circuit ofΩ0 is sβu0(λs)−γv0

−1 for some subwords u0 and v0 of some cyclic conjugates of u and v,respectively. Killing p, q and s, we get that λγ equals in Λ a word of length at most n, beingthat it is obtained from v0

−1u0 by deleting some letters. So |γ| ≤ DistΛ〈λ〉(n), and thereforeCLΣ(u, v) ≤ 2DistΛ〈λ〉(n) + n, which is within the required bound (1).

Secondly, suppose Ω has a radial p-corridor C but no q-corridor. There is no s-annulus inΩ since it cannot involve a cell labelled by the relator [q, λs], and so would have the sameword along its inner and outer boundary. It could then be excised and the boundaries glued,reducing the area of Ω. Any s-corridor in Ω starts and ends on the boundary, so there areat most n/2 such corridors. We claim that each s-corridor intersects C at most twice. AsC consists entirely of cells labelled [p, s], this will then imply that CLΣ(u, v) is at most thelength of C (i.e. number of cells it contains or, equivalently, the length of the word alongboth of its sides) plus n, and so at most 2n, which will also be within the bound (1).

To prove that an s-corridor can intersect C at most twice, first note that the word alongthe side of C is a power of s (since Ω is minimal area and so reduced). Therefore if ans-corridor crosses C, it must spiral around the diagram before crossing it again.

Next suppose an s-corridor D intersects C at least three times as illustrated in the rightannulus in Figure 2. Then there is a sub-van Kampen diagram (shown shaded darker in thefigure) whose boundary is labelled by a word x1w1x−1

2 w−12 where w1 and w2 are words on

λ, p±1 read along portions of the sides of D and x1 and x2 are words on s±1 read alongportions of the sides of C. Moreover w1, w2, x1, and x2 are reduced words since Ω isminimal area (and so reduced). As there are no q-corridors, x1w1x−1

2 w−12 = 1 in

Λ′′ := 〈 A, λ, p, s | R, [λ, s], [p, s] 〉.

On account of the retraction Λ′′ →→ 〈s〉 = Z we have that x1 = x2 (as words). And since theletters of wi commute with the letters of xi and Λ′′ also retracts onto the free product of Λ


with Z = 〈p〉 (on killing s), we deduce that w1 = w2 (as words). But then we could cut thisquadrilateral out of Ω and obtain a smaller area annular diagram for u ∼ v by identifyingthe x1 and x2 paths and identifying the w1 and w2 paths, contrary to the minimality of thearea of Ω. We conclude that indeed an s-corridor can intersect C at most twice.

p

p

p

p

x

x1

x2y

s

s

s

Figure 2. Prohibited s-corridors in Ω.

Finally, suppose Ω has a radial q-corridor but no p-corridors. Let s = λs. Then

Σ = 〈 A, λ, p, q, s | R, [λ, s], [p, λ−1 s], [q, s] 〉

since the relations [λ, s] = 1 and [λ, s] = 1 are equivalent. Let u and v be u and v (respec-tively) with λ−1 s in place of each letter s. Then u and v both include the letter q, but notp. We can therefore find a minimal area annular diagram Ω for u ∼ v with respect to thenew presentation for Σ. This annular diagram Ω will have a radial q-corridor, but no p-corridors. The argument now follows that given above for when Ω has a radial p-corridor,but no q-corridor, interchanging the roles of p and q, and replacing s with s. Translatingback to the original presentation by replacing letters s by λs, our length estimates maydouble, leading to the bound CLΣ(u, v) ≤ 4n, which is again within (1).

5. Conjugator length of BS(1,m): proof of Theorem 3

In this section we give a proof of Theorem 3: for all m ≥ 2, the conjugator length functionof

G = BS(1,m) = 〈a, s | s−1as = am〉

satisfies CL(n) ' n.

The relations a±1s = sa±m and s−1a±1 = a±ms−1 can be used to shuffle each letter s in aword on a, s±1 to the front and each s−1 to the end without changing the element of G itrepresents. Thus we can rewrite a word w in the form spaqs−r for some p, q, r ∈ Z withp, r ≥ 0, p + r ≤ `(w) and |q| ≤ m`(w). Similar calculations show that for all k ∈ Z,

∣∣∣ak∣∣∣ is at

most a constant times ln(|k| + 1).

The lower bound CL(n) n is witnessed by the conjugate elements snas−n and a. The obvi-ous conjugator is sn, for which we note |sn| = n. The centraliser of a is spaqs−p | p, q ∈ Z, p ≥ 0.


Thus any conjugator between snas−n and a is of the form sn+paqs−p, and |sn+paqs−p| is min-imised when p = q = 0.

As for proving CL(n) n, suppose u and v are words on a, s±1 such that u ∼ v in G. Weaim to show that CL(u, v) is at most a constant times n := `(u) + `(v).

Rewriting u and v in the form spaqs−r as above, and then replacing each with a suitablecyclic conjugate of itself, we can assume u = sαaβ and v = sγaδ in G for some α, β, γ, δ ∈ Zwith |α| + |γ| + ln(|β| + 1) + ln(|δ| + 1) at most a constant times n. The fact that u and v areconjugate implies that α = γ and, by replacing u and v by their inverses if necessary, wemay assume α ≥ 0.

Suppose we have an element of G expressed as w = spaqs−r where p, q, r ∈ Z and p, r ≥ 0.As α, p, r ≥ 0,

uw = sαaβspaqs−r = sα+paβmp+qs−r

wv = spaqs−r sαaδ = sα+paqmα+δmrs−r

in G. So, as a has infinite order in G, the condition uw = wv in G is equivalent to

(2) q(mα − 1) = βmp − δmr.

In the case α = 0, this reduces to βmp = δmr and we see that there exists k ∈ Z such thatusk = skv in G and |k| is at most a constant times ln(|β| + 1) + ln(|δ| + 1) ≤ n, as required.

By interchanging the roles of u and v and swapping w for w−1 if necessary, we may assumep ≥ r. Then (2) implies that mr divides q and so sp−raq/mr

conjugates u to v, so wemay assume that r = 0. Since for all k ∈ Z we have u(ukw) = (ukw)v in G, we mayreplace w with w′ = ukw, choosing k so that p′ := p + kα satisfies α > p′ ≥ 0. Thenw′ = (sαaβ)k spaq = sp′aq′ in G, where q′ = (βmp′ − δ)/(mα − 1) by (2). But then∣∣∣w′∣∣∣ ≤ p′ + ln

(∣∣∣q′∣∣∣ + 1)< α + ln

(βmp′ + |δ| + 1

),

which is at most a constant times n, as required.

6. Conjugator length of extensions

Our aim is to construct groups Λ that we can plug into Theorem 2 to obtain groups withparticular conjugator length functions. To do this, we begin with a group H that will beeither one of the Baumslag-Solitar or free-by-cyclic groups of Theorems 3 and 4, and takean HNN-extension G of H. Then Λ will be a central extension of G. This is explainedin detail at the start of Section 7. For this to work, we need to obtain the bound on CLΛ

required by Theorem 2. This is achieved in Lemma 7.1, but for that we need to boundCLG, which is the objective of this section.

We begin though by proving Theorem 5: the conjugator length function of Stallings’ groupgrows quadratically. To obtain an upper bound on its conjugator length function, we viewStallings’ group as an HNN-extension of a group H in which the stable letter t commuteswith all elements of a subgroup K. This is similar to how we view the groups G in theconstruction for the proof of Theorem 1. The case of Stallings’ group is the most straight-forward, and so serves as an introduction.

In each case our strategy will be to reduce the argument to estimating CL(u, v) in the specialcase where u ∼ v via an annular diagram with only radial t-corridors (and at least one such


corridor), by applying Lemma 3.2. Then we will take a short word conjugating the imagesof u and v in H under the map that kills t, and will adapt that to a short conjugator in G.

6.1. The conjugator length function of Stallings’ group. We will use the description ofStallings’ group from [BBMS97, Ger95] as the HNN-extension

(3) G = H∗K =⟨

a, b, c, d, t∣∣∣ [a, c], [a, d], [b, c], [b, d], ta = tb = tc = td

⟩of H = F(a, b) × F(c, d) where the stable letter t commutes with all elements of the sub-group K consisting of elements that can be represented by words on a, b, c, d±1 of zeroexponent-sum.

Proof that CLG(n) n2. Suppose u and v are words on a, b, c, d, t±1 representing non-identity conjugate elements in S and `(u) + `(v) ≤ n. We aim to show that CLG(u, v) n2.

In the light of Lemma 3.2 and the fact that CLH(n) ' n (as CLA×B ' maxCLA,CLB), itsuffices to consider only the case where u and v admit an annular diagram ∆ over presenta-tion (3) for G in which there is at least one t-corridor, and all the t-corridors are radial andpair off the t±1 in u with t±1 in v.

u

v

u1 u2 ul

v1 v2 vl

w1 w2 wl−1wl wl wl

tε1 tε2 tεl−1 tεl

tε1 tε2 tεl−1 tεl

Figure 3. A van Kampen diagram for uw = wv in G.

Replacing u and v by suitable cyclic conjugates, we have that u = u1tε1 · · · ultεl and v =

v1tε1 · · · vltεl , for some ε1, . . . , εl = ±1, where u := u1 · · · ul and v := v1 · · · vl are the wordsobtained from u and v by deleting all t±1, and l ≥ 1 is the number of t-corridors in ∆.Moreover, there exist words w1, . . . ,wl on a, b, c, d±1 representing elements of H (wordslabelling the sides of the t-corridors in ∆) such that ui+1wi+1 = wivi+1 in H for all i (indicesmod l). This is illustrated in Figure 3. In fact, by looking at the relators in G involving t, wesee w1, . . . ,wl ∈ K. We also have uwl = wlv in G and uwl = wlv in H = F(a, b) × F(c, d).

Let z : H → Z be the map sending a, b, c, d 7→ 1. So K = Ker z.

Let θ1 ∈ F(a, b), φ1 ∈ F(c, d) be such that θ1φ1 is a minimal length conjugator for u and vin H. In particular, `(θ1) + `(φ1) ≤ n. Let θ2 ∈ F(a, b), φ2 ∈ F(c, d) be such that v = θ2φ2.We also have `(θ2) + `(φ2) ≤ n. The set of all w ∈ H such that uw = wv in H is

W =θ1θ

p2φ1φ

q2

∣∣∣ p, q ∈ Z.

We know there is some such θ1θp2φ1φ

q2 in K, namely wl. So the equation

p z(θ2) + q z(φ2) = −z(θ1) − z(φ1)


has a solution p, q ∈ Z. When such a linear Diophantine equation has a solution, it hasone with |p| , |q| ≤ max |z(θ2)| , |z(φ2)| , |−z(θ1) − z(φ1)| (see e.g. [BFRT89]), and thus with|p| , |q| ≤ n. Define w′ := θ1θ

p2φ1φ

q2 for these p and q, which has length at most a constant

times n2.

Suppose i ∈ 1, . . . , l. Since w ∈ K and u−1i · · · u

−11 wv1 · · · vi = wi ∈ K, we deduce that

z(u−1i · · · u

−11 v1 · · · vi) = 0, and therefore, as w′ ∈ K, we learn that w′i := u−1

i · · · u−11 w′v1 · · · vi ∈

K also. Since t commutes with K, we get uw′ = w′v in G, via a diagram like that in Fig-ure 3, but with wi replaced by w′i and t-corridors replaced by van Kampen diagrams fortεi w′i = w′i t

εi , for all i.

Let F = F(a, b). For the quadratic lower bound on CLG(n) we return to the description ofStallings’ group as the kernel of the sum of the three coordinate maps from D = F × F × Fto Z each of which sends a and b to a generator.

Proof that CLG(n) n2. We take an arbitrary integer n and consider

u =(anb−n, (ab−1)na, (ab−1)na−1

)v =

(b−nan, (ab−1)na, (ab−1)na−1

)w = (an, 1, 1) .

So u and v are elements of G and uw = wv in D. Let C be the centraliser of u in D. The setof all conjugators for u ∼ v in D is

Cw =

g =

((anb−n)ran, ((ab−1)na)s, ((ab−1)na−1)t

) ∣∣∣∣ r, s, t ∈ Z,

which intersects G in g | n + s − t = 0. So u ∼ v in G and any conjugator g ∈ G has either|s| ≥ n/2 or |t| ≥ n/2. Either way, we get that |g|D ≥ n2. It follows that |g|G n2 (withrespect to any fixed finite generating set for G) since |h|G |h|D for all h ∈ G.

6.2. Extensions of a family of free-by-cyclic groups. Fix m ≥ 1 and define

H := F oϕ Z = 〈a1, . . . , am, s | s−1ais = ϕ(ai) ∀i〉

where F = F(a1, . . . , am) is the rank-m free group, ϕ(ai) = aiai−1 for i , 1, and ϕ(a1) = a1.We will promote the conjugator length bound CLH(n) n for H of Theorem 4 to theextension

G = H∗F = 〈a1, . . . , am, s, t | s−1ais = ϕ(ai), t−1ait = ai ∀i〉.

We will need the following statement that has been extracted from Proposition 10.1 of[BRS]. It tells us that if two elements in H are conjugate via an element of the subgroupF, then there is a short conjugator from F. Rather than a linear bound on its length though,it will be quadratic.

Proposition 6.1 ([BRS, Proposition 10.1]). There exists K > 0 such that whenever α, β ∈F and p ∈ Z have the property that there exists γ ∈ F with αspγ = γβsp in H, there existsγ ∈ F such that αspγ = γβsp in H and which in addition satisfies

|γ|H ≤ K(|αsp|H + |βsp|H

)2 .

Proposition 6.2. The conjugator length function of G satisfies CLG(n) n2.


Proof. Given that CLH(n) n, by Lemma 3.2 it suffices to consider the situation illustratedin Figure 3: u and v are words on a1, . . . , ak, s, t±1 representing conjugate elements in G,with `(u) + `(v) ≤ n, and admitting an annular diagram ∆ in which there is at least one t-corridor and all the t-corridors are radial. Moreover, u = u1tε1 · · · ultεl and v = v1tε1 · · · vltεl ,for some ε1, . . . , εl = ±1, where u := u1 · · · ul and v := v1 · · · vl are the words obtained fromu and v by deleting all t±1 and l ≥ 1 is the number of t-corridors in ∆. Also w1, . . . ,wl arewords on a1, . . . , am

±1 such that for all i we have ui+1wi+1 = wivi+1 in H (indices mod l),and defining w := wl, we have uw = wv in G and uw = wv in H.

In a van Kampen diagram for ui+1wi+1 = wivi+1, no s-corridor starts or ends on a part ofthe boundary labelled wi or wi+1. So, for each i, the exponent sum of s in ui+1 equals thatof vi+1.

Let α, β, γ ∈ F and p ∈ Z be such that u = αsp, v = βsp, and γ = w in H. Then αspγ = γβsp

in H. So, for a constant K > 0 as per Proposition 6.1, and because |αsp| ≤ `(u) ≤ `(u), and|βsp|H ≤ `(v) ≤ `(v), and `(u) + `(v) ≤ n, there exists γ ∈ F such that uγ = γv in H and

|γ|H ≤ K(∣∣∣αsk

∣∣∣H +

∣∣∣βsk∣∣∣H

)2≤ Kn2.

As the exponent sum of the s in ui equals that in vi for all i, we have u−11 γv1 ∈ F, and hence

u−12 u−1

1 γv1v2 ∈ F, and so on until we get u−1l · · · u

−11 γv1 · · · vl ∈ F, and so all commute

with t in G. In the light of this, uγ = γv in H implies uγ = γv in G, and we haveCLG(u, v) ≤ |γ|G ≤ |γ|H ≤ Kn2, as required.

6.3. An extension of the Baumslag-Solitar group. In this section we have

H = BS(1, 2) = 〈a, s | s−1as = a2〉.

We know that CLH(n) n by Theorem 3. Before proving a linear bound on conjugatorlength of the corresponding extension of the Baumslag-Solitar group, we give the followingtechnical result that will be of use below.

Lemma 6.3. Suppose p, p′, q, q′, r, r′, λ, µ ∈ Z with p, p′, r, r′ ≥ 0. The following areequivalent

(i) spaqs−raλ = aµsp′aq′ s−r′ in H,(ii) p − r = p′ − r′ and µ = λ2r−p + q2−p − q′2−p′ .

Proof. As s is the stable letter of an HNN-extension, condition (i) implies that p − r =

p′ − r′. For all λ, µ ∈ Z, spaqs−raλ = spaq+λ2rs−r and aµsp′aq′ s−r′ = sp′aq′+µ2p′

s−r′ in H. Socondition (i) is equivalent to in H,

sp−p′aq+λ2rs−(r−r′) = aq′+µ2p′

.

Assume p − r = p′ − r′. If p − p′ ≤ 0, then (i) is further equivalent to

a(q+λ2r)2−(p−p′ )= aq′+µ2p′

,

and if p − p′ ≥ 0, then toaq+λ2r

= a(q′+µ2p′ )2(p−p′).

In either case equating the exponents and rearranging gives the second equation of condi-tion (ii). So, as a has infinite order in H, we have that (i) and (ii) are equivalent.


Proposition 6.4. The conjugator length function of G = 〈H, t | t−1at = a〉 satisfiesCLG(n) n.

Proof. Our proof follows a similar strategy to that for Proposition 6.2. It is simplified bycalculations in 〈a〉 = Z being easier than in F(a1, . . . , am), but is complicated by a 7→ a2 notextending to an automorphism of 〈a〉 = Z, which will impact how we represent elementsof H.

Again, in the light of Lemma 3.2 and the fact CLH(n) n, it suffices to bound the conjuga-tor lengths of pairs of words u and v on a, s, t±1 that admit an annular diagram ∆ in whichthere is at least one t-corridor and all the t-corridors are radial.

Replacing u and v by suitable cyclic conjugates, we have that u = u1tε1 · · · ultεl and v =

v1tε1 · · · vltεl , for some ε1, . . . , εl = ±1, where u := u1 · · · ul and v := v1 · · · vl are the wordsobtained from u and v by deleting all t±1 and l ≥ 1 is the number of t-corridors in ∆.Moreover, there exist k1, . . . , kl ∈ Z (so that the words aki label the sides of the t-corridorsin ∆) such that ui+1aki+1 = aki vi+1 in H for all i (indices mod l). Also, uakl = akl v in G anduakl = akl v in H. This is, again, as in Figure 3, except with aki replacing wi.

As per Section 5, the relation s−1as = a2 implies that a±1s = sa±2 and s−1a±1 = a±2s−1, andso can be used to shuffle all the letters s in a word on a, s, t±1 to the front and all the s−1

to the end without changing the element of H it represents. Accordingly, in H we have u =

spaqs−r and v = sp′aq′ s−r′ , and also ui = spi aqi s−ri and vi = sp′i aq′i s−r′i , where i = 1, . . . , land p, p′, pi, p′i , q, q

′, qi, q′i , r, r′, ri, r′i are integers such that p, p′, pi, p′i , r, r

′, ri, r′i ≥ 0 and

p + p′ + r + r′ ≤ `(u) + `(v),(4)l∑

i=1

(pi + ri + p′i + r′i ) ≤ `(u) + `(v),(5) ∣∣∣q∣∣∣ +∣∣∣q′∣∣∣ ≤ 2`(u)+`(v), and(6)

l∑i=1

(∣∣∣qi

∣∣∣ +∣∣∣q′i ∣∣∣) ≤ 2`(u)+`(v).(7)

As uakl = akl v in H, Lemma 6.3 tells us that p− r = p′− r′ and kl = kl2r−p +q2−p−q′2−p′ ,or equivalently,

q − q′2p−p′ = kl(2p − 2r).

If p , r, we have kl = (q − q′2p−p′ )/(2p − 2r) and then |kl| is at most a constant times2`(u)+`(v) by (4) and (6). And, as for all kl ∈ Z we have that

∣∣∣akl∣∣∣H is at most a constant times

ln(|kl|+ 1), it follows that∣∣∣akl

∣∣∣H , and therefore

∣∣∣akl∣∣∣G, is at most a constant times `(u) + `(v),

as required.

Assume, then, we are in the remaining case: p = r (equivalently, p′ = r′). Then a com-mutes with u, since a commutes with s−masm for all m ∈ Z. (In fact, as uakl = akl v in H,we have u = v in H.)

Given that ui+1aki+1 = aki vi+1 in H, Lemma 6.3 tells us that pi+1 − ri+1 = p′i+1 − r′i+1 and

(8) ki+1 = ki2ri+1−pi+1 + qi+12−pi+1 − q′i+12−p′i+1 .


Define L to be the product of all the 2pi−ri such that i ∈ 1, . . . , l and pi − ri > 0. For aninteger m that we will determine below, define

(9) k′i = ki + mL2(r1−p1)+···+(ri−pi)

for i = 1, . . . , l. Doing so gives us

k′i+1 = k′i 2(ri+1−pi+1) + qi+12−pi+1 − q′i+12−p′i+1 for i = 1, . . . , l − 1,

which implies, by Lemma 6.3, that

(10) ui+1ak′i+1 = ak′i vi+1 in H for i = 1, . . . , l − 1.

We also have that u1ak′1 = ak′l v1 in H for the following reasons. We know that uak′l = ak′l vin H—that is, u1 · · · ulak′l = ak′l v1 · · · vl. In light of (10) this implies that u1 · · · ul−1ak′l−1 vl =

ak′l v1 · · · vl and so u1 · · · ul−1ak′l−1 = ak′l v1 · · · vl−1, and then u1 · · · ul−2ak′l−2 = ak′l v1 · · · vl−2likewise, and so on until finally u1ak′1 = ak′l v1.

We can now deduce that uak′l = ak′l v in G. Indeed, we can build a van Kampen dia-gram for uak′l = ak′l v in G by replacing the sub-diagrams for ui+1aki+1 = aki vi+1 in ∆ bydiagrams for ui+1ak′i+1 = ak′i vi+1 in H, and changing the t-corridors accordingly. Hereis the argument in algebraic terms. Combine u1ak′1 = ak′l v1 with t−1at = a to get thatu1tε1 ak′1 = ak′l v1tε1 in G. Then use (10) in the case i = 1 and t−1at = a to deduce thatu1tε1 u2tε2 ak′2 = ak′l v1tε1 v2tε2 . Then we apply (10) in the case i = 2 and t−1at = a, and so onuntil we have u1tε1 · · · ultεl ak′l = ak′l v1tε1 · · · vltεl , which is uak′l = ak′l v as required.

As p = r, we have∑

i(ri − pi) = 0, and so (9) gives k′l = kl + mL. We complete our proofby choosing m to be the integer such that 0 ≤ k′l < L. Then log2 L ≤

∑li=1 |pi − ri|, which is

at most `(u) + `(v) by (5), and so∣∣∣ak′l

∣∣∣G is at most a constant times `(u) + `(v).

7. Proof of Theorem 1

We return to the general setting, slightly modified from Section 3.2: we have a finitelypresented group H = 〈A ∪ B | R〉 and the HNN-extension

G = H∗〈A〉 = 〈A ∪ B ∪ t | R, t−1at = a ∀a ∈ A〉.

The groups Λ we will feed into Theorem 2 to get Theorem 1 will be central extensions ofsuch G of the form

Λ := 〈A ∪ B ∪ t, λ | R, [t, a] = λ ∀a ∈ A, λ central〉.

The conjugator length function of G acts as an upper bound for that of Λ.

Lemma 7.1. With Λ and G as above, CLΛ(n) ≤ CLG(n) for all n.

Proof. Suppose words u and v represent conjugate elements of Λ and `(u) + `(v) ≤ n.Delete all λ±1 in u and v to obtain words u and v, respectively, representing conjugateelements of G. Let w be a minimal length word such that uw = wv in G. So `(w) ≤ CLG(n)since `(u)+`(v) ≤ `(u)+`(v) ≤ n. We claim that uw = wv in Λ also. We may write u = uλp

and v = vλq for some p, q ∈ Z. We must have p = q since λ generates a Z quotient. Then

uw = uλpw = uwλp = wvλp = wv.

So CLΛ(u, v) ≤ `(w) ≤ CLG(n).


The next three results establish conditions under which the distortion function of 〈A〉 in Hdetermines that of the central Z-subgroup 〈λ〉 of Λ.

Lemma 7.2. With Λ and H as above, DistΛ〈λ〉(n) n + AreaG(n).

Proof. Suppose k ∈ Z and w is a word on (A ∪ B ∪ t, λ)±1 such that λk = w in Λ. Let l bethe exponent sum of the λ in w and let w be w with all the λ±1 deleted. So w = λlw in Λ.

Then w = 1 in G, which means that

(11) w freely equalsN∏

i=1

u−1i riui

for some N ≤ AreaG(`(w)), some words ui on (A ∪ B ∪ t)±1, and some ri ∈ R±1 ∪

[t, a] | a ∈ A ±1.

If ri ∈ R±1, then define ri = ri. And if ri = [t, a]±1, then define ri = riλ∓1. As λ is central in

Λ, (11) implies

w = λMN∏

i=1

u−1i riui in Λ,

for some integer M such that |M| ≤ N. As the ri are relations in Λ, it follows that w = λM

in Λ, and so λk = w = λlw = λl+M in Λ. So k = l + M and

|k| ≤ |l| + |M| ≤ `(w) + AreaG(`(w)) ≤ `(w) + AreaG(`(w)),

and the result follows.

We get a lower bound on DistΛ〈λ〉(n) when the distortion function DistH〈A〉(n) is realized by

positive words on A. This is guaranteed, for example, when A is a singleton set and 〈A〉 =

Z.

Lemma 7.3. Suppose that for all n ∈ N there exist a word un on (A ∪ B)±1 and a positiveword αn on A such that `(un) ≤ n, `(αn) = DistH

〈A〉(n), and un = αn in G. Then n DistH〈A〉(n)

DistΛ〈λ〉(n).

Proof. The word [un, tn] has length at most 4n and equals [αn, tn] in Λ, and so equals λ tothe power of the exponent sum of αn times n. As αn is a positive word this is n `(αn) =

n DistH〈A〉(n) and the result follows.

Corollary 7.4. If AreaG(n) n DistH〈A〉(n) and the distortion function DistH

〈A〉(n) is realizedby positive words on A, then DistΛ〈λ〉(n) ' n DistH

〈A〉(n).

Our next lemma concerns a special case of the G and H discussed above.

Lemma 7.5. Suppose F = F(a1, . . . , am) is free and α : F → F is a monomorphism.Define HNN-extensions of F,

H = 〈 a1, . . . , am, s | s−1ais = α(ai), ∀i 〉,

G = 〈 a1, . . . , am, s, t | s−1ais = α(ai), t−1ait = ai, ∀i 〉.

The Dehn function Area : N→ N of G satisfies Area(n) n DistHF (n).


Proof. Suppose w is a word of length n on a1, . . . , am, s, t±1 which represents the identityin G. Let Ω be a minimal area van Kampen diagram for w over the given presentationfor G. There are no s- or t-annuli in Ω as if there were, then the word around the outerboundary of an ‘innermost’ one would represent 1 in the free group F, and the annulusand the subdiagram it enclosed could be excised and the hole closed up, decreasing thediagram’s area. So the number of s- and t-corridors is at most n/2 since each one connectstwo edges in ∂Ω. Along one side of each such corridor ρ is a word u on a1, . . . , am

±1

which equals some subword w′ of w in G—and, indeed equals in H = F oϕ Z the wordobtained from w′ by deleting all t±1—so, a word of length at most n. Either this wordu or the word along the other side of ρ must be reduced (else Ω would not be a reducedvan Kampen diagram and so not of minimal area). So, as the word along one side of ρequals in H a word of length no more than n, the number of faces in ρ is at most DistH

F (n).The result follows.

The free-by-cyclic groups F oϕ Z of Section 6.2 are of the form of H of Lemma 7.5, as isBS(1, 2). So, given that DistFoϕZ

Fn(n) ' nm and DistBS(1,2)

〈a〉 (n) ' 2n, Corollary 7.4 gives us:

Corollary 7.6. Let H = 〈A ∪ B | R〉 and Λ := 〈H, t, λ | [t, a] = λ∀a ∈ A, λ central〉.

(1) Let m be a positive integer, and H = F oϕ Z be the free-by-cyclic group of Sec-tion 6.2, with A = a1, . . . , am, B = s, and R = s−1ais = ϕ(ai) | i = 1, . . . ,m.Then DistΛ〈λ〉(n) ' nm+1.

(2) Let H be the Baumslag–Solitar group BS(1, 2) of Section 6.3, with A = a, B =

s, and R = s−1as = a2. Then DistΛ〈λ〉(n) ' 2n.

We are now ready to exhibit finitely presented groups which have integer power polynomialand exponential conjugator length functions.

Proof of Theorem 1. If Σ is a non-abelian free group, then its conjugator length functionsatisfies CL(n) ' n. The groups Σ of Theorem 2, when constructed from those groups Λ

of Corollary 7.6, supply all the remaining examples posited in Theorem 1. Theorem 2 re-quires CLΛ(n) DistΛ〈λ〉(n). This condition holds because n2 DistΛ〈λ〉(n) by Corollary 7.6,and CLΛ(n) n2: we have CLΛ(n) ≤ CLG(n) by Lemma 7.1 and CLG(n) n2 by Propo-sitions 6.2 and 6.4. So Theorem 2 tells us that CLΣ ' DistΛ〈λ〉. Corollary 7.6 then identifiesthe growth of DistΛ〈λ〉 as required.

8. Alternative approaches and future directions

It seems natural to believe that finitely presented groups with prescribed conjugator lengthfunctions (and, at the same time, small Dehn functions) might be constructed by means ofthe S -machine technology in [SBR02] and fibre product constructions. Specifically, onemight modify the main construction of [SBR02] by excluding the ‘hub’ in the manner of theproof of Corollary 1.2 in that paper. However, the difficulties to overcome are considerable.

In [OS04] Olshanskii and Sapir embed finitely generated groups H with solvable conjugacyproblem into finitely presented groups G with solvable conjugacy problem, but they do notconstrain the complexity of the conjugacy problem in G in terms of the complexity of theconjugacy problem in H in a quantified way. Sapir identifies the fundamental difficulty at


the end of Section 4 of [Sap06]—solving the conjugacy problem in some cases requires“solving systems of equations in free groups (i.e. the Makanin–Razborov algorithm)”. Un-derstanding the sizes of solutions of such systems is the corresponding difficulty that islikely to impede the search for upper bounds on conjugator length functions of groupsinvolving S -machines.

As a final remark, here are some groups which exhibit subgroups with interesting distor-tion function and which might be used to construct, using the methods of this paper, finitelypresented groups whose conjugator length functions are those interesting functions: iter-ated Baumslag–Solitar groups such as 〈a, s1, s2 | s−1

1 as1 = a2, s−12 s1s2 = s2

1〉, Baumslag’s1-relator group, hydra groups (these three appear in the survey [Ril17]), snowflake groups[BBFS09], and Olshanskii’s examples [Ol′97]. We do not have the upper bounds on theconjugator length functions of all of these groups (that is, analogs of Propositions 6.2 and6.4) that would be required, but we believe that in some cases such bounds are not out ofreach. An additional obstacle with snowflake and hydra groups is that the unusual distor-tion is not witnessed by positive words, and so analogs of the calculations of the lowerbounds on DistΛ〈λ〉 in Lemma 7.4 would break down.

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http://arxiv.org/abs/1503.03888



Martin R. BridsonMathematical Institute, 24-29 St Giles’ Oxford, OX1 3LB, United [email protected], https://people.maths.ox.ac.uk/bridson/

Timothy R. RileyDepartment of Mathematics, 310 Malott Hall, Cornell University, Ithaca, NY 14853, [email protected], http://www.math.cornell.edu/∼riley/

Andrew W. SaleDepartment of Mathematics, University of Hawaii at Manoa, Honolulu, HI 96822, [email protected], http://math.hawaii.edu/∼andrew/

http://www2.maths.ox.ac.uk/~bridson/

http://www.math.cornell.edu/~riley/

http://math.hawaii.edu/~andrew/

Documents

CONJUGATOR LENGTH IN FINITELY PRESENTED GROUPSpi.math.cornell.edu/.../Conjugator_Length.pdf · CONJUGATOR LENGTH IN FINITELY PRESENTED GROUPS M. R. BRIDSON, T. R. RILEY AND A. W