JSJ decompositions and polytopes for two-generator one-relator groups
aa r X i v : . [ m a t h . G R ] J a n JSJ DECOMPOSITIONS AND POLYTOPES FORTWO-GENERATOR ONE-RELATOR GROUPS
GILES GARDAM, DAWID KIELAK, AND ALAN D. LOGAN
Abstract.
We provide a direct connection between the Z max (oressential) JSJ decomposition and the Friedl–Tillmann polytope ofa hyperbolic two-generator one-relator group with abelianisationof rank 2.We deduce various structural and algorithmic properties, likethe existence of a quadratic-time algorithm computing the Z max -JSJ decomposition of such groups. Dedicated to the memory of Stephen J. Pride Introduction
A two-generator one-relator group is a group which admits a presen-tation of the form h a, b | R i . One-relator groups are a cornerstone ofGeometric Group Theory (see, for example, the classic texts in com-binatorial group theory [MKS04] [LS77] or the early work of Mag-nus [Mag30] [Mag32]), and they continue to have applications today, forexample in 3-dimensional topology and knot theory. In particular, mo-tivated by the Thurston norm of a 3-manifold, Friedl–Tillmann intro-duced a polytope for presentations h a, b | R i with R ∈ F ( a, b ) ′ [FT20],which has subsequently been shown to be a group invariant [HK] (an-other proof of this fact follows from the work of Friedl–L¨uck [FL17]combined with a more recent result of Jaikin-Zapirain–L´opez- ´Alvarezon the Atiyah conjecture [JZLA20]).In this paper we focus on hyperbolic one-relator groups, and also analgebraic generalisation of these groups. JSJ-theory rose to prominencedue to Sela’s work on Z max -JSJ decompositions (originally called “es-sential” JSJ decompositions) of hyperbolic groups, which are graph of Mathematics Subject Classification.
Key words and phrases.
One-relator group, JSJ decomposition, polytope. groups decompositions encoding all the “important” virtually- Z split-tings of the group. These Z max -JSJ decompositions are significant be-cause they are a group invariant (up to certain moves), and they governfor example the model theory [Sel09] and (coarsely) the outer auto-morphism group [Sel97] [Lev05a] of the group. Moreover, computing Z max -JSJ decompositions is a key step in the algorithm to solve theisomorphism problem for hyperbolic groups [Sel95] [DG08] [DG11].Our main theorem connects Z max -JSJ decompositions and Friedl–Tillmann polytopes [FT20]. This gives a connected between JSJ de-compositions and these polytopes, as JSJ decompositions are refine-ments of Z max -JSJ decompositions [GL17, Section 9.5]. This connec-tion is significant because it means that, under our assumptions, the Z max -JSJ decomposition of h a, b | R i can be understood simply byinvestigating the relator R , which yields fast algorithmic results (seebelow).We say that a one-ended group has trivial Z max -JSJ decomposition ifit has a Z max -JSJ decomposition which is a single vertex with no edges,and non-trivial otherwise (see Convention 2.7). Theorem A (Theorem 5.4) . Let G be a hyperbolic group admittinga two-generator one-relator presentation P = h a, b | R i with R ∈ F ( a, b ) ′ \ { } . The following are equivalent.(1) G has non-trivial Z max -JSJ decomposition.(2) There exists a word T of shortest length in the Aut( F ( a, b )) -orbit of R such that T ∈ h a, b − ab i but T is not conjugate to [ a, b ] k for any k ∈ Z .(3) The Friedl–Tillmann polytope of P is a straight line, but not asingle point. We illustrate this theorem with an example.
Example 1.1.
Let G be the group defined by the presentation h a, b | ( a b a − b − a − b − ) n i where n > . By using Whitehead’s algorithm, itcan be see that (2) of Theorem A does not hold, and so G has trivial Z max -JSJ decomposition. Alternatively one can consider the Friedl–Tillmann polytope, which we see from Figure 1 is a triangle. There-fore, (3) of Theorem A does not hold, and so G has trivial Z max -JSJdecomposition. The proof of Theorem A splits into two cases: either G is a one-relator group with torsion, or is torsion-free. The difficulty lies in thetorsion-free case, and here Theorem A is a special case of Theorem 5.4,which applies more generally to “restricted Gromov groups”, or “RG SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 3
Figure 1.
To obtain the Friedl–Tillmann polytope, tracethe word ( a b a − b − a − b − ) n on the ab -plane to obtain aclosed loop γ , as in the first diagram (this is independentof n ). Take the convex hull of γ , as in the second diagram;this is a polytope P ′ . Then take the bottom-left corner ofall squares in γ that touch the vertices of P ′ , as in the thirddiagram. The Friedl–Tillmann polytope P is the polytopewith these points as vertices, as in the fourth diagram.Note that the Friedl–Tillmann polytope is in fact a“marked” polytope, but we only care about the shapeso we have omitted these details from this example. Inthe third diagram we took the bottom-left corner of thesquares; this is different from Friedl and Tillmann who takethe centre points of these squares, but this is not an issuebecause the polytope is only well-defined up to translation.groups”. Such groups generalise torsion-free hyperbolic groups, andthey have a pleasingly simple algebraic definition (see below).If G does not have Z abelianisation, so R F ( a, b ) ′ , then the Friedl–Tillmann polytope is less useful (it is always a straight line!). However,we can still ask if (1) and (2) from Theorem A are equivalent. To provethis, we could try to apply the machinery underlying these polytopes,which is that of the universal L torsion, but unfortunately this lies be-yond our current understanding of these L invariants. We can howeveruse more classical techniques to prove this equivalence for one-relatorgroups with torsion. One-relator groups with torsion.
The easier case in the proof ofTheorem A is that of one relator groups with torsion. Such groupsare characterised by having presentations h x | S n i where n >
1. Theyare always hyperbolic [LS77, Theorem IV.5.5] (Brodda has put thisfact in a historical context [Bro20]), and as such these groups are ofgreat interest as test-cases for both hyperbolic groups and one-relatorgroups. For example, one-relator groups with torsion are residually
GILES GARDAM, DAWID KIELAK, AND ALAN D. LOGAN finite [Wis12], while it is an open problem of Gromov whether or notall hyperbolic groups are residually finite, and one-relator groups withtorsion are coherent [LW21], while it is an open problem of G. Baumslagwhether or not all one-relator groups are coherent.Our results for one-relator groups with torsion include the case of R F ( a, b ) ′ . A primitive element of F ( a, b ) is an element which ispart of a basis for F ( a, b ). Theorem B (Theorem 3.4) . Let G be a group admitting a two-generatorone-relator presentation P = h a, b | R i where R = S n in F ( a, b ) with n > maximal and S ∈ F ( a, b ) is non-empty and non-primitive. Thefollowing are equivalent.(1) G has non-trivial Z max -JSJ decomposition.(2) There exists a word T of shortest length in the Aut( F ( a, b )) -orbit of S such that T ∈ h a, b − ab i but T is not conjugate to [ a, b ] ± . The conditions on S are because Z max -JSJ decompositions are onlymeaningful for one-ended groups, and if S is primitive or empty thenthe group defined by h a, b | S n i is not one-ended (it is a free product ofcyclic groups, Z ∗ C n or Z ∗ Z ). In contrast, all the groups in TheoremA are one-ended. Forms of Z max -JSJ decompositions. The following corollary de-scribes the Z max -JSJ decompositions of the groups from Theorems Aand B as HNN-extensions of one-relator groups; this description is es-sentially a reworking of (2) from these theorems.In the torsion-free case, a result of Kapovich and Weidmann describessuch Z max -JSJ decompositions as HNN-extensions [KW99a, TheoremA]; Corollary C further says that the base groups are one-relator groups,which is new and surprising. In the corollary, T ( a, b − ab ) correspondsto the word T in the above theorems, and we view R as a power S n for n > R like this because Theorem B deals withthe root S of the relator R = S n . We lose nothing by doing this as ifthe group in the corollary is torsion-free then n = 1 and R = S . Weuse | W | to denote the length of a word W ∈ F ( x ). Corollary C (Corollary 6.1) . Let the group G and the presentation P be as in Theorem A or B, and write R = S n for n > maximal.Suppose that G has non-trivial Z max -JSJ decomposition Γ . Then thegraph underlying Γ consists of a single rigid vertex and a single loopedge. Moreover, the corresponding HNN-extension has vertex group SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 5 h a, y | T n ( a, y ) i , stable letter b , and attaching map given by y = b − ab ,where T ( a, b − ab ) is the word T from the theorem. Finally, | T | < | S | . In the case of Theorem B, the base group h a, y | T ( a, y ) i also satisfiesthe conditions of Theorem B. Therefore, the “ | T | < | S | ” conditiongives a strong accessability result, similar to the more general resultof Louder–Touikan [LT17], but where all the groups involved are one-relator groups. If we are in the case of Theorem A then T ( a, y ) maynot be in the derived subgroup, and so we do not obtain the analogousresult. Example 1.2.
Set G = h a, b | ( a − b − a b ) n i , n > . Clearly, G isan HNN-extension of the group H = h a, y | ( a − y ) n i , with attachingmap y = b − ab . In fact, this obvious decomposition of G as an HNN-extension corresponds to its Z max -JSJ decomposition, by Corollary C. This example illustrates an important point: we can spot Z max -JSJdecompositions of the groups in Theorem A or B, since if a decom-positions “looks” like a Z max -JSJ decomposition then it is indeed a Z max -JSJ decomposition (see Theorems 2.8 and 3.2). Now, the group G from Example 1.2 can also be viewed as an HNN-extension of thegroup H ′ = h a, z | ( a − z ) n i , with attaching map z = b − a b . This isnot a Z max -splitting (as h z i is not maximal in G ), but it “looks” like itis a JSJ decomposition of G . However, we were unable to prove ana-logues of Theorems 2.8 and 3.2 for JSJ decompositions, and so we areunable to conclude that this splitting is in fact a JSJ decomposition. Algorithmic consequences.
Computing Z max -JSJ decompositionshas implications for the isomorphism problem, gives information aboutouter automorphism groups, and is potentially an important first stepfor many algorithmic questions about the elementary theory of hyper-bolic groups [DG08]. However, current algorithms for computing JSJdecompositions or Z max -JSJ decompositions have bad computationalcomplexity.For example, algorithms of Barrett [Bar18], Dahmani–Groves [DG08],Dahmani–Guirardel [DG11], and Dahmani–Touikan [DT19] work invery general settings, but when applied to hyperbolic groups they eachhave no recursive bound on their time complexity. For the algorithmsof Barrett, Dahmani–Groves, and Dahmani–Guirardel, this is becausethey require computation of the hyperbolicity constant δ , and there isno recursive bound on the time complexity for computing δ (as hyper-bolicity is undecidable). The algorithm of Dahmani–Touikan requiresa solution to the word problem, but all known general solutions forhyperbolic groups require preprocessing for which there is no recursive GILES GARDAM, DAWID KIELAK, AND ALAN D. LOGAN bound on the time complexity (for example, computing an automaticstructure or Dehn presentation). Even if we assume an oracle givesus δ , these algorithms each proceed by detecting a splitting and thensearching blindly through all presentations of the given group to findsome presentation which realises the detected splitting, and this pro-cedure clearly has an awful time complexity. So far as the authors areaware, the only algorithm which computes the JSJ decompositions fora class of hyperbolic groups and which has a known (reasonable) boundon its time complexity is due to Suraj Krishna [SK20], but here thecomputed bound is doubly exponential and is not in general applicableto one-relator groups.Theorems A and B can be applied to give fast algorithms for both de-tecting and computing Z max -JSJ decompositions in our setting. Firstly,there is a quadratic-time algorithm to find the Z max -JSJ decomposi-tion of a given group (essentially, the algorithm is to compute all of theshortest possible element of the Aut( F ( a, b ))-orbit of the relator R ). Corollary D (Corollary 6.2) . There exists an algorithm with input apresentation P = h a, b | R i of a group G from Theorem A or B, andwith output the Z max -JSJ decomposition of G .This algorithm terminates in O ( | R | ) -steps. In the case of Theorem A, the polytope allows us to detect a non-trivial Z max -JSJ decomposition in linear time (essentially, the algorithmis to draw the Friedl–Tillmann polytope). Corollary E (Corollary 6.3) . There exists an algorithm with input apresentation P = h a, b | R i of a group G from Theorem A, and withoutput yes if the group G has non-trivial Z max -JSJ decomposition and no otherwise.This algorithm terminates in O ( | R | ) -steps. Detecting non-trivial Z max -JSJ decompositions is useful in its ownright. We demonstrate this with the following application of CorollariesD and E. Corollary F (Corollary 6.4) . There exists an algorithm with inputa presentation P = h a, b | R i of a group G from Theorem A or B,and which detemines if the outer automorphism group of G is finite,virtually- Z , or GL ( Z ) .If G is as in Theorem A, this algorithm terminates in O ( | R | ) -steps.Else, it terminates in O ( | R | ) -steps. Relationships between outer automorphism groups.
Write G k SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 7 for the group defined by h a, b | S k i , where k > S isfixed. In general there is very little relationship between Out( G ) andOut( G n ) for n >
1. For example, if S = b − a ba − then G = BS(2 , G n ) for n > Z [Log16b]. In contrast, Theorems Aand B imply that if S ∈ F ( a, b ) ′ and G is hyperbolic then the trivialityof the Z max -JSJ decomposition of G k depends solely on the word S ; theexponent k is irrelevant. We can then apply the relationship between Z max -JSJ decompositions and outer automorphism groups to see thatOut( G m ) and Out( G n ) are commensurable for m, n >
1. Our nextcorollary says that this relationship is much stronger.
Corollary G (Corollary 6.5) . Write G k for the group defined by h a, b | S k i , where k > is maximal. If S ∈ F ( a, b ) ′ and G is hyperbolic then:(1) Out( G m ) ∼ = Out( G n ) for all m, n > .(2) Out( G n ) embeds with finite index in Out( G ) . The isomorphism of (1) is essentially already known, and holds un-der the more general restriction that S is non-primitive [Log16b]; weinclude it for completeness. All the maps here are extremely natural,and in particular the embedding Out( G n ) ֒ → Out( G ) is induced bythe natural map G n → G with kernel normally generated by S . What about JSJ decompositions?
Most of our main results canbe stated in terms of JSJ decompositions, but we state them in termsof Z max -JSJ decompositions as this gives a smoother exposition. Noth-ing is lost by doing this as Z max -JSJ decompositions encode all theinformation we are interested in and motivated by (outer automor-phism groups, isomorphism problem), and in fact we gain CorollaryD, which we are unable to rephrase in terms of JSJ decompositions.Additionally, when we rephrase Corollary C in these terms then thephrase “ T ( a, b − ab ) is the word T from the theorem” is replaced with“ T ( a, b − ab ) is some word”, so we lose the nice connection with thetheorems. Restricted Gromov groups.
It is unclear how intrinsic hyperbolic-ity is to this paper. Indeed, an analogue of Theorem A will hold for anyclass of one-relator groups for which Theorem 4.1 on Friedl–Tillmannpolytopes is applicable, and which satisfies a certain description of split-tings given by Kapovich–Weidmann [KW99a, Theorem 3.9]. (Indeed, Z max -JSJ decompositions are not required. They just give context anda convenient language for our results, which instead can be stated interms of “essential Z -splittings”, as in Theorem 2.8.) GILES GARDAM, DAWID KIELAK, AND ALAN D. LOGAN
In their aforementioned work, Kapovich–Weidmann’s main technicalresult is for a more general class of groups than torsion-free hyperbolicgroups, and most of the above results extend to this class too: A restricted Gromov group , or
RG group , is a finitely generated, torsion-free group G such that for all g, h ∈ G either h g, h i is cyclic or thereexists some i ∈ Z such that h g i , h i i is free of rank two, and everyelement of G is contained in a maximal cyclic subgroup of G . Torsion-free hyperbolic groups are RG groups, as are their finitely generatedsubgroups (so there are finitely presented RG groups which are non-hyperbolic [Bra99]).This paper proves results for RG groups, and obtains the above re-sults as special cases. The only exceptions are Theorem B (as RGgroups are torsion-free), and Corollaries F and G, on outer automor-phism groups.Therefore, it seems like hyperbolicity is not intrinsic to this paper.On the other hand, it is a famous conjecture of Gersten that everyone-relator group with no Baumslag–Solitar subgroups is hyperbolic.RG groups contain no Baumslag–Solitar subgroups, and so Gersten’sconjecture suggests that hyperbolicity is intrinsic to this paper.On balance, because the results of this paper are so strong and read-ily generalise to RG groups, we claim that our results give evidencetowards the following weak form of Gersten’s conjecture: Conjecture 1.3 (Weak Gersten conjecture) . Every one-relator RGgroup is hyperbolic.
Outline of the paper.
In Section 2 we build a theory of Z max -JSJdecompositions applicable to two-generated RG groups, and we provea useful theorem which allows us to “spot” Z max -JSJ decompositionsof two-generated RG groups (Theorem 2.8). In Section 3 we proveTheorem B. In Section 4 we prove our main technical result involvingpolytopes, Theorem 4.1, which takes as input a pair of “compatible”presentations of some group G , one having the form h a, b | R i with R ∈ F ( a, b ) ′ and the other the form h a, b | s i with s ⊂ h a, b − ab i , andproves that the Friedl–Tillmann polytope of G is a straight line. InSection 5 we prove our main theorem, Theorem 5.4, which is a generalform of Theorem A incorporating RG groups. In Section 6 we provegeneral versions of Corollaries C–F, again incorporating RG groups. Acknowledgements.
We are very grateful to Nicholas Touikan forseveral helpful comments and suggestions. This work has received
SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 9 funding from the European Research Council (ERC) under the Euro-pean Union’s Horizon 2020 research and innovation programme, Grantagreement No. 850930, from the UK’s Engineering and Physical Sci-ences Research Council (EPSRC), grant EP/R035814/1, and from theDeutsche Forschungsgemeinschaft (DFG, German Research Founda-tion) – Project-ID 427320536 – SFB 1442, as well as under Germany’sExcellence Strategy EXC 2044–390685587, Mathematics M¨unster: Dyn-amics–Geometry–Structure.2.
JSJ-theory and RG groups
JSJ-theory plays a key role in the theory of hyperbolic groups [RS97][Sel95] [Lev05a] [Sel09] [DG11]. Mirroring the JSJ decomposition of3-manifolds, the JSJ and Z max -JSJ decompositions of a one-ended hy-perbolic group are a graph of groups decompositions where all edgegroups are virtually- Z , and are (in an appropriate sense) unique if cer-tain vertices, called “flexible vertices”, are treated as pieces to be leftintact and not decomposed. Z max -JSJ decompositions were originallycalled “essential JSJ decompositions” by Sela, and they contain allthe important information used in the applications of JSJ-theory citedabove. We refer the reader to Guirardel–Levitt’s monograph for fur-ther background and motivation [GL17], in particular their Section 9.5on Z max -JSJ decompositions.In this section we define JSJ decompositions and Z max -JSJ decompo-sitions. We then build a theory of Z max -JSJ decompositions relevant tothe RG groups considered in this paper. Our main result here is The-orem 2.8, which can be summarised by: when a decomposition lookslike a Z max -JSJ decomposition, it is indeed a Z max -JSJ decomposition. Z max -JSJ decompositions are a certain kind of graph of groups split-ting. In a graph Γ, every edge e , with intial and terminal vertices ι ( e ) and τ ( e ), respectively has an associated reverse edge e such that ι ( e ) = τ ( e ) and τ ( e ) = ι ( e ). We use Γ to denote a graph of groups,with a connected underlying graph Γ, associated vertex groups (or ver-tex stabilisers ) { G v | v ∈ V Γ } and edge groups (or edge stabilisers ) { G e | e ∈ E Γ , G e = G e } , and set of monomorphisms θ e : G e → G ι ( e ) .For notation and background on graphs of groups we refer the readerto Serre’s book [Ser03]. JSJ decompositions. A VC -tree of G is a tree T equipped with anaction of G whose edge stabilisers are virtually- Z . We define Z -trees analogously, by requiring the edge stabilisers to be infinite cyclic. Asubgroup H G is elliptic in T if it fixes a point in T (and hence is contained in a vertex stabiliser of T ), and universally elliptic if itis elliptic in every VC -tree of G . A VC -tree T is universally elliptic ifits edge stabilisers are universally elliptic. A VC -tree T dominates the VC -tree T ′ if every subgroup of G which is elliptic in T is elliptic in T ′ .A VC -JSJ tree T of G is a VC -tree such that:(1) T is universally elliptic, and(2) T dominates every other universally elliptic VC -tree T ′ .The quotient graph of groups Γ = T /G is a VC -JSJ decomposition of G . All the trees we consider are VC , so we shall abbreviate “ VC -tree”,“ VC -JSJ tree” and “ VC -JSJ decomposition” to simply tree , JSJ tree and
JSJ decomposition , respectively.A vertex of a tree T is elementary if its group is virtually- Z , whilea non-elementary vertex is rigid if its group is universally elliptic, and flexible otherwise. Flexible vertices are the “pieces to be left intactand not decomposed” mentioned above and understanding their vertexgroups plays a central role in JSJ-theory, although in this paper flexiblevertices play a very minor role. Z max -JSJ decompositions. A Z max -subgroup of G is a maximalvirtually- Z subgroup of G with infinite centre. A Z max -tree of G isa tree T equipped with an action of G where edge stabilisers are in Z max . A Z max -tree T is universally Z max -elliptic if its edge stabilisersare elliptic in every Z max -tree of G . A Z max -JSJ tree T of G is a Z max -tree such that:(1) T is universally Z max -elliptic, and(2) T dominates every other universally Z max -elliptic tree T ′ .The quotient graph of groups Γ = T /G is a Z max -JSJ decomposition of G . In this context, a non-elementary vertex of a Z max -tree is rigid ifits group is universally Z max -elliptic, and flexible otherwise.The theory of Z max -JSJ decompositions for hyperbolic groups is es-pecially powerful because there are canonical structures: for each group G there is a unique “deformation space” of Z max -JSJ trees, and thereis a canonical Z max -JSJ tree [GL17, Section 9.5]. We wish to extendthese results to Z max -JSJ trees of RG groups. To facilitate this, weconsider CSA Z groups.2.1. CSA Z groups. A CSA group is a group G with the property thatevery maximal abelian subgroup of G is malnormal. A CSA Z group is aCSA group where every non-trivial abelian subgroup is infinite cyclic.We are interested in CSA Z groups because, as we shall soon see, RGgroups are CSA Z , and because we can readily study their Z max -JSJ SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 11 decompositions. The starting point for this study is the existing theoryof JSJ decompositions of CSA groups [GL17, Theorem 9.5].We have kept this section short, and only prove the few results weneed. In particular, we focus on CSA Z groups rather than CSA groups,and we only prove that Z max -JSJ decompositions exist under certainconditions. We suspect that this section can be extended into a com-plete Z max -JSJ theory for CSA groups. As these results are alreadyknown for hyperbolic groups, the reader who is only interested in hy-perbolic groups may skip this section. Existence.
We start by proving that certain CSA Z groups have Z max -JSJ decompositions. To do this we follow the analogous existence prooffor hyperbolic groups [GL17, Section 9.5] [DG11, Section 4]. We re-strict ourselves to those CSA Z groups whose JSJ decompositions haveno flexible vertices. This sidesteps subtleties in the proof for hyperbolicgroups (see the discussion following Remark 9.29 of [GL17], or [DG11,Remark 4.13]), and is the only situation used in this paper. Our ex-istence result actually gives the existence of a canonical Z max -JSJ tree T Z max which has certain nice properties (e.g. invariant under groupautomorphisms).Let C be a non-trivial cyclic subgroup of a CSA Z group G . Zorn’slemma guarantees the existence of a maximal abelian subgroup A of G containing C . Since G is CSA Z , the group A is infinite cyclic. Let A ′ beanother maximal abelian subgroup of G containing C . Take a ∈ A ′ . As a centralises C , we have that C is a non-trivial subgroup of aAa − , andso by malnormality of A we have that a ∈ A . Hence A ′ = A , and wehave shown that if C is a non-trivial cyclic subgroup of a CSA Z group G then there exists the unique maximal cyclic subgroup containing C ;we will denote this subgroup by ˆ C .Given two distinct edges e , e of a tree T such that ι ( e ) = ι ( e )then we can fold these edges together by identifying τ ( e ) to τ ( e ), e to e , and the reverse edges e to e , and we obtain a new tree T / [ e = e ] [Sta83]. If G acts on T and E is some set of edges of T with the same initial vertex then an orbit-fold of the edges E is thetree T / [ G y E ] obtained by folding all the edges in g · E for all g ∈ G .This ensures that G acts on the new tree T / [ G y E ]. Proposition 2.1.
Let G be a one-ended CSA Z group whose JSJ treeshave no flexible vertices. Then G has a canonical Z max -JSJ tree T Z max ,and this tree has no flexible vertices.Proof. The group G admits a canonical JSJ tree T c , namely the (abelian)“tree of cylinders” of Guirardel–Levitt [GL17, Theorem 9.5]. This is a bipartite tree with vertex set V ( T c ) ⊔ V ( T c ), where V ( T c ) consists ofrigid vertices and V ( T c ) consists of elementary vertices. Note that edgegroups are subgroups of finite index in the vertex groups of elementaryvertices, essentially since non-trivial subgroups of Z are of finite index.Let T ′ c be the the quotient of T c by the smallest equivalence relationsuch that, for all edges e of T and all h ∈ ˆ G e , we have h · e ∼ e . We shalluse a contructive definition of T ′ c , which is equivalent [GL17, Proof ofLemma 9.27] (see also the “ Z max -fold” of [DG11, Section 4.3]). The tree T ′ c is the G -tree constructed from T c by iteratively orbit-folding togetheredges as follows: Let e be an edge such that G e = ˆ G e . If one of theendpoints of e is fixed by ˆ G e , then orbit-fold together the set of edges inthe ˆ G e -orbit of e . If not, then as ˆ G e is cyclic with generator x , say, suchthat x n fixes an edge for some n > G e = h x n i ), we havethat the subtree Fix( ˆ G e ) fixed by ˆ G e is non-empty [Ser03, Proposition25]. Let e ′ be the first edge in the shortest path joining Fix( ˆ G e ) to e ,so ι ( e ′ ) ∈ Fix( ˆ G e ) but ι ( e ′ ) Fix( ˆ G e ). Then orbit-fold together the setof edges in the ˆ G e ′ -orbit of e ′ . The first description of T ′ c gives us that,because T c is canonical, the tree T ′ c is canonical. Now, let T Z max be theminimal G -invariant subtree of T ′ c ; this tree again is canonical, and weshall see that it is a Z max -JSJ tree with the required properties.The tree T Z max is universally Z max -elliptic [GL17, Lemma 9.27.(3)].It is also a Z max -tree: Orbit-folds preserve the bipartite structure of thegraph, and so every edge of T Z max ⊆ T ′ c is adjacent to a non-elementaryvertex, and hence every edge has stabiliser which is a maximal infinitecyclic subgroup of G [DG11, Lemma 4.10.1].We now prove that T Z max dominates every other universally Z max -elliptic tree S , and that it has no flexible vertices. If v ′ is an elementaryvertex of T Z max ⊆ T ′ c then consider a vertex v ∈ T c such that v ′ is theimage of v under the natural map T c → T ′ c . As orbit-folds preservethe group action, the vertex v satisfies G v G v ′ , so G v ′ contains anedge stabiliser G e of T c with finite index. As T c is a JSJ-tree, G e iselliptic in S , and so G v ′ is elliptic in S [GL17, Remark 9.26]. If v ′ is notan elementary vertex, then there exists a vertex v ∈ T c with adjacentedges e , . . . , e n such that G v ′ has stabiliser a multiple amalgam (i.e.a tree of groups) G v ′ = G v ( ∗ G ei ˆ G e i ) ni =1 [DG11, Lemma 4.10.3]. Thegroup G v is elliptic in S as v is rigid, and so G v ′ is also elliptic in S [GL17, Remark 9.26]. This also implies that the vertex v ′ is rigid,and the result follows. (cid:3) Uniqueness.
JSJ trees or Z max -JSJ trees are not in general unique, SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 13 but instead unique up to certain moves. More formally: The defor-mation space of a tree T is the set of trees T ′ such that T dominates T ′ and T ′ dominates T . Equivalently, two trees are in the same de-formation space if and only if they have the same elliptic subgroups.Elements of a deformation space are connected by moves of a certaintype [GL17, Remark 2.13] [CF09, Theorem 1.1]. Therefore, the follow-ing says that Z max -JSJ decompositions of a CSA Z group are unique, upto the moves cited above. The proof is clear because, by the definitionof Z max -JSJ trees, they pairwise dominate one another and so all lie inthe same deformation space. Proposition 2.2.
Let G be a finitely generated one-ended CSA Z group.If G has a Z max -JSJ tree, then all Z max -JSJ trees of G lie in the samedeformation space D Z max . RG groups.
The remainder of this section relies on work ofKapovich–Weidmann, who studied cyclic splittings of two-generatedtorsion-free hyperbolic groups, but their main result is for RG groupsas defined in the introduction [KW99a, Theorem 3.9]. Within the classof torsion-free groups, we have the following:hyperbolic ⊂ RG ⊂ CSA Z The first inclusion, that torsion-free hyperbolic groups are RG, is dueto Kapovich–Weidmann [KW99a, Lemma 3.6.1]. The second inclusionis similar to one of their observations [KW99a, Proposition 3.4.3], butwe include it for completeness. This second inclusion means that thereexists a JSJ-theory of RG groups.
Lemma 2.3.
RG groups are CSA Z .Proof. Let G be an RG group, and A an abelian subgroup of G . Everypair of elements of A generate a cyclic subgroup, and so A is locallycyclic. As every element of G is contained in a maximal cyclic subgroupof G , we further have that A is cyclic, generated by an element a ∈ A .Finally, A is malnormal as if g ∈ G is such that g − a m g = a n where m, n = 0, then no non-trivial powers of g and a generate a non-cyclicfree subgroup, and so h a, g i is cyclic, hence g ∈ A by maximality of A . (cid:3) As RG groups are CSA Z , they have JSJ decompositions. However,we wish to study their Z max -JSJ decompositions. Existence of thesedecompositions for two-generated one-ended RG groups is obtained byapplying Proposition 2.1 to the following. The proof uses the canonicalJSJ tree T c of G [GL17, Theorem 9.5] which we previously used in theproof of Proposition 2.1. However, here we have no restriction on flexible vertices so our description changes slightly: The canonical JSJtree T c is a bipartite tree with vertex set V ( T c ) ⊔ V ( T c ), where V ( T c )consists of rigid and flexible vertices, and V ( T c ) consists of elementaryvertices. Lemma 2.4.
Let G be a two-generated one-ended RG group. Then noJSJ tree of G has a flexible vertex.Proof. By Lemma 2.3, G is CSA Z and so has a (canonical) JSJ tree T c [GL17, Theorem 9.5]. This tree can be altered to give a new tree T c which has no elementary vertices: on the level of T c /G , slide everyelementary vertex of degree 1 into the unique adjacent vertex group,while as G is an RG group every elementary vertex v of T c which isof degree > e with ι ( e ) = v and such that G e = G v , so slide every such elementary vertex v into the vertex τ ( e ).The quotient graph T c either consists of a single vertex and no edges,or a single vertex and a single loop edge [KW99a, Theorem 3.9].Then by reversing the above moves, and using the bipartite structureof T c , we have that T c /G consists of a single non-elementary vertex v ,possibly an elementary vertex u with incident edges e, f each connectedto v (this corresponds to the loop edge), and possibly some elementaryvertices of degree one each connected to v .It follows that the vertex group G v is two-generated [KW99a, Propo-sitions 3.7 & 3.8] (we apply one of these citations for each elementaryvertex). Suppose v is flexible. Then G v is the fundamental groupsof a compact surface [GL17, Theorem 9.5], and hence is a Fuchsiangroup. This is a contradiction as every two-generated Fuchsian groupcontains torsion [Ros86, Lemma 1], but G v is torsion-free. Therefore, T c does not have a flexible vertex, and as flexible vertices are definedby a universal property it follows that no JSJ tree of G has a flexiblevertex. (cid:3) We now summarise what we know so far.
Proposition 2.5.
Let G be a two-generated one-ended RG group. Then G has a canonical Z max -JSJ tree T Z max , and all Z max -JSJ trees of G liein the same deformation space D Z max .Moreover, no Z max -JSJ tree of G has a flexible vertex.Proof. By Lemmas 2.3 and 2.4, we can apply Proposition 2.1 to geta canonical Z max -JSJ tree T Z max for G , and this tree has no flexiblevertices. The result then follows by Proposition 2.2, and because one Z max -JSJ tree has a flexible vertex if and only if they all have a flexiblevertex. (cid:3) SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 15
Two-generated groups.
We now consider two-generated groups.Firstly, we define the Z max -JSJ decomposition of a two-generated one-ended RG group, and of a two-generator one-relator group with torsion;this is Convention 2.7, but we require a result first which justifies theword “the”, above. We then prove a result which allows us to deter-mine if a given splitting is, in fact, the Z max -JSJ decomposition of suchan RG group.An essential Z max -tree is a Z max -tree such that no edge group has fi-nite index in an adjacent vertex group, while an essential Z max -splitting of a group is a corresponding graph of groups decomposition. We de-fine essential Z -trees and essential Z -splittings analogously, for Z -trees.Note that a Z max -JSJ decomposition with no elementary vertices is anessential Z max -splitting.The following proof uses a connection between outer automorphismgroups and essential Z -splittings. (Really this connection is for essential Z max -splittings, which is well-studied for hyperbolic groups [Lev05a],but we only need it for Z -splittings.) Every edge e of an essential Z -splitting of a group G corresponds to a single-edge graph of groupssplitting by collapsing all the other edges of the graph. These splittingshave two forms, and in each case we can define an automorphism of G : A ∗ g = h B A ∗ t − gt = h φ e : a a ∀ a ∈ A φ e : a a ∀ a ∈ Ab b h ∀ b ∈ B t gt The maps φ e are called Dehn twists about edges, and Dehn twistsof single-edge splittings generate infinite order subgroups of Out( G )which pairwise intersect trivially. As flexible vertices of Z max -JSJ de-compositions have groups which are not elliptic in every Z max -tree,they contribute at least two Dehn twists to Out( G ). In particular, if a Z max -JSJ decomposition of G contains a flexible vertex then Out( G ) isnot virtually- Z . Proposition 2.6.
Let G be a two-generated one-ended RG group, ora two-generator one-relator group with torsion which is not isomorphicto h a, b | [ a, b ] n i for any n > .If G has an essential Z max -tree T which contains at least one edge,then this tree is unique and is a Z max -JSJ tree of G . The Z max -JSJdecomposition T /G consists of a single rigid vertex with a single loopedge.Proof. If G is a one-relator group with torsion then every edge groupof T is infinite cyclic [Log16a, Lemmas 5.1 & 5.5], while the same is clearly true if G is an RG group as G is torsion-free. Recall fromProposition 2.5 that if G is an RG group then no Z max -JSJ tree of G has a flexible vertex. The same is true if G is a one-relator group withtorsion, as here Out( G ) is virtually- Z [Log16b, Theorem A], but if T JSJ is a Z max -JSJ tree with a flexible vertex then Out( G ) is not virtually- Z as the multiple splittings at this vertex contribute multiple Dehn twiststo Out( G ), a contradiction.Let T be as in the statement. We first prove that T /G consistsof a single vertex v and a single loop edge. If G is an RG groupthen this is known [KW99a, Theorem 3.9], so suppose G is a two-generator one-relator groups with torsion. Here, Out( G ) is virtually- Z [Log16b, Theorem A], and as noted above every Z max -splitting is a Z -splitting [Log16a, Lemmas 5.1 & 5.5]. If T /G contains more thanone edge then Out( G ) is not virtually- Z as both splitting contributeDehn twists to Out( G ), a contradiction. Therefore, T /G consists of asingle vertex with a single loop edge.We now apply a result of Levitt on reduced trees, which are trees T where if an edge e has the same stabiliser as one of its endpoints,then both endpoints of e are in the same G -orbit of T (so e projectsonto a loop in the quotient graph T /G ). If G is a hyperbolic or RGgroup and the tree is a Z max -tree then being reduced is equivalent tobeing an essential splitting (as if T is a reduced non-essential Z max -treethen T /G contains a loop edge with endpoints an elementary vertex,which gives a Baumslag–Solitar (in fact Z ) subgroup of G ). Writing v for the vertex of T /G , and e and e for the directed edges, because G is either hyperbolic or an RG group, and because edge stabilisers areinfinite cyclic, we have that there exists no g ∈ G v such that g − G e g ∩ G e is non-trivial, and therefore T is the unique reduced tree in itsdeformation space [Lev05b, Theorem 1] (see also [DG11, Theorem 5.13]for hyperbolic groups), and hence T is the unique essential tree in itsdeformation space.We now prove that any tree T as in the statement is a Z max -JSJtree. Let T Z max be the canonical Z max -JSJ tree guaranteed to exist byProposition 2.5. As this is a Z max -tree, if e is an edge incident to anelementary vertex v of T Z max then G e = G v . Therefore, we can slide anorbit of elementary vertex of T Z max into an orbit of adjacent vertices (soon the level of T Z max /G , slide a single elementary vertex into an adjacentvertex), and the resulting tree is still a Z max -JSJ tree of G . As thereare finitely many orbits of elementary vertices of T Z max , this processterminates at a tree T JSJ which has no elementary vertices. There arethen two options: either T JSJ consists of a single rigid vertex, or T JSJ contains at least one edge. If T JSJ consists of a single rigid vertex
SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 17 then then the whose group G is elliptic in every Z max -splitting of G ,and therefore G has no Z max -splittings, contradicting the existence ofthe tree Z max -tree T which has an edge. Therefore, T JSJ contains anedge, so is as in the statement, and so by the above T JSJ /G consistsof a single rigid vertex and a single loop edge. As T JSJ is universally- Z max elliptic, it is elliptic with respect to T , and so there exists arefinement ˆ T JSJ of T JSJ which dominates T and such that every edgestabiliser of T contains an edge stabiliser of ˆ T JSJ [GL17, Proposition2.2], and moreover this tree is itself a Z max -tree [GL17, Discussionfollowing Definition 9.25]. However, we have already seen that T JSJ has no flexible vertices, so as ˆ T JSJ is a refinement of T JSJ it is obtainedby adding elementary vertices to T JSJ . Therefore, as T JSJ and ˆ T JSJ are both Z max -trees, the edge stabilisers of ˆ T JSJ are precisely the edgestabilisers of T JSJ , and therefore every edge stabiliser of T contains anedge stabiliser of T JSJ . As these edge stabilisers are Z max , it followsthat edge stabilisers of T are edge stabilisers of T JSJ , and so T isuniversally Z max -elliptic. Then T is elliptic with respect to T JSJ , soas above but with T and T JSJ swapped we obtain a refinement ˆ T of T which dominates T JSJ and which is a Z max -tree. Now, every edgestabiliser of ˆ T fixes an edge of T or of T JSJ [GL17, Proposition 2.2],and so as ˆ T is a Z max -tree every edge stabiliser is equal to an edgestabiliser of one of these two trees, and therefore ˆ T is universally Z max -elliptic. As it dominates T JSJ , we have that ˆ T is a Z max -JSJ tree, andso contains a single non-elementary vertex. As ˆ T is a refinement of T across Z max -subgroups, this means that T itself is a Z max -JSJ tree, asrequired.Therefore, if T is as in the statement then it is the unique essentialtree in its deformation space, and is a Z max -JSJ tree. As argued above, T /G consists of a single vertex and a single loop edge, and as T is a Z max -JSJ tree the vertex is rigid. Now, Z max -JSJ trees always lie inthe same deformation space (as they dominate one another), and so T is the unique essential Z max -JSJ tree, and the result follows. (cid:3) Fuchsian groups are treated differently in the theory of JSJ decom-positions. For example, Bowditch excludes these groups from his maintheorem [Bow98, Theorem 0.1]. These groups play a minor role in thisarticle, and it is easiest if, similarly to Bowditch, we simply define suchgroups have trivial Z max -JSJ decomposition. Convention 2.7.
Let G be a two-generated one-ended RG group, ora two-generator one-relator group with torsion. If G is isomorphic to h a, b | [ a, b ] n i for n > , then “the Z max -JSJ decomposition of G ” is the graph of groups consisting a single vertex and no edges. Other-wise, “the Z max -JSJ decomposition of G ” is a Z max -JSJ decompositionwith no elementary vertices; by Proposition 2.6 this is unique (up toconjugation).Such a group G has trivial Z max -JSJ decomposition it is Fuchsian orits Z max -JSJ decomposition (as above) consists of a single vertex andno edges. Spotting Z max -JSJ decompositions. Is next result is only for RGgroups; the analogous result for one-relator groups with torsion has astronger statement bespoke for one-relator groups, so we postpone ituntil the next section. These results give conditions which are equiva-lent to one of our groups have non-trivial Z max -JSJ decomposition, andthese conditions work well with algorithms. In particular, algorithmscan easily detect Conditions (2) and (3) of the following theorem, andhence can detect a non-trivial Z max -JSJ decomposition, while cruciallyCondition (4) says that Condition (3) actually finds, rather than justdetects, the Z max -JSJ decomposition of G . Theorem 2.8.
Let G be a two-generated one-ended RG group with Z abelianisation. The following are equivalent.(1) G has non-trivial Z max -JSJ decomposition.(2) G has an essential Z -splitting.(3) G admits a presentation h a, b | t ( a, b − ab ) i for some subset t ( a, y ) ⊂ F ( a, y ) .(4) G has Z max -JSJ decomposition with a single rigid vertex and asingle loop edge, corresponding to an HNN-extension with stableletter b : h a, y, b | t ( a, y ) , y = b − ab i for some subset t ( a, y ) ⊂ F ( a, y ) .Moreover, the presentations in (3) and (4) are related in the obviousway, so in each presentation the letters a, b represent the same groupelements of G and the subsets t ( a, y ) are the same subsets of F ( a, y ) .Proof. We prove the chain of inclusions (4) ⇒ (2) ⇒ (1) ⇒ (4).(4) ⇒ (2) The splitting given by (4) is a Z -splitting, and is essentialas the base group H = h a, y i of the HNN-extension is torsion-free andnon-cyclic [KW99a, Proposition 3.8].(2) ⇒ (1) This follows from Proposition 2.6.(1) ⇒ (4) By Proposition 2.6, the Z max -JSJ decomposition of G consistsof a single rigid vertex v and a single loop edge e , as claimed. This SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 19 corresponds to an HNN-extension h H, t | t − p n t = q m i where h p i and h q i are maximal cyclic in G . As this is a Z max -JSJ decompositionwe have | m | = | n | = 1, so without loss of generality we may assume m = n = 1. Now, there exists some h ∈ H such that G = h th, p i and H = h p, h − qh i [KW99a, Proposition 3.8], and so by taking a = p , b = th and y = h − qh we obtain the following presentation for G , asan HNN-extension with stable letter b : h a, y, b | t ( a, y ) , y = b − ab i This still corresponds to the Z max -JSJ decomposition of G as the Bass–Serre tree is the same for the two HNN-extensions, and this tree isprecisely the Z max -JSJ tree (that this is the Z max -JSJ decompositionalso follows from Proposition 2.6). Hence, (4) holds.Finally we prove (4) ⇔ (3).(3) ⇒ (4) First, apply the Tietze transformation corresponding toadding the new generator y as b − ab to get from the presentation in(3) to the presentation in (4): h a, b | t ( a, b − ab ) i ∼ = h a, y, b | t ( a, y ) , y = b − ab i . As G has Z abelianisation, both the subgroups h a i and h y i are maximalinfinite cyclic subgroups of G . Therefore, the resulting presentationdescribes an HNN-extension, with base group H = h a, y | t ( a, y ) i andstable letter b , as claimed. Now, this HNN-extension corresponds tosome essential Z -splitting, because H is one-ended as G is one-ended[KW99a, Proposition 3.8], and so by Proposition 2.6 the vertex is rigidand this splitting is in fact the Z max -JSJ decomposition of G . Hence,(4) holds.(4) ⇒ (3) This is obvious, by reversing the Tietze transformation fromthe above equivalence. (cid:3) One-relator groups with torsion
A one-relator group h x | S n i , where n ≥ n > one relator group with torsion is precisely a one-relator group h x | S n i where n >
1; such groups are always hyperbolic [LS77, The-orem IV.5.5], and so when they are one-ended we can discuss their Z max -JSJ decompositions.In this section we prove Theorem 3.2; we also prove a result aboutthe automorphic orbits of elements in the subgroup h a, b − ab i which isalso used in the proof of Theorem A. Z max -JSJ decompositions are only defined for one-ended groups, whileFuchsian groups must be treated with care. The following classifica-tion of when two-generator one-relator groups are one-ended and areFuchsian is therefore useful. Recall that a primitive element of F ( a, b )is an element which is part of a basis for F ( a, b ). We say a free product A ∗ B is trivial if either A or B is trivial, and is non-trivial otherwise. Proposition 3.1.
The group defined by h a, b | S n i , n ≥ maximal, is:(1) one-ended if and only if S is non-primitive and non-empty.(2) one-ended and Fuchsian if and only if n > and S is conjugateto [ a, b ] ± .Proof. Let G = h a, b | S n i , n ≥ S is empty then G is free and so not one-ended, while if S is primitive then G ∼ = h a, b | a n i , so G is either two-ended (if n = 1)or infinitely ended (if n > G is one-ended then S is non-primitive and non-empty. For the other direction, if S is non-primitiveand non-empty then the group G defined by h a, b | S i does not splitnon-trivially as a free product [LS77, Proposition II.5.13] and is notcyclic [LS77, Proposition II.5.11], but is torsion-free [LS77, PropositionII.5.18]. Hence, G is one-ended, and so also G is one-ended [Log16a,Lemma 3.2].For (2), the group G is one-ended and Fuchsian if and only if n > G ∼ = h a, b | [ a, b ] n i [Ros86, Lemma 1], if and only if there exists anautomorphism φ ∈ Aut( F ( a, b )) such that φ ( S n ) = [ a, b ] ± n [Pri77], ifand only if S is conjugate to [ a, b ] ± [MKS04, Theorem 3.9], as required. (cid:3) Next we prove the analogue of Theorem 2.8 for two-generator one-relator groups with torsion. The main difference is that the set t ( a, y )is replaced with a single word T ( a, y ) n , and that Condition (3) givesinformation about this word. The restrictions on the word S in thistheorem correspond precisely to G being one-ended and non-Fuchsian,by Proposition 3.1. In the theorem, the pair of letters ( a, b ) in (4) and(5) represent the same pair of group elements of G . However, the pair( a, b ) from P are different, and are related to those in (4) and (5) viathe map ψ . Theorem 3.2.
Let G be a group admitting a two-generator one-relatorpresentation P = h a, b | R i where R = S n with n > maximal andwhere S ∈ F ( a, b ) is non-empty, non-primitive, and not conjugate to [ a, b ] ± . The following are equivalent.(1) G has non-trivial Z max -JSJ decomposition.(2) G has an essential Z -splitting. SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 21 (3) There exists an automorphism ψ ∈ Aut( F ( a, b )) such that ψ ( S ) ∈h a, b − ab i .(4) G admits a presentation h a, b | T ( a, b − ab ) n i for some word T ∈ F ( a, y ) .(5) G has Z max -JSJ decomposition with a single rigid vertex and asingle loop edge, corresponding to an HNN-extension with stableletter b : h a, y, b | T ( a, y ) n , y = b − ab i Moreover, the presentations in (4) and (5) are related in the obviousway, so in each presentation the letters a, b represent the same elementsof G and the words T ( a, y ) are the same, and indeed may be taken tobe such that T ( a, b − ab ) = ψ ( S ) , with ψ as in (3).Proof. We start with (1) ⇔ (2).(1) ⇒ (2) This is immediate from the definitions (see Convention 2.7).(2) ⇒ (1) This follows from Proposition 2.6.Next we prove (3) ⇔ (4).(3) ⇒ (4) It is a standard result in the theory of group presentationsthat h a, b | S n i ∼ = h a, b | ψ ( S n ) i , so the result follows by taking T ( a, y ) ∈ F ( a, y ) such that T ( a, b − ab ) = ψ ( S ).(4) ⇒ (3) This holds as the word S ′ := T ( a, b − ab ) ∈ h a, b − ab i is inthe Aut( F ( a, b ))-orbit of S [Pri77], as required.Finally we prove (5) ⇒ (1) ⇒ (4) ⇒ (5).(5) ⇒ (1) This is clear as the stated Z max -JSJ decomposition is non-trivial.(1) ⇒ (4) Suppose G has non-trivial Z max -JSJ decomposition. Then G admits an essential Z -splitting [Log16a, Theorem A], and hence hasinfinite outer automorphism group [Lev05a, Theorem 5.1]. Therefore,there exists S ′ ∈ h a, b − ab i such that G ∼ = h a, b | S ′ i [Log16b, Lemma5.1]. Setting T ( a, y ) to be the word such that T ( a, b − ab ) = S ′ givesthe result.(4) ⇒ (5) The group G has presentation h a, b | T ( a, b − ab ) n i ∼ = h a, b, y | T ( a, y ) n , y = b − ab i which has the claimed form. This is an HNN-extension with base group H = h a, y | T ( a, y ) n i and stable letter b as the subgroups h a i and h y i of H are isomorphic, because a , and hence its conjugate y = b − ab ,have infinite order in G by the B.B.Newman Spelling Theorem [LS77,Theorem IV.5.5]. Indeed, G has presentations h a, b | T ( a, b − ab ) n i and h y, b | T ( byb − , y ) n i , whence it follows that the subgroups h a i and h y i are maximal cyclic subgroups of G [New73, Lemma 2.1].Therefore, the HNN-extension describes a graph of groups decompo-sition Γ of G with a single vertex and a single loop edge, and where theedge groups are maximal cyclic in G . By Proposition 2.6, the vertex isrigid and Γ is the Z max -JSJ decomposition of G , as required. (cid:3) Before proving Theorem 3.4 (which corresponds to Theorem B) weneed the following result about automorphic orbits of elements of h a, b − ab i .We also use this result in the proof of Theorem 5.4 (Theorems 3.4 and5.4 combine to prove Theorem A). Proposition 3.3.
Let W ∈ h a, b − ab i . Then there exists a cyclic shift W ′ of W such that W ′ has shortest length in its Aut( F ( a, b )) -orbit,and W ′ ∈ h a, b − ab i .Proof. For a conjugacy class [ U ] of an element U ∈ F ( a, b ), we write | [ U ] | := min {| V | | V ∈ [ U ] } . So | [ U ] | is simply the length of U aftercyclic reduction. Therefore, the proposition says in particular that for W ∈ h a, b − ab i , there is no automorphism α ∈ Aut( F ( a, b )) such that | [ α ( W )] | < | [ W ] | . Clearly the result is true if [ W ] contains some powerof a . Therefore, assume that [ W ] ∩ h a i = ∅ , and suppose that thereexists an automorphism α ∈ Aut( F ( a, b )) such that | [ α ( W )] | < | [ W ] | .We seek a contradiction.By the “peak reduction” lemma from Whitehead’s algorithm [LS77,Proposition I.4.20], there exists a Whitehead automorphism β of thesecond kind (so a Whitehead automorphism which is not a permutationof { a, b } ± ) such that | [ β ( W )] | < | [ W ] | . Writing γ b ∈ Aut( F ( a, b )) forconjugation by b , the only four such Whitehead automorphisms whichdo not fix [ W ] are β : a ab , b b , β − : a ab − , b b , γ b β , and γ b β − .We find a contradition for the automorphisms β and β − ; the casesof γ b β and γ b β − follow immediately as [ γ b β ( W )] = [ β ( W )] and[ γ b β − ( W )] = [ β − ( W )]. Consider a cyclically reduced conjugate W ′ of W which begins with a and ends with b . Then W ′ ∈ h a, b − ab i ,and we write this word as a reduced word U ( a, b − ab ). Let X denotethe total number of x -terms in U ( x, y ), and Y the number of y -terms,and Syl( Y ) the total number of y -syllables in U ( x, y ) (a y -syllable isa maximal subword of the form y i ). Then | W ′ | = X + Y + 2 Syl( Y ).Now, [ β ( W ′ )] = [ U ( ba, ab )] and [ β − ( W ′ )] = [ U ( ab − , b − a )], and nofree reduction or cyclic reduction happens when forming U ( ba, ab ) or U ( ab − , b − a ), and so | [ β ( W ′ )] | = 2 X + 2 Y = [ β − ( W ′ )]. Now, clearly Y ≥ Syl( Y ), while Syl( X ) = Syl( Y ) as U ( x, y ) starts with an x -syllable SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 23 and ends with a y -syllable, by our choice of W ′ , and so X ≥ Syl( Y ).Therefore, 2 X + 2 Y ≥ X + Y + 2 Syl( Y ). However, by assumption2 X +2 Y < X + Y +2 Syl( Y ), so we have our promised contradiction. (cid:3) We now prove Theorem B.
Theorem 3.4 (Theorem B) . Let G be a group admitting a two-generatorone-relator presentation P = h a, b | R i where R = S n in F ( a, b ) with n > maximal and where S is non-empty and not a primitive elementof F ( a, b ) . The following are equivalent.(1) G has non-trivial Z max -JSJ decomposition.(2) There exists a word T of shortest length in the Aut( F ( a, b )) -orbit of S such that T ∈ h a, b − ab i but T is not conjugate to [ a, b ] ± .Proof. Suppose (1) holds. Then G is not Fuchsian and so, by Propo-sition 3.1, S is not conjugate to [ a, b ] ± . Then by Theorem 3.2 thereexists an automorphism ψ ∈ Aut( F ( a, b )) such that ψ ( S ) ∈ h a, b − ab i .By Proposition 3.3, there exists a cyclic shift T of ψ ( S ) which hasshortest length in its Aut( F ( a, b ))-orbit, and T ∈ h a, b − ab i . Moreover,as S is not conjugate to [ a, b ] ± , neither is T [MKS04, Theorem 3.9].Hence, the word T satisfies (2).If (2) holds then (1) follows from Theorem 3.2. (cid:3) Splittings and Friedl–Tillmann polytopes
Theorem 2.8 gives a clear link between Z max -JSJ decompositionsand presentations of the form P ′ = h x, y | t ( x, y − xy ) i for subsets t ( x, y − xy ) of h x, y − xy i . In this section we suppose that this JSJ-presentation P ′ for a group G is “close” to being one-relator (specifi-cally, the generating pair ( x, y ) also admits a one-relator presentation h x, y | S i ), and we prove that the Friedl–Tillmann polytope of G isa straight line (Theorem 4.1), and hence that any one-relator presen-tation for G has this JSJ-form h a, b | R ( a, b − ab ) i , up to a Nielsentransformation (Lemma 4.2).In Section 5 we prove that Theorem 4.1 applies to RG groups, andhence torsion-free hyperbolic groups. The proof of Theorem 4.1 reviewsthe notion of a Friedl–Tillmann polytope. Theorem 4.1.
Let G be a torsion-free two-generator one-relator groupwith Z abelianisation, and let P = h a, b | R i be any one-relator pre-sentation of G . Suppose there exist a normal subgroup N of F ( x, y ) such that:(1) F ( x, y ) /N ∼ = G , (2) N can be normally generated by a single element R N ∈ F ( x, y ) ,(3) There exists a subset t N of h x, y − xy i such that t N normallygenerates N .Then the Friedl–Tillmann polytope FT of P is a straight line.Proof. Let X denote the Cayley 2-complex associated to the one-relatorpresentation h x, y | R N i of G . Since G is torsion-free, X is aspherical[Coc54]. As G is finitely presentable, there exists a finite subset t ′ N of t N which also normally generates N . Therefore, G admits a finitepresentation h x, y | t ′ N ( x, y − xy ) , R N i , and let Y be the Cayley 2-complex of this presentation . Note that X is a subcomplex of Y . Moreover, since every relator r in t ′ N ( x, y − xy )corresponds to a closed loop γ r in X , we can form a 3-complex Z from Y by G -equivariantly gluing in 3-cells, one G orbit for each elementin t ′ N ( x, y − xy ), in such a way that the boundary of the 3-cell is theunion of two 2-discs glued along their boundary, where the first discfills in the relator r in Y , and the second fills in the loop γ r in X . It isclear that Z is a 3-dimensional cofinite G -complex which retracts onto X , and hence is aspherical.We will now use Z , together with the fact that G satisfies the Atiyahconjecture [JZLA20] and that it is L -acyclic [DL07], and compute the L -torsion polytope P (2) ( G ) of G , as defined by Friedl–L¨uck [FL17].Note that, in general, P (2) is not necessarily a polytope, but a formaldifference of two polytopes. Nevertheless, when G is a one-relator groupthen P (2) ( G ) coincides with the Friedl–Tillmann polytope by [FL17,Remark 5.5], and is a single polytope.We start by looking at the cellular chain complex C • of Z . For every n , the n -chains C n form a free Z G -module. We pick a natural cellularbasis for C • , namely: the vertex 1 forms the basis for C ; the edgesconnecting 1 to y and x form the basis of C ; the discs filling-in R N in X and the relators from t ′ N ( x, y − xy ) in Y form the basis for C ;and finally the 3-cells attached to the basic discs filling-in relators from t ′ N ( x, y − xy ) form the basis of C .We may now identify C • with Z G α → Z G α +1 → Z G → Z G where α = | t ′ N ( x, y − xy ) | .Before proceeding any further, we need to deal with the special casein which the Fox derivatives ∂r∂x are zero for all r ∈ t ′ N ( x, y − xy ). The SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 25 fundamental formula of Fox calculus [Fox53] tells us that ∂r∂x (1 − x ) + ∂r∂y (1 − y ) = 1 − r = 0in Z G . Hence ∂r∂y (1 − y ) = 0, which forces ∂r∂y = 0, as G satisfies theAtiyah conjecture and hence Z G has no non-trivial zero divisors. (Acareful reader might observe that for this argument we need y = 1 in G , which is true since otherwise the abelianisation of G would not be Z .) In fact, we can conclude that Z G does not have non-trivial zero-divisors from an earlier work of Lewin–Lewin [LL78]. Now, since bothFox derivatives of r vanish, the 1-cycle corresponding to r is trivial.Since R N lies in the normal closure of t ′ N ( x, y − xy ), the 1-cycle givenby R N must also be trivial. Hence, the differential C → C is zero.We know that C • is L -acyclic. For torsion-free groups satisfying theAtiyah conjecture, like G , this amounts to saying that D ( G ) ⊗ Z G C • is acyclic, where D ( G ) is the Linnell skew-field , a skew-field whichcontains Z G . In our case, if the differential C → C is trivial, thentensoring C • with D ( G ) yields the chain D ( G ) α −→ D ( G ) α +1 0 −→ D ( G ) −→ D ( G )which cannot be acyclic for dimension reasons. This is a contradiction.Let r ∈ t ′ N ( x, y − xy ) be such that the Fox derivative ∂r∂x is not zero.Let d denote the 2-chain given by the disc filling-in r in Y ; note that d is a basis element of C . We now look at the commutative diagramwith exact columns( † ) 0 (cid:15) (cid:15) / / (cid:15) (cid:15) / / (cid:15) (cid:15) / / (cid:15) (cid:15) (cid:15) (cid:15) / / h d i (cid:15) (cid:15) / / Z G (cid:15) (cid:15) / / Z G id (cid:15) (cid:15) Z G α / / id (cid:15) (cid:15) Z G α +1 / / (cid:15) (cid:15) Z G / / (cid:15) (cid:15) Z G (cid:15) (cid:15) Z G α / / (cid:15) (cid:15) Z G α +1 / h d i / / (cid:15) (cid:15) (cid:15) (cid:15) / / (cid:15) (cid:15) / / / / / / h d i denotes the Z G -span of d . Note that ( † ) is really a shortexact sequence of chain complexes. By assumption, the middle row of ( † ) is L -acyclic. We claim thatthe second row, which we will denote by B • , is also. To prove the claim,we need to look at the chain complex B • in more detail: h d i (cid:16) ∂r∂y ∂r∂x (cid:17) / / Z G − y − x / / Z G Since G is not cyclic, we have 1 − y = 0; we also have ∂r∂x = 0 byassumption. Now B • fits into the exact sequence of chain complexes( ‡ ) 0 (cid:15) (cid:15) / / (cid:15) (cid:15) / / (cid:15) (cid:15) (cid:15) (cid:15) / / Z G (cid:15) (cid:15) (1 − y ) / / Z G id (cid:15) (cid:15) h d i id (cid:15) (cid:15) / / Z G (cid:15) (cid:15) / / Z G (cid:15) (cid:15) h d i (cid:15) (cid:15) ( ∂r∂x ) / / Z G (cid:15) (cid:15) / / (cid:15) (cid:15) / / / / Z G and then the projection onto the second coordinate. Sinceboth horizontal differentials labelled in the diagram are multiplicationsby non-zero elements of Z G , they are invertible over D ( G ), and hencethe second and fourth rows of this commutative diagram are exactupon tensoring with D ( G ). By [FL17, Lemma 2.9], the middle rowalso becomes exact upon tensoring with D ( G ). This proves the claim.We are now back to examining ( † ). We have just shown that thesecond row is L -acyclic, and hence [FL17, Lemma 2.9] tells us that sois the fourth row, and that P (2) ( C • ) = P (2) ( B • ) + P (2) ( D • )where D • denotes the fourth row of ( † ). We can split P (2) ( B • ) furtherusing diagram ( ‡ ) and [FL17, Lemma 1.9], and obtain P (2) ( B • ) = P − Q where − P is the L -torsion polytope of Z G − y −→ Z G SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 27 and − Q of Z G ∂r∂x −→ Z G Now we need to recall how the polytope P (2) is actually constructed,at least in the above (simple) cases. When considering a chain complexof the form Z G z −→ Z G with z ∈ Z G r { } , the polytope is obtained by first taking the supportsupp( z ) ⊆ G , then taking the image of this set in H ( G ; R ), and thentaking the convex hull. At the end the polytope is given a sign, negativein this situation, and in general depending on whether the single non-trivial differential starts in an even or an odd dimension.With the construction in mind, it is now immediate that, since r isa word in x and y − xy , the polytope P is contained within the strip[ − , × R . The polytope Q is obtained from 1 − y , and hence it is thesegment [0 , × { } .Since D • is concentrated around a single differential from odd toeven degree chains, P (2) ( D • ) = − R where R is some polytope. Hence P (2) ( G ) = P (2) ( C • ) = P − Q − R Since G is a one-relator group, P (2) ( G ) is actually a single polytope.Hence, in particular, P − Q must be a single polytope, and thereforeit must lie on the line {− } × R . Subtracting a further polytope fromsuch a polytope does not alter this fact, and hence we have finishedthe proof. (cid:3) Next, we prove the converse to Theorem 4.1. This is applied to provethat (3) implies (2) in the proof of Theorem 5.4.
Lemma 4.2.
Let G be a group admitting a two-generator one-relatorpresentation P = h a, b | R i with R ∈ F ( a, b ) ′ \ { } . The Friedl–Tillmann polytope of P is a straight line if and only if there existsa word T of shortest length in the Aut( F ( a, b )) -orbit of R such that T ∈ h a, b − ab i .Proof. Suppose that the Friedl–Tillmann polytope (which coincideswith P (2) ) is a line. By applying a suitable Nielsen automorphism,we may assume that G is given by the presentation h x, y | S i and thatthe polytope lies on the line { } × R .The polytope is obtained either as described in the proof of Theorem4.1, following the Friedl–L¨uck procedure, or in a more direct fashion:first we form a polytope P by simply tracing the closed loop the word S gives in the Cayley graph of the free part of the abelianisation of G taken with respect to the image of { x, y } as a generating set, and then taking the convex hull of the image of the loop in H ( G ; R ). Then weobserve that P has to be the Minkowski sum of another polytope, say P ′ , and the square [ − , × [ − , P ′ is precisely theFriedl–Tillmann polytope. (Note that the polytope is only well-definedup to translation, so it does not matter which square we choose.)Now, we know that P ′ is a line, which forces P to lie inside the strip[ − , × R . Hence, the loop given by S must also lie in this strip. Since S ∈ F ( x, y ) ′ , it can be written as a product of right conjugates of x bypowers of y . We now see that the only conjugates of x appearing canbe x and y − xy . Therefore, there exists a word T ′ in the Aut( F ( a, b ))-orbit of R such that T ′ ∈ h a, b − ab i , and so by Proposition 3.3 thereexists a word T of shortest length in the Aut( F ( a, b ))-orbit of R with T ∈ h a, b − ab i , as required.Now suppose that there exists a word T of shortest length in theAut( F ( a, b ))-orbit of R such that T ∈ h a, b − ab i . Then we may assumethat G is given by a presentation h a, b | T ′ i with T ′ ∈ h a, bab − i . Itfollows that the loop given by T must lie inside the strip [ − , × R , andso P also lies inside this strip. As P is the Minkowski sum of P ′ andthe square [ − , × [ − , P ′ is a straight segment. (cid:3) If the polytope is a single point then the conditions on the relatorare much stronger. We use this lemma in the proof of Theorem A.
Lemma 4.3.
Let G be a group admitting a two-generator one-relatorpresentation P = h a, b | R i with R ∈ F ( a, b ) ′ \ { } . The Friedl–Tillmann polytope of P is a point if and only if R is conjugate to [ a, b ] k for some k ∈ Z .Proof. Suppose the Friedl–Tillmann polytope P ′ of P is a point. Asin the proof of Lemma 4.2, we may assume that G is given by thepresentation h x, y | S i and that the polytope lies on the line { } × R .As P ′ is a point, the polytope P is the square [0 , × [0 ,
1] (up totranslation). This means precisely that S = [ a, b ] k for some non-zero k ∈ Z .Now suppose that R is conjugate to [ a, b ] k for some k ∈ Z . Then wemay assume that G is given by the presentation h a, b | ( aba − b − ) k i ,and this presentation is easily seen to have Friedl–Tillmann polytopea point. (cid:3) The main theorem
We now prove Theorem 5.4, which is a more general form of TheoremA applying also to RG groups. Our first lemma allows us to applyTheorem 4.1.
SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 29
Lemma 5.1.
Let G be a two-generator one-relator RG group with Z abelianisation. Suppose G admits an HNN-presentation h H, t | t − p n t = q i where p, q are nontrivial elements of H and the subgroups h p i and h q i are malnormal in H . Then there exist a normal subgroup N of F ( x, y ) such that:(1) F ( x, y ) /N ∼ = G ,(2) N can be normally generated by a single element R N ∈ F ( x, y ) ,(3) There exists a finite subset t N of h x, y − xy i such that t N nor-mally generates N .Proof. Suppose that G admits a one-relator presentation h a, b | R i . Thesubgroups h p i and h q i are “conjugacy separated” in H [KW99a, Lemma3.6.2], and so there exists a Nielsen transformation ψ of F ( a, b ) andsome g ∈ G such that ψ ( a ) = G g − p i g and ψ ( b ) = G g − thg for some i ∈ Z and some h ∈ H [KW99b, Corollary 3.1]. Therefore, writing x := p and y := th , the pair ( x, y ) generates G . But also the pair ( x i , y )generates G , and as G has abelianisation Z we have that | i | = 1. Notethat as the generating pair ( a, b ) of G admits a one-relator presentation,and as ψ is a Nielsen transformation, we have that the pair ( x, y ) alsoadmits a one-relator presentation h x, y | R N i . Hence, points (1) and(2) of the theorem hold for some subgroup N ≤ F ( x, y ).We now establish (3) for N ≤ F ( x, y ). Firstly, note that H = h p, h − qh i by [KW99a, Proposition 3.8], and so H has presentation h x, z | s ( x, z ) i where x := p (as above) and z := h − qh . Therefore, G has presentation h x, z, t | s ( x, z ) , t − x n t = w ( x, z ) zw ( x, z ) − i where w ( x, z ) represents the element h ∈ H . As remarked above the theorem,because G is finitely presentable we may assume that the set s ( x, z )is finite. As y = th , we can apply Tietze transformations as follows,where the generators x and y correspond precisely to the generators x and y in the previous paragraph: h x, z, t | s ( x, z ) , t − x n t = w ( x, z ) zw − ( x, z ) i∼ = h x, y, z, t | s ( x, z ) , t − x n t = w ( x, z ) zw − ( x, z ) , y = tw ( x, z ) i∼ = h x, y, z, t | s ( x, z ) , y − x n y = z, y = tw ( x, z ) i∼ = h x, y, z | s ( x, z ) , y − x n y = z i∼ = h x, y | s ( x, y − x n y ) i = h x, y | t N i for some finite subset t N of h x, y − xy i . As the symbols x and y in thispresentation of G and the one-relator presentation h x, y | R N i of G both represent the same elements of G , it follows that N is the normalclosure of the set t N in F ( x, y ), as required. (cid:3) Our next result is the torsion-free part of our main theorem. It isworth stating separately because it does not include the exceptionalcases of points (2) and (3).
Theorem 5.2.
Let G be an RG group admitting a two-generator one-relator presentation P = h a, b | R i with R ∈ F ( a, b ) ′ \ { } . The follow-ing are equivalent.(1) G has non-trivial Z max -JSJ decomposition.(2) There exists a word T of shortest length in the Aut( F ( a, b )) -orbit of R such that T ∈ h a, b − ab i .(3) The Friedl–Tillmann polytope of P is a straight line.Proof. (2) and (3) are equivalent by Lemma 4.2.(2) ⇒ (1) The word T gives a presentation for G : h a, b | T i = h a, b | T ( a, b − ab ) i . Then (1) follows by Theorem 2.8.(1) ⇒ (3) By Theorem 2.8 the conditions of Lemma 5.1 are satisfied,so we may apply Theorem 4.1, and (3) follows. (cid:3) We wish to combine Theorems 5.2 and Theorem B to prove our maintheorem, Theorem 5.4. However, if R = S n with n > S rather than the relator R = S n . The followingobservation is therefore useful. Lemma 5.3. If n > and S ∈ F ( a, b ) , then S has shortest lengthin its Aut( F ( a, b )) -orbit if and only if S n has shortest length in its Aut( F ( a, b )) -orbit.Proof. Note that such shortest-length elements are cyclically reduced,so | S n | = n | S | and also | α ( S n ) | = n | α ( S ) | for any automorphism taking S or S n to the shortest length in its Aut( F ( a, b ))-orbit. Therefore, if S has shortest length in its Aut( F ( a, b ))-orbit then | S | = | α ( S ) | , andso | S n | = | α ( S n ) | by these equalities. On the other hand, if S n hasshortest length in its Aut( F ( a, b ))-orbit then | S n | = | α ( S n ) | , and so | S | = | α ( S ) | by these equalities. (cid:3) We now combine Theorems 5.2 and Theorem B to prove our maintheorem, which immediately implies Theorem A as torsion-free hyper-bolic groups are RG.
SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 31
Theorem 5.4 (Theorem A) . Let G be a hyperbolic group or an RGgroup admitting a two-generator one-relator presentation P = h a, b | R i with R ∈ F ( a, b ) ′ \ { } . The following are equivalent.(1) G has non-trivial Z max -JSJ decomposition.(2) There exists a word T of shortest length in the Aut( F ( a, b )) -orbit of R such that T ∈ h a, b − ab i but T is not conjugate to [ a, b ] k for any k ∈ Z .(3) The Friedl–Tillmann polytope of P is a straight line, but not asingle point.Proof. If there exists a word T in the Aut( F ( a, b ))-orbit of R whichis conjugate to [ a, b ] k for some k ∈ Z , then G is either abelian (when | k | = 1) or contains torsion (when k > P is a single point that G is either abelian orcontains torsion. Therefore, if G is an RG group then the result followsby Theorem 5.2.If G is not RG then G contains torsion, so R = S n in F ( a, b ) forsome n > ⇔ (3) This follows from Lemmas 4.2 and 4.3.(1) ⇒ (2) If G has non-trivial Z max -JSJ decomposition then, by The-orem B, there exists a word T ′ of shortest length in the Aut( F ( a, b ))-orbit of S such that T ′ ∈ h a, b − ab i but T ′ is not conjugate to [ a, b ] ± .Then the word T := ( T ′ ) n has shortest length in its Aut( F ( a, b ))-orbit,by Lemma 5.3, and hence in the Aut( F ( a, b ))-orbit of R , is containedin h a, b − ab i , and is not conjugate to [ a, b ] ± n (we are using uniquenessof roots in free groups here). As n is maximal, T is not conjugate toany other power of [ a, b ] ± , and so (2) holds.(2) ⇒ (1) For T as in (2) , write ψ ∈ Aut( F ( a, b )) for the automorphismsuch that ψ ( R ) = T , and set T ′ := ψ ( S ). Then T ′ has shortest lengthin the Aut( F ( a, b ))-orbit of S , by Lemma 5.3, is in h a, b − ab i and isnot conjugate to [ a, b ] ± . Therefore, by Theorem B, (1) holds. (cid:3) Consequences of the main theorem
We now prove corollaries of Theorems 5.4 and B. These corollariescorresponds to Corollaries C–F from the introduction, but in most casesthey additionally apply to RG groups. Therefore, most of these corol-laries refer to Theorem 5.4 in their statement, rather than TheoremA.
Forms of Z max -JSJ decompositions. We first prove Corollary C,which describes the Z max -JSJ decompositions of our groups. Corollary 6.1 (Corollary C) . Let the group G and the presentation P be as in Theorem 5.4 or B, and write R = S n for n ≥ maximal.Suppose that G has non-trivial Z max -JSJ decomposition Γ . Then thegraph underlying Γ consists of a single rigid vertex and a single loopedge. Moreover, the corresponding HNN-extension has vertex group h a, y | T n ( a, y ) i , stable letter b , and attaching map given by y = b − ab ,where T ( a, b − ab ) is the word T from the respective theorem. Finally, | T | < | S | .Proof. By (2) of Theorem 5.4 or B, as appropriate, G admits a presen-tation of the form h a, b | T n ( a, b − ab ) i , n ≥
1, where T n ( a, b − ab ) and T n ( a, b ) are the same words. Hence, G has a presentation h a, b, y | T n ( a, y ) , b − ab = y i , which is an HNN-extension with stable letter b , and corresponds to agraph of groups decomposition Γ as in the statement of the corollary.This decomposition corresponds to the Z max -JSJ decomposition of G and has rigid vertex by Theorem 2.8, if G is as in Theorem 5.4, and byTheorem 3.2 otherwise.Finally, as T is not a power of a primitive we have that | T | < | T | ,and so | T | < | T | ≤ | S | , as required. (cid:3) Algorithmic consequences.
We now prove Corollary D, on comput-ing Z max -JSJ decompositions. Note that it is possible to calculate aprecise bound for the time complexity of this algorithm, as the con-stants are known for the time complexity of Whitehead’s algorithm in F ( a, b ) [Kha04] [CR15, Theorem 1.1]. Corollary 6.2 (Corollary D) . There exists an algorithm with input apresentation P = h a, b | R i of a group G from Theorem 5.4 or B, andwith output the Z max -JSJ decomposition for G .This algorithm terminates in O ( | R | ) -steps.Proof. Write R as S n with n ≥ O S for the set ofshortest elements in the Aut( F ( a, b ))-orbit of S .Firstly, use Whitehead’s algorithm to compute the set O S ; this takes O ( | S | ) ≤ O ( | R | ) steps [Kha04]. Then compute T := O S ∩ h a, b − ab i ;Khan gave a linear bound on the cardinality of O S [Kha04], so thistakes O ( | S | ) ≤ O ( | R | ) steps. If the intersection T is empty then, byTheorem 5.4 or B as appropriate, the Z max -JSJ decomposition of G is trivial. Therefore, output as the Z max -JSJ decomposition of G thegraph of groups consisting of a single vertex, with vertex group G , andno edges. SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 33 If T is non-empty then take some T ∈ T . Rewrite this word T asa word T ( a, b − ab ); this takes O ( | T | ) ≤ O ( | R | ) steps. By Corollary6.1, the Z max -JSJ decomposition of G is the graph of groups with asingle vertex, a single loop edge, vertex group h a, y | T n ( a, y ) i andthe attaching map a = byb − , and so output this as the Z max -JSJdecomposition of G . (cid:3) We next prove Corollary E, on detecting non-trivial Z max -JSJ de-compositions. Corollary 6.3 (Corollary E) . There exists an algorithm with input apresentation P = h a, b | R i of a group G from Theorem 5.4, and withoutput yes if the group G has non-trivial Z max -JSJ decomposition and no otherwise.This algorithm terminates in O ( | R | ) -steps.Proof. It takes O ( | R | ) steps to draw the Friedl–Tillmann polytope of G , and O ( | R | )-steps to further determine if it is a straight line, but nota point. The result then follows from Theorem 5.4. (cid:3) Finally, we prove Corollary F, on determining the commensurabilityclasses of outer automorphism groups.
Corollary 6.4 (Corollary F) . There exists an algorithm with inputa presentation P = h a, b | R i of a group G from Theorem A or B,and which determines if the outer automorphism group of G is finite,virtually Z , or GL ( Z ) .If G is as in Theorem A, this algorithm terminates in O ( | R | ) -steps.Else, it terminates in O ( | R | ) -steps.Proof. If R is conjugate to [ a, b ] n for some n ∈ Z then Out( G ) ∼ =GL ( Z ) [Log16b, Theorem A]. Else if G has non-trivial Z max -JSJ de-composition then, by Theorems 2.8 and 3.2, G has an essential Z -splitting, and so Out( G ) is virtually Z [Lev05a, Theorem 5.1]. Else,by Theorems 2.8 and 3.2, G admits no essential Z -splittings, and soOut( G ) is finite [Lev05a, Theorem 5.1].Our algorithm is therefore as follows: Firstly, determine whether R is conjugate to [ a, b ] n for some n ∈ Z ; this takes O ( | R | ) steps. Ifit is, then Out( G ) ∼ = GL ( Z ). Otherwise determine whether G hastrivial Z max JSJ decomposition; this takes O ( | R | ) steps if R ∈ F ( a, b ) ′ ,by Corollary E, and otherwise takes O ( | R | ) steps, by Corollary D. If G has non-trivial Z max -JSJ decomposition Out( G ) is virtually Z , elseOut( G ) is finite. (cid:3) Relationships between outer automorphism groups.
We nowprove Corollary G, on embeddings of outer automorphism groups.
Corollary 6.5 (Corollary G) . Write G k for the group defined by h a, b | S k i , where k ≥ is maximal. If S ∈ F ( a, b ) ′ and G is hyperbolic then:(1) Out( G m ) ∼ = Out( G n ) for all m, n > .(2) Out( G n ) embeds with finite index in Out( G ) .Proof. We first prove (1). For k >
1, every automorphism φ of G k is induced by a Nielsen transformation φ t of F ( a, b ) [Pri77, PrincipalLemma]. This gives a homomorphism θ k : H k ։ Out( G k ) for somesubgroup H k of Out( F ( a, b )). Now, such a map φ t sends S to a con-jugate of S or of S − in F ( a, b ) [MKS04, Theorem N5], so the mapsare dependent only on the word S , and independent of the exponent k in S k , and therefore H m = H n for all m, n >
1. Moreover, themaps θ k are all isomorphisms [Log16b, Theorem 4.2], and so we haveOut( G m ) ∼ = H m = H n ∼ = Out( G n ) as required.For (2), as we observed in the previous paragraph, the automor-phisms of G n depend solely on S and so we have a sequence of homo-morphisms H n θ n −→ Out( G n ) ξ n −→ Out( G ) where θ n is injective (it is anisomorphism in fact). Then ξ n θ n is injective, by an identical proof tothe proof that θ n is injective [Log16b, Lemma 4.4], and so ξ n is injective.To see that im( ξ n ) has finite index in Out( G ) note that because G is hyperbolic, S is not conjugate to [ a, b ] ± , and so G has non-trivial Z max -JSJ decomposition if and only if G n has non-trivial Z max -JSJ de-composition by Theorems A and B. If both Z max -JSJ decompositionsare non-trivial then, by Corollary C, both Z max -JSJ decomposition areessential Z -splittings consisting of a single rigid vertex with a single loopedge, and so both Out( G ) and Out( G n ) are virtually Z [Lev05a, The-orem 5.1], and the result follows. If both Z max -JSJ decompositions aretrivial then both Out( G ) and Out( G n ) are finite [Lev05a, Theorem5.1], and the result follows. (cid:3) The above proof uses the fact that S ∈ F ( a, b ) ′ in two places; firstlyto prove that Out( G n ) embeds in Out( G ) (which is hidden in thecitation), and secondly to obtain finite index (by using Theorem A).If S F ( a, b ) ′ then we can still ask if Out( G n ) embeds in Out( G ).Indeed, we still obtain a sequence H n θ n −→ Out( G n ) ξ n −→ Out( G ), buthere the proof that θ n is injective does not extend to the map ξ n θ n [Log16b, Lemma 4.6] (as the proof uses the theory of one-relator groupswith torsion, in the form of a strengthened version of the B.B. Newman SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 35
Spelling Theorem). Therefore, we are currently unable to prove thatOut( G n ) embeds in Out( G ) in general. References [Bar18] Benjamin Barrett,
Computing JSJ decompositions of hyperbolic groups ,J. Topol. (2018), no. 2, 527–558. MR 3828057[Bow98] Brian H. Bowditch, Cut points and canonical splittings of hyperbolicgroups , Acta Math. (1998), no. 2, 145–186. MR 1638764[Bra99] Noel Brady,
Branched coverings of cubical complexes and subgroups ofhyperbolic groups , J. London Math. Soc. (2) (1999), no. 2, 461–480.MR 1724853[Bro20] Carl-Fredrik Nyberg Brodda, The B. B. Newman spelling theorem ,arXiv:2004.01484 (2020).[CF09] Matt Clay and Max Forester,
Whitehead moves for G -trees , Bull. Lond.Math. Soc. (2009), no. 2, 205–212. MR 2496498[CL83] D.J. Collins and F. Levin, Automorphisms and Hopficity of certainBaumslag-Solitar groups , Arch. Math. (Basel) (1983), no. 5, 385–400.MR 707725[Coc54] W. H. Cockcroft, On two-dimensional aspherical complexes , Proc. Lon-don Math. Soc. (3) (1954), 375–384. MR 63042[CR15] Bobbe Cooper and Eric Rowland, Classification of automorphic conju-gacy classes in the free group on two generators , Algorithmic problems ofgroup theory, their complexity, and applications to cryptography, Con-temp. Math., vol. 633, Amer. Math. Soc., Providence, RI, 2015, pp. 13–40. MR 3364219[DG08] Fran¸cois Dahmani and Daniel Groves,
The isomorphism problem for toralrelatively hyperbolic groups , Publ. Math. Inst. Hautes ´Etudes Sci. (2008),no. 107, 211–290. MR 2434694[DG11] Fran¸cois Dahmani and Vincent Guirardel,
The isomorphism problem forall hyperbolic groups , Geom. Funct. Anal. (2011), no. 2, 223–300.MR 2795509[DL07] Warren Dicks and Peter A. Linnell, L -Betti numbers of one-relatorgroups , Math. Ann. (2007), no. 4, 855–874. MR 2285740[DT19] Fran¸cois Dahmani and Nicholas Touikan, Deciding isomorphy using Dehnfillings, the splitting case , Invent. Math. (2019), no. 1, 81–169.MR 3904450[FL17] Stefan Friedl and Wolfgang L¨uck,
Universal L -torsion, polytopes andapplications to 3-manifolds , Proc. Lond. Math. Soc. (3) (2017), no. 6,1114–1151. MR 3661347[Fox53] Ralph H. Fox, Free differential calculus. I. Derivation in the free groupring , Ann. of Math. (2) (1953), 547–560. MR 53938[FT20] Stefan Friedl and Stephan Tillmann, Two-generator one-relator groupsand marked polytopes , Ann. Inst. Fourier (Grenoble) (2020), no. 2,831–879. MR 4105952[GL17] Vincent Guirardel and Gilbert Levitt, JSJ decompositions of groups ,Ast´erisque (2017), no. 395, vii+165. MR 3758992 [HK] Fabian Henneke and Dawid Kielak,
The agrarian polytope of two-generator one-relator groups , Journal of the London Mathematical Soci-ety n/a , no. n/a.[JZLA20] Andrei Jaikin-Zapirain and Diego L´opez- ´Alvarez,
The strong Atiyah andL¨uck approximation conjectures for one-relator groups , Math. Ann. (2020), no. 3-4, 1741–1793. MR 4081128[Kha04] Bilal Khan,
The structure of automorphic conjugacy in the free groupof rank two , Computational and experimental group theory, Contemp.Math., vol. 349, Amer. Math. Soc., Providence, RI, 2004, pp. 115–196.MR 2077762[KW99a] Ilya Kapovich and Richard Weidmann,
On the structure of two-generatedhyperbolic groups , Math. Z. (1999), no. 4, 783–801. MR 1709496[KW99b] ,
Two-generated groups acting on trees , Arch. Math. (Basel) (1999), no. 3, 172–181. MR 1705011[Lev05a] Gilbert Levitt, Automorphisms of hyperbolic groups and graphs of groups ,Geom. Dedicata (2005), 49–70. MR 2174093[Lev05b] ,
Characterizing rigid simplicial actions on trees , Geometric meth-ods in group theory, Contemp. Math., vol. 372, Amer. Math. Soc., Prov-idence, RI, 2005, pp. 27–33. MR 2139674[LL78] Jacques Lewin and Tekla Lewin,
An embedding of the group algebra ofa torsion-free one-relator group in a field , J. Algebra (1978), no. 1,39–74. MR 485972[Log16a] Alan D. Logan, The JSJ-decompositions of one-relator groups with tor-sion , Geom. Dedicata (2016), 171–185. MR 3451463[Log16b] ,
The outer automorphism groups of two-generator, one-relatorgroups with torsion , Proc. Amer. Math. Soc. (2016), no. 10, 4135–4150. MR 3531167[LS77] Roger C. Lyndon and Paul E. Schupp,
Combinatorial group theory , Clas-sics in Mathematics, Springer-Verlag, Berlin, 1977, Reprint of the 1977edition. MR 1812024[LT17] Larsen Louder and Nicholas Touikan,
Strong accessibility for finitely pre-sented groups , Geom. Topol. (2017), no. 3, 1805–1835. MR 3650082[LW21] Larsen Louder and Henry Wilton, One-relator groups with torsion arecoherent , Math. Res. Letters (to appear), arXiv:1805.11976 (2021).[Mag30] Wilhelm Magnus, ¨Uber diskontinuierliche Gruppen mit einer definieren-den Relation. (Der Freiheitssatz) , J. Reine Angew. Math. (1930),141–165. MR 1581238[Mag32] ,
Das Identit¨atsproblem f¨ur Gruppen mit einer definierenden Re-lation , Math. Ann. (1932), no. 1, 295–307. MR 1512760[MKS04] Wilhelm Magnus, Abraham Karrass, and Donald Solitar,
Combinatorialgroup theory , second ed., Dover Publications, Inc., Mineola, NY, 2004,Presentations of groups in terms of generators and relations. MR 2109550[New73] B. B. Newman,
The soluble subgroups of a one-relator group with tor-sion , J. Austral. Math. Soc. (1973), 278–285, Collection of articlesdedicated to the memory of Hanna Neumann, III. MR 0338188[Pri77] Stephen J. Pride, The isomorphism problem for two-generator one-relatorgroups with torsion is solvable , Trans. Amer. Math. Soc. (1977), 109–139. MR 430085
SJ DECOMPOSITIONS AND POLYTOPES FOR ONE-RELATOR GROUPS 37 [Ros86] Gerhard Rosenberger,
All generating pairs of all two-generator Fuchsiangroups , Arch. Math. (Basel) (1986), no. 3, 198–204. MR 834836[RS97] E. Rips and Z. Sela, Cyclic splittings of finitely presented groups andthe canonical JSJ decomposition , Ann. of Math. (2) (1997), no. 1,53–109. MR 1469317[Sel95] Z. Sela,
The isomorphism problem for hyperbolic groups. I , Ann. of Math.(2) (1995), no. 2, 217–283. MR 1324134[Sel97] ,
Structure and rigidity in (Gromov) hyperbolic groups and dis-crete groups in rank Lie groups. II , Geom. Funct. Anal. (1997), no. 3,561–593. MR 1466338[Sel09] , Diophantine geometry over groups. VII. The elementary theoryof a hyperbolic group , Proc. Lond. Math. Soc. (3) (2009), no. 1, 217–273. MR 2520356[Ser03] Jean-Pierre Serre, Trees , Springer Monographs in Mathematics,Springer-Verlag, Berlin, 2003, Translated from the French original byJohn Stillwell, Corrected 2nd printing of the 1980 English translation.MR 1954121[SK20] Meda Satish Suraj Krishna,
Immersed cycles and the JSJ decomposition ,Algebr. Geom. Topol. (2020), no. 4, 1877–1938. MR 4127086[Sta83] John R. Stallings, Topology of finite graphs , Invent. Math. (1983),no. 3, 551–565. MR 695906[Wis12] Daniel T. Wise, From riches to raags: 3-manifolds, right-angled Artingroups, and cubical geometry , CBMS Regional Conference Series inMathematics, vol. 117, Published for the Conference Board of the Math-ematical Sciences, Washington, DC; by the American Mathematical So-ciety, Providence, RI, 2012. MR 2986461
Mathematisches Institut, Einsteinstr. 62, 48149 M¨unster, Germany
Email address : [email protected] University of Oxford, Oxford, OX2 6GG, UK
Email address : [email protected] Heriot-Watt University, Edinburgh, EH14 4AS, UK
Email address ::