aa r X i v : . [ m a t h . G R ] O c t Large Groups of De(cid:28) ien y 1J. O. ButtonSelwyn CollegeUniversity of CambridgeCambridge CB3 9DQU.K.jb128dpmms. am.a .uk1 INTRODUCTION 2Abstra tWe prove that if a group possesses a de(cid:28) ien y 1 presentation whereone of the relators is a ommutator then it is the integers times theintegers, is large, or is as far as possible from being residually (cid:28)nite.Then we use this to show that a mapping torus of an endomorphism ofa (cid:28)nitely generated free group is large if it ontains the integers timesthe integers as a subgroup of in(cid:28)nite index, as well as showing thatsu h a group is large if it ontains a Baumslag-Solitar group of in(cid:28)niteindex and has a (cid:28)nite index subgroup with (cid:28)rst Betti number at least2. We give appli ations to free by y li groups, 1 relator groups andresidually (cid:28)nite groups.1 Introdu tionRe all [45℄ that a (cid:28)nitely generated group G is large if it has a (cid:28)nite indexsubgroup possessing a homomorphism onto a non-abelian free group. This isa strong property and implies that G ontains a non-abelian free subgroup[42℄, G is SQ-universal [45℄ (every ountable group is a subgroup of a quo-tient of G ), G has (cid:28)nite index subgroups with arbitrarily large (cid:28)rst Bettinumber [37℄, G has uniformly exponential word growth [25℄, as well as hav-ing subgroup growth of stri t type n n (whi h is the largest possible growthfor (cid:28)nitely generated groups) [38℄, and the word problem for G is solvablestrongly generi ally in linear time [32℄. Thus on proving that G is large weobtain all these other properties for free.There have been a range of results that give riteria for (cid:28)nitely generatedor (cid:28)nitely presented groups to be large. Starting with B. Baumslag andS. J. Pride [2℄ whi h showed that groups with a presentation of de(cid:28) ien y atleast 2 are large, we then have in [23℄ a ondition that implies this result,as well as a proof that a group with a de(cid:28) ien y 1 presentation in whi hone of the relators is a proper power is large. This latter result was alsoindependently derived by Stöhr in [48℄ and was followed by onditions for agroup with a de(cid:28) ien y 0 presentation where some of the relators are properpowers to be large, due to Edjvet in [19℄. Then further onditions for a(cid:28)nitely presented group to be large, all of whi h imply the Baumslag-Prideresult, are by Howie in [29℄, G. Baumslag in [6℄ and a hara terisation byLa kenby in [34℄. In Se tion 2 we give a riterion, based on the Howie result,for a (cid:28)nitely presented group G to be large whi h is purely in terms of the INTRODUCTION 3Alexander polynomial of G and is straightforward to use in pra ti e. Thisresult is parti ularly powerful in the ase of de(cid:28) ien y 1 groups whi h arethen our fo us for mu h of the rest of the paper. Of ourse unlike groups ofde(cid:28) ien y 2 or higher, not all groups of de(cid:28) ien y 1 are large: think of Z orthe soluble Baumslag-Solitar groups given by the presentations h x, y | xyx − = y m i for m ∈ Z \{ } . Other examples of non-large de(cid:28) ien y 1 groups weregiven by Pride and Edjvet in [20℄ onsisting of those Baumslag-Solitar groups h x, y | xy l x − = y m i for l, m = 0 where l and m are oprime, as well someHNN extensions of these, and one an (cid:28)nd the odd further example in theliterature.As for large groups of de(cid:28) ien y 1, we have already mentioned those witha relator that is a proper power and we again have examples in [20℄ withTheorem 6 stating that the group h x, y | x n y l x − n = y m i for l, m, n = 0 islarge if | n | > or if l and m are not oprime. Further results of a morete hni al nature whi h give largeness for some other 2 generator 1 relatorpresentations are in [18℄. At this point it seems di(cid:30) ult to say onvin inglyeither way whether groups of de(cid:28) ien y 1 are generally large. In this paperwe hope to o(cid:27)er substantial eviden e that largeness is a natural property toexpe t in a de(cid:28) ien y 1 group. Although we will display a few new groupsof de(cid:28) ien y 1 whi h are not large in Example 3.5(ii), our main results areon establishing families of de(cid:28) ien y 1 groups whi h are all large. In Se tion3 we introdu e the on ept of a non abelian residually abelianised (NARA)group and this has a number of equivalent de(cid:28)nitions, one of whi h is that itis (cid:28)nitely generated and non-abelian but has no non-abelian (cid:28)nite quotients;the idea being that a NARA group G is as far from being residually (cid:28)niteas possible be ause we annot distinguish G from its abelianisation G/G ′ byjust looking at (cid:28)nite index subgroups. We obtain Theorem 3.6 whi h statesthat if G has a de(cid:28) ien y 1 presentation in whi h one of its relators is a ommutator then G = Z × Z or G is NARA with abelianisation Z × Z or G is large.In [18℄ from 1984 it is asked if those groups whi h are an extension ofa (cid:28)nitely generated non-abelian free group by Z are large. They are er-tainly torsion free groups with a natural de(cid:28) ien y 1 presentation and arealso alled mapping tori of (cid:28)nitely generated non-abelian free group auto-morphisms. These groups appear to make up a sizeable lass of de(cid:28) ien y 1groups but we an expand this lass onsiderably by allowing arbitrary endo-morphisms in pla e of automorphisms to obtain groups whi h are as endingHNN extensions of (cid:28)nitely generated free groups. Su h groups have been the INTRODUCTION 4attention of mu h re ent resear h where signi(cid:28) ant progress has been made.In parti ular these groups have been shown to be oherent (every (cid:28)nitelygenerated subgroup is (cid:28)nitely presented) in [21℄, Hop(cid:28)an in [22℄ and evenresidually (cid:28)nite in [13℄. If largeness were added to this list (on removing theobvious small ex eptions) then it would show that su h an HNN extension,indeed even a group whi h is virtually su h an HNN extension, has all theni e properties that one ould reasonably hope for.In Se tion 4 we apply our results to show that for G a mapping torusof a (cid:28)nitely generated free group endomorphism, we have G is large if it ontains a Z × Z subgroup of in(cid:28)nite index. Also G is large if it ontains aBaumslag-Solitar subgroup and has a (cid:28)nite index subgroup H ( = Z × Z ) with β ( H ) ≥ . Of ourse if β ( H ) = 1 for all H then we would have an exampleof su h a G whi h is not large. However we know of no examples apart fromthe soluble Baumslag-Solitar groups themselves, and it seems believable thatno other G has this property.In Se tion 5 we restri t to groups G of the form F -by- Z where F is free.By Se tion 4 G is large if it ontains Z × Z and F = F n is of (cid:28)nite rank n ≥ . It is known by [8℄, [9℄ and [14℄ that these are exa tly the groupsof the form F n -by- Z whi h are not word hyperboli . We also show that if F is of in(cid:28)nite rank but G is (cid:28)nitely generated then G is in fa t large. By ombining these results with known fa ts about word hyperboli groups, thisallows us to prove in Theorem 5.4 that if G is any (cid:28)nitely generated groupwhi h is virtually free-by- Z then (apart from the obvious small ex eptions) G is SQ-universal, has uniformly exponential growth and has a word problemthat is solvable strongly generi ally in linear time. This is also true for the(cid:28)nitely generated subgroups of G .Se tion 6 looks at 1 relator groups G , where we need only onsider the ase where G has a 2 generator 1 relator presentation. We know that bySe tion 3 we obtain largeness unless G = Z × Z (whi h is easily dete tedin the 1 relator ase) or G is NARA. It is true that 2 generator 1 relatorgroups whi h are NARA exist, but if we insist that the relator is a produ tof ommutators then no examples are known; indeed it was only re entlythat non residually (cid:28)nite examples of su h groups were given in [43℄ Problem(OR7). Moreover if the relator is a single ommutator then no examples areknown that fail even to be residually (cid:28)nite (this is Open Problem (OR8) in[43℄) so in this ase not being NARA and hen e large seems very likely. Wegive methods that show this in pra ti e for a given presentation. Finally inSe tion 7 we make the straightforward but useful observation that a group G A CONDITION FOR LARGENESS 5is large if and only if the quotient of G by its (cid:28)nite residual is large, suggestingthat the best setting in whi h to examine largeness is the residually (cid:28)nite ase. We prove that a residually (cid:28)nite group with in(cid:28)nitely many ends islarge (this is most de(cid:28)nitely not true if residual (cid:28)niteness is removed) andexamine (cid:28)nitely presented groups whi h are LERF, whi h is a strengtheningof being residually (cid:28)nite.The author would like to a knowledge helpful omments from IlyaKapovi h, Gilbert Levitt and Ale Mason, as well as thanking the referee fora thorough reading of the paper.2 A Condition for LargenessWe quote the following fa ts that we will need about the Alexander polyno-mial of a (cid:28)nitely presented group; see [35℄. Let G be given by a (cid:28)nite presenta-tion h x , . . . , x n | r , . . . , r m i and let G ′ be the derived ( ommutator) subgroupof G . Then the abelianisation G = G/G ′ is a (cid:28)nitely generated abelian group Z β ( G ) × T for T the torsion subgroup whereas the free abelianisation ab ( G ) is Z β ( G ) . On taking any surje tive homomorphism χ : G → Z , we have theAlexander polynomial ∆ G,χ ∈ Z [ t ± ] whi h is a Laurent polynomial up to theambiguity of multipli ation by the units ± t k for k ∈ Z . It is de(cid:28)ned in thefollowing way: on taking t to be an element in G with χ ( t ) = 1 we have that t a ts by onjugation on H ( ker χ ; Z ) , so H ( ker χ ; Z ) is a module over thegroup ring Z [ t ± ] of the integers. It is easy to see that this is a (cid:28)nitely pre-sented module, for instan e we ould use the Reidemeister-S hreier rewritingpro ess to obtain a presentation of ker χ from that of G and then abelianise,whi h would result here in an ( n − × m presentation matrix. Thus wehave the (cid:28)rst elementary ideal whi h is generated by the maximal minors,these being the determinants of the matri es left over when we ross o(cid:27) the orre t number of olumns to make the resulting matrix square (here we areassuming there are at least as many olumns as rows, or else we let the (cid:28)rstelementary ideal and the Alexander polynomial be zero). Note that this idealis independent of the parti ular presentation matrix hosen for H ( ker χ ; Z ) .The de(cid:28)nition of the Alexander polynomial ∆ G,χ ( t ) is then the generator (upto units) of the smallest prin ipal ideal ontaining the (cid:28)rst elementary ideal,or equivalently the highest ommon fa tor of the maximal minors.The next point is the ru ial fa t whi h allows us to use the Alexanderpolynomial to dete t largeness. A CONDITION FOR LARGENESS 6Theorem 2.1 If G is a (cid:28)nitely presented group whi h has a homomorphism χ onto Z su h that ∆ G,χ = 0 then G is large.Proof. We have seen that H ( ker χ ; Z ) is a (cid:28)nitely presented module over Z [ t ± ] but we an also take rational oe(cid:30) ients and use the fa t that H ( ker χ ; Z ) ⊗ Z Q = H ( ker χ ; Q ) is a (cid:28)nitely presented module over Q [ t ± ] where t a ts in the same way, and we even have the same presentation matrix.Thus we an de(cid:28)ne the Alexander polynomial over Q exa tly as above interms of the (cid:28)rst elementary ideal, and it will be the same polynomial as for Z , ex ept that now it is only de(cid:28)ned up to units of Q [ t ± ] whi h are now qt ± n for q ∈ Q \{ } . However note that ∆ G,χ is zero over Z if and only if it is zeroover Q . The advantage of moving to rational oe(cid:30) ients is that Q [ t ± ] is aprin ipal ideal domain, so by the stru ture theorem it is a dire t sum of y li modules. Thus the presentation matrix P an be put into anoni al form inwhi h all o(cid:27)-diagonal entries are zero and the diagonal entries are d , . . . , d k for d i ∈ Q [ t ± ] . By evaluating the (cid:28)rst elementary ideal we see that theAlexander polynomial over Q is d . . . d k and this is zero if and only if some d i is zero whi h happens if and only if H ( ker χ ; Q ) has a free Q [ Z ] -moduleof at least rank 1 in its de omposition.Now we invoke Howie's ondition for largeness in [29℄ Se tion 2. Adoptingthat notation, we let K be the standard onne ted 2- omplex obtained fromour (cid:28)nite presentation of G , with N = ker χ and K the 2- omplex whi h isthe regular overing of K orresponding to N so that π ( K ) = N . Let F bea (cid:28)eld: on following through the proof of [29℄ Proposition 2.1, we see that if H ( K ; F ) ontains a free F [ Z ] -module of rank at least 1 then the on lusionof the proposition holds. But this is the hypothesis of [29℄ Theorem 2.2 whi hproves that for any su(cid:30) iently large n the (cid:28)nite index subgroup N G n admitsa homomorphism onto the free group of rank 2.In our ase we have on setting F = Q that H ( K ; Q ) = H ( ker χ ; Q ) soif ∆ G,χ = 0 we on lude that G is large. ✷ Note: the above also works if we take F to be Z /p Z and ∆ G,χ vanishesover this (cid:28)eld.Corollary 2.2 If G is a (cid:28)nitely presented group possessing a homomorphismto Z with kernel having in(cid:28)nite rational (cid:28)rst Betti number then G is large.Proof. We have by de(cid:28)nition that β ( ker χ ; Q ) is the dimension of H ( ker χ ; Q ) as a ve tor spa e over Q . It is also the degree of the Alexan-der polynomial ∆ G,χ (where the degree of a Laurent polynomial in t is the DEFICIENCY 1 GROUPS 7degree of the highest non-zero power of t minus the degree of the lowest) by[35℄ Theorem 6.17 or [40℄ Se tion 4. In parti ular ∆ G,χ = 0 if and only if β ( ker χ ; Q ) is in(cid:28)nite, so this laim now follows dire tly from Theorem 2.1. ✷ Note: The Corollary is most de(cid:28)nitely not true for all (cid:28)nitely generatedgroups; we do require a (cid:28)nite number of relators too, as an be seen by theexample of the restri ted wreath produ t Z ≀ Z .3 De(cid:28) ien y 1 groupsThe de(cid:28) ien y of a (cid:28)nite presentation is the number of generators minus thenumber of relators and the de(cid:28) ien y def ( G ) of a (cid:28)nitely presented group G is the maximum de(cid:28) ien y over all presentations. (It is bounded above by β ( G ) so is (cid:28)nite.) We know that groups of de(cid:28) ien y at least 2 are largeso it seems reasonable to ask whether we an use our riterion to obtainlarge groups with lower de(cid:28) ien ies, for instan e de(cid:28) ien y 1. In fa t this ase turns out to be a very fruitful hoi e, both from the point of view that al ulating the Alexander polynomial of a de(cid:28) ien y 1 group is more e(cid:30) ientthan for lower de(cid:28) ien ies, and be ause of the behaviour of de(cid:28) ien y in(cid:28)nite overs. Given a presentation for a group G with n generators and m relators and an index i subgroup H of G , we an use Reidemeister-S hreierrewriting to obtain a presentation for H of G with ( n − i + 1 generatorsand mi relators, thus the de(cid:28) ien y of H is at least ( def ( G ) − i + 1 . Soif def ( G ) = 1 then either def ( H ) = 1 for all H ≤ f G or H , and thus G , islarge anyway by [2℄.Theorem 3.1 If G is a group with a de(cid:28) ien y 1 presentation h x , . . . , x n | r , . . . , r n − i where one of the relators is of the form x i x j x − i x − j then G islarge if the subgroup of ab ( G ) generated by the images of x i and x j has in(cid:28)niteindex.Proof. Without loss of generality we an reorder the generators and sowe an assume we have the relator x x x − x − . As ab ( G ) is a free abeliangroup Z β ( G ) of (cid:28)nite rank, we have that x and x generate a free abeliansubgroup of stri tly smaller rank. Therefore there must exist a surje tivehomomorphism χ : G → Z with x and x in the kernel, as well as oprimeintegers k , . . . , k n su h that k χ ( x ) + . . . + k n χ ( x n ) = 1 . Therefore thereexists a matrix M ∈ GL ( n − , Z ) su h that its (cid:28)rst olumn is ( k , . . . , k n ) DEFICIENCY 1 GROUPS 8and this gives rise to an automorphism β of Z n − sending the standard basis e , . . . , e n (where we think of e i as the image of the generator x i in theabelianisation Z n of the free group F n ) to a new basis b , . . . , b n . Now by [39℄I.4.4, we have an automorphism α of F n that (cid:28)xes x , x and indu es β on e , . . . , e n . On rewriting our presentation in terms of y = α ( x ) , . . . , y n = α ( x n ) , we now have χ ( y ) = 1 and so we an regard H ( ker χ ; Z ) as a Z [ t ± ] module where t is equal to y and a ts by onjugation. We an obtain apresentation matrix P for this module by performing Reidemeister-S hreierrewriting on G using { t j : j ∈ Z } as a S hreier transversal. We (cid:28)nd that ouroriginal relation x x x − x − be omes the set of group relations x ,j x ,j x − ,j x − ,j where x ,j = t j x t − j and x ,j = t j x t − j . To obtain the equivalent relation for P , we abelianise and regard ea h of these group relations as the same modulerelation multiplied by powers of t . But this be omes zero, thus giving us azero olumn in P .The ru ial point about the group presentation having de(cid:28) ien y one isthat this makes P a square matrix (of size n − ). This means that theAlexander polynomial ∆ G,χ is merely the determinant of P , whi h must bezero owing to the zero olumn, hen e we have largeness of G by Theorem2.1. ✷ Corollary 3.2 If G = h x , . . . , x n | r , . . . , r n − i has a de(cid:28) ien y 1 presenta-tion with a relator r = x x x − x − and the abelianisation G = Z × Z × Z /m Z for m ≥ then G is large.Proof. We are done by Theorem 3.1 unless the images x , x in G generatea (cid:28)nite index subgroup S of G , but if so then S must have Z -rank equalto that of G , whi h is 2. However S is generated by two elements so inthis ase S an only be isomorphi to Z × Z . Now take a homomorphism θ from G onto Z /j Z for some j ≥ su h that S is in the kernel. We requireanother generator g ∈ { x , . . . , x n } su h that θ ( g ) generates Im θ but this an be a hieved by taking an appropriate automorphism of the free groupof rank n that (cid:28)xes x and x , just as in Theorem 3.1. We now performReidemeister-S hreier rewriting to obtain from our original presentation of G a de(cid:28) ien y 1 presentation for ker θ onsisting of nj + 1 generators and nj relators. We have g i , ≤ i < j as a S hreier transversal for ker θ in G and on setting x ,i = g i x g − i and x ,i = g i x g − i , whi h will all beamongst the generators for our presentation of ker θ given by this pro ess DEFICIENCY 1 GROUPS 9(be ause x , x ∈ S ≤ ker θ ), our original relator r gives rise to j relators x ,i x ,i x − ,i x − ,i in the presentation for our subgroup. As these disappear whenwe abelianise, we see that β ( ker θ ) is at least j + 1 and we are done byTheorem 3.1. ✷ It might be felt that requiring two generators to ommute in a de(cid:28) ien y1 presentation is rather restri tive but most of the rest of our results arebased on (cid:28)nding de(cid:28) ien y 1 groups G whi h have a (cid:28)nite index subgroup H possessing su h a presentation. This means β ( H ) ≥ and Corollary 3.2 willapply unless the abelianisation H = Z × Z . We now dis uss a generalisation ofthe property of being residually (cid:28)nite whi h allows us to avoid this ex eption.Re all that a group G is residually (cid:28)nite if the interse tion R G over all the(cid:28)nite index subgroups F ≤ f G is the trivial group I . Although this worksperfe tly well as a general de(cid:28)nition, it is most useful when G is (cid:28)nitelygenerated and that will be our assumption here. Our motivation for the nextde(cid:28)nition is to ask: how badly an a group fail to be residually (cid:28)nite andwhat is the worst possible ase? The (cid:28)rst answer that would ome to mind iswhen G ( = I ) has no proper (cid:28)nite index subgroups at all, but we are dealingwith groups possessing positive (cid:28)rst Betti number and hen e in(cid:28)nitely manysubgroups of (cid:28)nite index. By noting that elements outside the ommutatorsubgroup G ′ annot be in R G , we obtain our ondition.De(cid:28)nition 3.3 We say that the (cid:28)nitely generated group G is residuallyabelianised if G ′ = \ F ≤ f G F. If further G is non-abelian then we say it is NARA (non-abelian residu-ally abelianised).Note that by ex luding G being abelian, we have that G residually (cid:28)niteimplies G is not NARA. The de(cid:28)nition has many equivalent forms but thegeneral idea is that a NARA group annot be distinguished from its abeliani-sation if one only uses standard information about its (cid:28)nite index subgroups.Proposition 3.4 Let G be (cid:28)nitely generated and non-abelian with ommu-tator subgroup G ′ , abelianisation G = G/G ′ and let R G be the interse tion ofthe (cid:28)nite index subgroups of G . The following are equivalent: DEFICIENCY 1 GROUPS 10(i) G is NARA.(ii) G has no non-abelian (cid:28)nite quotient.(iii) G has no non-abelian residually (cid:28)nite quotient.(iv) If a n ( G ) denotes the number of (cid:28)nite index subgroups of G having index n then a n ( G ) = a n ( G ) for all n .(v) For all F ≤ f G we have F ′ = G ′ .(vi) For all F ≤ f G we have F ∩ G ′ = G ′ .(vii) For all F ≤ f G we have F ′ = F ∩ G ′ .Proof. The equivalen e of (i) with (ii) is immediate on dropping down to a(cid:28)nite index normal subgroup. We have (iii) implies (ii) and (i) implies (iii)as any residually (cid:28)nite image of G must fa tor through G/R G . As for (iv),this is just using the index preserving orresponden e between the subgroupsof G and the subgroups of G ontaining G ′ .As for the rest, we have that F ′ ≤ F ∩ G ′ ≤ G ′ whenever F is a subgroupof G . If (i) holds for G with F a (cid:28)nite index subgroup then R F = R G = G ′ but R F is inside F ′ so F ′ and G ′ are equal, giving (v). This immediatelyimplies (vi) and (vii) so we just require that these two in turn imply (i). Thisis obvious for (vi) and for (vii) we an adopt the proof of [36℄ Theorem 4.0.8whi h states that if Γ is a residually (cid:28)nite group then for ea h of its (nontriv-ial) y li subgroups there exists a homomorphism onto another (nontrivial) y li group whi h an be extended to a (cid:28)nite index subgroup of Γ . If (i)fails then take F ≤ f G and g ∈ G ′ but g / ∈ F . Dropping down to N ≤ F with N ✂ f G , we have H = N h g i ≤ f G and H/N ∼ = h g i / ( N ∩ h g i ) . Thus g / ∈ H ′ be ause by being outside N it survives under a homomorphism from H to an abelian group. But g is ertainly in H ∩ G ′ . ✷ The importan e of ondition (vii) holding for G is that we fail to pi k upextra abelianisation in (cid:28)nite overs F ≤ f G sin e F/F ′ is just F/ ( F ∩ G ′ ) ∼ = F G ′ /G ′ ≤ f G/G ′ . In parti ular β ( F ) = β ( G ) so G is not large.Example 3.5(i) The Thompson group T is NARA. This group has a 2 generator 2 relatorpresentation with abelianisation Z × Z and its ommutator subgroup T ′ hasno proper (cid:28)nite index subgroups as T ′ is in(cid:28)nite and simple; see [17℄. Butfor F ≤ f T we have F ∩ T ′ ≤ f T ′ thus T ′ ≤ F .(ii) If G is in(cid:28)nite but has no proper (cid:28)nite index subgroups then G is NARA. DEFICIENCY 1 GROUPS 11Moreover for any su h G and any residually abelianised group A we have Γ = G ∗ A is NARA be ause if N ✂ f Γ then N ∩ G ✂ f G so G ≤ N . Thisimplies that the normal losure C of G is in N so Γ /N must be abelian as itis a (cid:28)nite quotient of Γ /C ∼ = A . A famous example that will do for G is theHigman group H with 4 generators and 4 relators as introdu ed in [27℄. Ithas β ( G ) = 0 so its de(cid:28) ien y must be zero. Thus H ∗ H is NARA so it toohas no proper (cid:28)nite index subgroups, sin e it is in(cid:28)nite and equals its own ommutator subgroup. By repeating this onstru tion we obtain H ∗ . . . ∗ H using n opies of H whi h gives us examples of NARA groups G n whi h anhave arbitrarily many generators (by the Grushko-Neumann theorem) andwith β ( G ) = 0 and de(cid:28) ien y zero. In order to obtain examples of de(cid:28) ien y1 NARA groups we an take the free produ t of G n with Z or Z × Z so thatthe resulting groups need arbitrarily many generators and have their (cid:28)rstBetti number equal to 1 or 2. Of ourse there are no groups of de(cid:28) ien ytwo or higher whi h are NARA be ause they are all large.(iii) There are 1 relator groups whi h are NARA: the (cid:28)rst example dates ba kto a short paper [3℄ of G. Baumslag in 1969 entitled (cid:16)A non- y li one-relatorgroup all of whose (cid:28)nite quotients are y li (cid:17) with the group in questionbeing h a, b | a = a − b − a − bab − ab i . However Baumslag-Solitar groups are not NARA, as an be seen by takingquotients onto dihedral groups.In terms of its wide appli ation, the following is our main result on large-ness of de(cid:28) ien y 1 groups.Theorem 3.6 If G has a de(cid:28) ien y 1 presentation h F n | R i where one of therelators is a ommutator in F n then exa tly one of these o urs:(i) G = Z × Z .(ii) G is NARA with abelianisation Z × Z .(iii) G is large.In parti ular if there exists H ≤ f G with H = Z × Z then G is large.Proof. If our relator r = uvU V for u, v words in the generators for F n thenwe an regard r as the ommutator of two generators simply by adding u and v to the generators and their de(cid:28)nitions to the relators, noting that this doesnot hange the de(cid:28) ien y. We must have β ( G ) ≥ and if β ( G ) ≥ (or ifthe subgroup generated by the images of u and v has Z -rank less than 2) then G is large by Theorem 3.1. Moreover if G has non-trivial torsion then we have DEFICIENCY 1 GROUPS 12largeness by Corollary 3.2 so the only ase in whi h the given presentation for G does not show largeness is when G = Z × Z with the subgroup S = h u, v i (where u and v are the images of u and v in G ) being of (cid:28)nite index. Ifneither (i) nor (ii) are true then Proposition 3.4 tells us that none of thegiven seven onditions hold for G , so the failure of (vii) means that there isa subgroup L ≤ f G with γ in L ∩ G ′ but not in L ′ . Consequently L = L/L ′ is an abelian group whi h surje ts to L/ ( L ∩ G ′ ) with γ in the kernel. But L/ ( L ∩ G ′ ) is isomorphi to LG ′ /G ′ whi h is a (cid:28)nite index subgroup of G andthus is equal to Z × Z . As this is a Hop(cid:28)an group, we on lude that β ( L ) ≥ but L = Z × Z . We know that L also has a de(cid:28) ien y 1 presentation whi hwe an obtain from G by rewriting and if one of the relations in su h apresentation for L were a ommutator then we ould on lude by Theorem3.1 or Corollary 3.2 applied to this presentation that L , and hen e G , islarge. In fa t although we annot guarantee this, we now show that there isa (cid:28)nite index subgroup of L whi h has su h a presentation, along with thene essary abelianisation. We do this by keeping tra k of what happens toour original relator r when we perform Reidemeister-S hreier rewriting ondropping to a (cid:28)nite index subgroup. The pro ess of rewriting for a subgroup H in G = h g , . . . , g n i involves taking a S hreier transversal T to obtain agenerating set for H of the form tg i ( tg i ) − , where t ∈ T and x is the elementof T in the same oset as x . In parti ular by taking t equal to the identitywe have that if a generator g i of G is in H then it be omes a generator of H .Moreover as the relators for H are obtained by expressing the relators tr j t − in terms of these generators, any relator made up solely from those g i whi hare ontained in H will remain un hanged in the presentation for H .First we take H to be the normal subgroup of (cid:28)nite index in G whi his the inverse image of h u, v i under the abelianisation map from G to G .Then the de(cid:28) ien y 1 presentation for H has r as one of its relators thuswe have largeness for H and G unless H = Z × Z . However if so then wemust have H ′ = H ∩ G ′ . This is be ause otherwise we have a surje tive butnon-inje tive homomorphism from H/H ′ to H/ ( H ∩ G ′ ) ∼ = Z × Z , thus we an use Hop(cid:28) ity again. Moreover as G ′ ≤ H we get H ′ = G ′ .Let k, l be the minimum positive integers su h that a = u k and b = v l are in L . We set N = h H ′ , a, b i whi h is a (cid:28)nite index normal subgroup of H and of G , with G/N abelian. We rewrite for N in H in two stages; (cid:28)rst wedrop to the subgroup with exponent sum of u equal to 0 mod k and rewriteusing the transversal u i , ≤ i < k , and then we do the same with v . In bothof these stages a and b will be amongst the generators for N and our relator ASCENDING HNN EXTENSIONS OF FREE GROUPS 13 uvU V in H gives rise to a relator ava − v − after the (cid:28)rst rewrite, and thenthis be omes abAB . Thus we have G ′ ≤ N ✂ f G with N having a de(cid:28) ien y1 presentation whi h in ludes generators a, b and the relator abAB .Finally we go from N to the subgroup L ∩ N ≤ f G whi h on rewritingwill keep a and b be ause they are generators in the presentation for N whi halso lie in L ∩ N , and onsequently abAB remains too. Now our γ ∈ L frombefore whi h is in G ′ \ L ′ is also in N as G ′ ≤ N . But from above we havea surje tive homomorphism from L to Z × Z × Z /j Z for some j ≥ with γ mapping onto the Z /j Z fa tor. But then we an restri t this surje tion tothe (cid:28)nite index subgroup L ∩ N whi h also ontains γ so L ∩ N has the rightpresentation and the right homology to obtain largeness. ✷ Note that Example 3.5 (ii) shows that ase (ii) in Theorem 3.6 an o urbut this is the only type of example known to us.4 As ending HNN extensions of free groupsA wide and important lass of de(cid:28) ien y 1 groups is obtained by taking afree group F n with free basis x , . . . , x n and an endomorphism θ of F n to reate the mapping torus G = h x , . . . , x n , t | tx t − = θ ( x ) , . . . , tx n t − = θ ( x n ) i . We all su h a presentation a standard presentation for G . We do not assumethat θ is inje tive or surje tive. However there is a neat way of sidesteppingthe non-inje tive ase using [31℄ where it is noted that G is isomorphi toa mapping torus of an inje tive free group homomorphism ˜ θ : F m → F m where m ≤ n . Of ourse it might be that F n is non-abelian but m = 0 or 1, however this would mean that G = Z or h a, t | tat − = a k i for k = 0 .However in these ases G is soluble and so is de(cid:28)nitely not large. Thereforewe will assume throughout that θ is inje tive, whereupon G is also alled anas ending HNN extension of the free group F n , where we onjugate the base F n to an isomorphi subgroup of itself using the stable letter t .We have that our base F n = h x , . . . , x n i embeds in G and we will referto this opy of F n in G as Γ . Then t Γ t − = θ (Γ) whi h is equal to Γ if andonly if θ is surje tive in whi h ase G is free by Z . Otherwise θ (Γ) < Γ , with θ (Γ) being isomorphi to F n meaning that it has in(cid:28)nite index in Γ . ASCENDING HNN EXTENSIONS OF FREE GROUPS 14On e an as ending HNN extension G is formed, there is an obvious ho-momorphism χ from G onto Z asso iated with it whi h is given by χ ( t ) = 1 and χ (Γ) = 0 , so that ker χ = ∞ [ i =0 t − i Γ t i . In the ase of an automorphism ker χ is just Γ but otherwise θ (Γ) = t Γ t − < Γ so that ker χ is a stri tly as ending union of free groups, thus is in(cid:28)nitelygenerated and lo ally free, but never free be ause β ( ker χ ; Q ) ≤ β (Γ; Q ) .The following result, whi h is Lemma 3.1 in [22℄, allows us to re ogniseas ending HNN extensions (cid:16)internally(cid:17).Lemma 4.1 A group G with subgroup Γ is an as ending HNN extensionwith Γ as base if and only if there exists t ∈ G with(1) G = h Γ , t i ;(2) t k Γ for any k = 0 ;(3) t Γ t − ≤ Γ .A strong property that as ending HNN extensions of free groups possessis that they are oherent by [21℄, in fa t the result is more general and givesus this des ription whi h is Theorem 1.2 and Proposition 2.3 in [21℄.Theorem 4.2 If G = h t, F i is an as ending HNN extension of the (possiblyin(cid:28)nitely generated) free group F with asso iated homomorphism χ and H is a (cid:28)nitely generated subgroup of G then H has a (cid:28)nite presentation of theform h s, a , . . . , a k , b , . . . , b l | sa s − = w , . . . , sa k s − = w k i where a i , b j ∈ ker χ and k, l ≥ , with w , . . . , w k words in the a i and the b j .The next proposition gives us standard but useful properties of as endingHNN extensions.Proposition 4.3 Let G be an as ending HNN extension h x , . . . , x n , t | tx t − = θ ( x ) , . . . , tx n t − = θ ( x n ) i with respe t to the inje tive endomorphism θ of the (cid:28)nitely generated freegroup Γ = F n with free basis x , . . . , x n and let χ be the asso iated homomor-phism.(i) Ea h element g of G has an expression of the form g = t − p γt q for p, q ≥ ASCENDING HNN EXTENSIONS OF FREE GROUPS 15and γ ∈ Γ .(ii) For ea h j ∈ N we have for s = t j the normal subgroup G j = h Γ , s i ofindex j in G with G/G j ∼ = Z /j Z whi h has presentation h x , . . . , x n , s | sx s − = θ j ( x ) , . . . , sx n s − = θ j ( x n ) i . (iii) If H ≤ f G then H is also an as ending HNN extension of a (cid:28)nitely gen-erated free group with respe t to the (restri tion to H of the) same asso iatedhomomorphism χ .(iv) If ∆ ≤ f Γ then H = h ∆ , t i has (cid:28)nite index in G = h Γ , t i .Proof. (i) is [21℄ Lemma 2.2 (1).(ii) This is [30℄ Lemma 2.2 (1).(iii) This an be proved dire tly but we may use the fa t that H has apresentation as in Theorem 4.2. We now show that G and all its (cid:28)niteindex subgroups have de(cid:28) ien y exa tly 1 so we must have l = 0 in thispresentation and then the result follows. This also demonstrates that inproving ertain as ending HNN extensions of free groups are large, we aregenuinely (cid:28)nding new examples as opposed to groups that ould be provedlarge by the Baumslag-Pride result [2℄.We know that H has a de(cid:28) ien y 1 presentation by Reidemeister-S hreierrewriting the standard presentation for G . By [46℄ Proposition 3.6 (ii) wehave that the 2- omplex C asso iated to this standard presentation of G isaspheri al and the Euler hara teristi χ ( C ) = 1 − ( n + 1) + n = 0 . Thereforethe (cid:28)nite over ˜ C of C with fundamental group H has χ ( ˜ C ) = 0 and isalso aspheri al. Now by the Hopf formula we have that an upper bound forthe de(cid:28) ien y of any (cid:28)nitely presented group Γ is β (Γ) − β (Γ) . As ˜ C is a K ( H, spa e we have β ( H ) − β ( H ) = β ( ˜ C ) − β ( ˜ C ) = 1 − χ ( ˜ C ) = 1 andso our de(cid:28) ien y 1 presentation for H is best possible.(iv) Let γ , . . . , γ d be a transversal for ∆ in Γ . If we an show that H ∩ K ≤ f K for K the kernel of the asso iated homomorphism then we are done, despitethe fa t that K and H ∩ K are in(cid:28)nitely generated, be ause t ∈ H and any g ∈ G is of the form kt m for k ∈ K .The set S = { t − m γ i t m : m ∈ N , ≤ i ≤ d } ontains an element of every oset of H ∩ K in K . This an be seen by writing k ∈ K as t − m γt m for γ ∈ Γ using (i). Then there is γ i su h that γγ i = δ ∈ ∆ .This means that kt − m γ i t m = t − m δt m whi h is in H and in K . We now show ASCENDING HNN EXTENSIONS OF FREE GROUPS 16that the index of H ∩ K in K is at most d . Note that for q > p , any elementof the form t − p γ i t p is in the same oset as some element of the form t − q γ j t q be ause θ q − p ( γ i ) γ − j = δ for some δ ∈ ∆ and some j ∈ { , . . . , d } , thusgiving t − p γ i t p ( t − q γ j t q ) − = t − q δt q whi h is in H ∩ K . Therefore we pro eedas follows: S is a set indexed by ( l, i ) ∈ N × Z /d Z and we refer to l as thelevel. Choose a transversal T for H ∩ K in K from S whi h a priori ould bein(cid:28)nite and let g be the element in T with smallest level l (and smallest i ifne essary). Then for ea h level l + 1 , l + 2 , . . . above l there is an elementin S with this level that is in the same oset of H ∩ K as g and so annotbe in T . Cross these elements o(cid:27) from S and now take the next element g in T a ording to our ordering of S . Certainly g with level l has not been rossed o(cid:27) and we repeat the pro ess of removing one element in ea h levelabove l ; as these are in the same oset as g they too have not been erasedalready. Now note that we an go no further than g d be ause then we willhave rossed o(cid:27) all elements from all levels above l d ; thus we must have atransversal for H ∩ K in K of no more than d elements. ✷ Let G = h F n , t i be the mapping torus of an inje tive endomorphism θ of the free group F n . We say that θ has a periodi onjuga y lass if thereexists i > , k ∈ Z and w ∈ F n \{ } su h that θ i ( w ) is onjugate to w k in F n . If this is so with θ i ( w ) = vw k v − then let us take the endomorphism φ of F n su h that φ = ι − v θ i where we use ι v to denote the inner automorphismof F n that is onjugation by v . We have on setting ∆ = h w i and s = v − t i that the subgroup h ∆ , s i of G is an as ending HNN extension with base ∆ and stable letter s by Lemma 4.1. Consequently it has the presentation h s, w | sws − = w k i . These presentations are part of the famous family of 2generator 1 relator subgroups known as the Baumslag-Solitar groups. Wede(cid:28)ne the Baumslag-Solitar group BS ( j, k ) = h x, y | xy j x − = y k i for j, k = 0 (and without loss of generality j > ). We have that G ontains BS (1 , k ) for some k if and only if G has a periodi onjuga y lass where θ i ( w ) is onjugate to w k . Furthermore if there exists i, j > , k ∈ Z and w ∈ F n \{ } with θ i ( w j ) onjugate in F n to w k then k = dj and θ i ( w ) is onjugate to w d so that θ has a periodi onjuga y lass. Indeed G annot ontain a subgroupisomorphi to a Baumslag-Solitar group BS ( j, k ) unless j = 1 (or j = k inwhi h ase G ontains BS (1 , anyway).We an now deal with as ending HNN extensions of free groups whi h ontain Z × Z . ASCENDING HNN EXTENSIONS OF FREE GROUPS 17Theorem 4.4 If θ is an inje tive endomorphism of the free group Γ of rank n with w ∈ F n \{ } su h that θ ( w ) = w then there is a (cid:28)nite index subgroup ∆ of Γ and j ≥ su h that ∆ has a free basis in luding w , and su h that θ j (∆) ≤ ∆ .Proof. We use the lassi result [24℄ of Marshall Hall Jnr. that if L is anon-trivial (cid:28)nitely generated subgroup of the non-abelian free group F n thenthere is a (cid:28)nite index subgroup F of F n su h that L is a free fa tor of F . Wejust need to put L = h w i so that F = h w i ∗ C for some C ≤ F n with w abasis element for F . The se ond ondition is the ru ial part. The aim is torepeatedly pull ba k F ; although we do not have F ≤ θ − ( F ) in general asthis is equivalent to θ ( F ) ≤ F whi h would mean we are done, we do (cid:28)nd thatthe index is non-in reasing. To see this note that θ − ( F ) = θ − ( F ∩ θ (Γ)) and θ − θ (Γ) = Γ as θ : Γ → θ (Γ) is an isomorphism. Now the index of F ∩ θ (Γ) in θ (Γ) is preserved by applying θ − to both sides, so it is equal to the index of θ − ( F ) in Γ . But the index of F ∩ θ (Γ) in θ (Γ) is no more than that of F in Γ , thus [Γ : θ − i ( F )] gives us a non-in reasing sequen e whi h must stabiliseat N with value k . When it does we have for i ≥ that θ − ( i + N ) ( F ) is justmoving around the (cid:28)nitely many index k subgroups. Although it happensthat θ − does not in general permute these index k subgroups, we must landon some su h subgroup ∆ twi e so we have j ≥ with θ − j (∆) = ∆ , giving ∆ ≥ θ j (∆) .We now show that, although the rank of θ − i ( F ) redu es whenever theindex redu es, we an keep w as an element of a free basis ea h time we pullba k. This time we restri t θ to an inje tive homomorphism from θ − ( F ) to F with image θθ − ( F ) . As θθ − ( F ) is a (cid:28)nitely generated subgroup of F ontaining a free basis element w of F , we an ensure w is in a free basis for θθ − ( F ) (for instan e see [39℄ Proposition I.3.19). Now θ − ( F ) and θθ − ( F ) are isomorphi via θ with inverse φ say, so a basis b , . . . , b r for the lattergives rise to a basis φ ( b ) , . . . , φ ( b r ) for θ − ( F ) and if b = w then φ ( b ) = w . ✷ Corollary 4.5 If G = h Γ , t i is a mapping torus of an inje tive endomor-phism θ of the free group Γ of rank n and Z × Z ≤ G then we have H ≤ f G su h that H has a de(cid:28) ien y 1 presentation h x , . . . , x m , s | r , . . . , r m i in lud-ing a relator of the form sx s − x − .Proof. As BS (1 , ≤ G we have w ∈ Γ \{ } with θ i ( w ) = vwv − for some v ∈ Γ , thus on dropping to the index i subgroup H of G given by H = h Γ , t i i ASCENDING HNN EXTENSIONS OF FREE GROUPS 18and setting φ to be ι − v θ i where ι v ( x ) = vxv − , we an assume that thereis w ∈ Γ \{ } with φ ( w ) = w and that H is an as ending HNN extensionof Γ via the inje tive endomorphism φ and with stable letter t H say. So byTheorem 4.4 we have ∆ ≤ f Γ with ∆ having a free basis w, x , . . . , x m and j ≥ with φ j (∆) ≤ ∆ . Thus by Proposition 4.3 (ii) and (iv) we have that L = h ∆ , s | s = t jH i has (cid:28)nite index in G and by Lemma 4.1 L is an as endingHNN extension with base ∆ and stable letter s . Thus on taking the standardpresentation for L given by onjugation of s on this free basis for ∆ , we seethat it has de(cid:28) ien y 1 with a relator equal to sws − w − . ✷ We an now gain largeness for a range of mapping tori.Corollary 4.6 If G = h Γ , t i is the mapping torus of an endomorphism θ ofthe free group Γ of rank n and Z × Z ≤ G then G = h x, y | xyx − = x ± i or islarge.Proof. We an assume without loss of generality that θ is inje tive be auseif not then we an repla e θ with ˜ θ whi h is an inje tive endomorphism of afree group F m with m ≤ n , and then G is still equal to the mapping torus of F m using ˜ θ and this will ontain Z × Z . Hen e we are in the ase of Corollary4.5 whi h allows us to apply Theorem 3.6 to H ≤ f G . As Z × Z ≤ G , wedo not have m = 0 and only the two groups above for m = 1 . Otherwise G and hen e H ontain a non-abelian free group for m ≥ so H is not in ase(i) of Theorem 3.6. By Proposition 4.3 (iii) H is an inje tive mapping torusof a (cid:28)nitely generated free group endomorphism and so the re ent result [13℄of Borisov and Sapir tells us that H is residually (cid:28)nite, so it is not NARA.Thus H and G are large. ✷ Although it might be said that one only requires the NARA property toapply Theorem 3.6 and not the full for e of residual (cid:28)niteness, it should bepointed out that there are mapping tori G of inje tive endomorphisms of thefree group F su h that the abelianisation G = Z × Z and su h that for any(cid:28)nite index subgroup N whi h is normal in G with G/N soluble, we have N = Z × Z .We (cid:28)nish this se tion by looking at those mapping tori G of endomor-phisms of free groups whi h ontain an arbitrary Baumslag-Solitar subgroup.Our results are not quite de(cid:28)nitive be ause we need β ( G ) ≥ in order toapply our methods and we annot show that G ne essarily has a (cid:28)nite index ASCENDING HNN EXTENSIONS OF FREE GROUPS 19subgroup with that property. However this is the only obsta le to largeness.Theorem 4.7 If G = h Γ , t i is a mapping torus of an endomorphism θ of thefree group Γ of rank n whi h ontains a Baumslag-Solitar subgroup BS ( j, k ) then either G is large or G = BS (1 , k ) or β ( H ) = 1 for all H ≤ f G .Proof. As usual we assume that θ is inje tive. We know that G an only ontain Baumslag-Solitar subgroups of type BS (1 , k ) or BS ( k, k ) for k = 0 and as we have already overed those whi h ontain BS (1 , , we need only onsider BS (1 , k ) ≤ G for k = ± . If there is some H ≤ f G with β ( H ) ≥ then we an repla e G by H be ause H is a mapping torus by Proposition4.3 (iii) and BS (1 , k ) ∩ H ≤ f BS (1 , k ) so H ontains some Baumslag-Solitargroup too. Therefore we are looking at the situation where we have a periodi onjuga y lass of the form w ∈ F n \{ } and i > with θ i ( w ) onjugate to w d for some d = ± . Just as in the Z × Z ase, we drop down to a (cid:28)niteindex subgroup and hange our automorphism by an inner automorphism,so we an assume that θ ( w ) = w d . Now we follow the proof of Theorem4.4 to get F ≤ f Γ with h w i a free fa tor of F , observing that w ∈ θ − ( F ) so that we keep w as we pull ba k F . Note that we an assume w is not aproper power by the omment before Theorem 4.4 so we an also preserve w in a free basis ea h time be ause w d ∈ θθ − ( F ) and if w c ∈ θθ − ( F ) for < | c | < | d | then the element u ∈ θ − ( F ) with θ ( u ) = w c annot be a powerof w but θ ( u d ) = θ ( w c ) , hen e ontradi ting inje tiveness. Thus w d an beextended to a free basis for θθ − ( F ) by [39℄ Proposition I.3.7 meaning that w will be in the orresponding basis for θ − ( F ) .We an now work to obtain an equivalent version of Corollary 4.5. Havinggone from G to the (cid:28)nite index subgroup H whi h is an as ending HNNextension of Γ via the inje tive homomorphism θ , we see as before that byrepeatedly pulling ba k F we obtain ∆ ≤ f Γ whi h has a free basis in luding w and with θ j (∆) ≤ ∆ . Hen e the HNN extension J of ∆ using θ j withstable letter s has (cid:28)nite index in H , as well as a de(cid:28) ien y 1 presentationthat in ludes the relator sws − w − e for e = 0 , ± . We will also require laterthat e = 2 and this an be obtained by taking the subgroup of J of index 2as in Proposition 4.3 (ii) so that now the relator would be sws − w − . Now onsider taking a surje tive homomorphism χ from J to Z (whi h must send w to 0). If β ( J ) were 1 then the only available χ would be the homomorphismasso iated to this HNN extension so it would send s to 1 (or − ). Howeverif not then we an (cid:28)nd χ ′ = χ as we have β ( H ) ≥ and hen e β ( J ) ≥ FREE BY CYCLIC GROUPS 20be ause J ≤ f H . Hen e we have a non-trivial homomorphism χ ′ − kχ thatsends s to 0 (whi h an be made surje tive by multiplying by the right onstant) where k is χ ′ ( s ) . On evaluating the Alexander polynomial of J withrespe t to this homomorphism, we pro eed as in Theorem 3.1 and dis overthat the group relation sws − w − e be omes the module relation (1 − e ) w whenrewritten and abelianised. Thus we have a olumn in our square presentationmatrix onsisting of all zeros ex ept − e in the row orresponding to thegenerator w . Thus if we apply Theorem 2.1 using the (cid:28)eld Z /p Z with p a prime dividing − e then our Alexander polynomial is zero so we havelargeness for J , and hen e for G . ✷ Although we do not have a proof that a mapping torus of a free groupendomorphism ontaining a Baumslag-Solitar subgroup of in(cid:28)nite index hasa (cid:28)nite index subgroup with (cid:28)rst Betti number at least two, the statementof Theorem 4.7 is still useful in a pra ti al sense be ause if we are presentedwith a parti ular group G of this form that we would like to prove is large,we an enter the presentation into a omputer and ask for the abelianisationof its low index subgroups. As soon as we see one with (cid:28)rst Betti number atleast two, we an on lude largeness. Note that in [30℄ it is onje tured that amapping torus of a free group endomorphism is word hyperboli if it does not ontain Baumslag-Solitar subgroups, and if this and the above question onhaving a (cid:28)nite index subgroup with (cid:28)rst Betti number at least two are bothtrue then we have proved largeness for all the non word hyperboli as endingHNN extensions of (cid:28)nitely generated free groups (with the obvious ex eptionof the soluble Baumslag-Solitar groups).We an even say something if G is a mapping torus of an inje tive endo-morphism of an in(cid:28)nitely generated free group in the ase when G is (cid:28)nitelygenerated, thanks to the power of [21℄ by using Theorem 4.2. We immedi-ately see that either G has de(cid:28) ien y at least two and so is large, or l = 0 in whi h ase G is also a mapping torus of an endomorphism of a (cid:28)nitelygenerated free group and so the results of this se tion apply.5 Free by Cy li GroupsIf in the previous se tion we use an automorphism α of a free group F toform our mapping torus, we obtain a semidire t produ t F ⋊ α Z and everyfree-by- Z group is of this form. We already have largeness for a range of FREE BY CYCLIC GROUPS 21these groups.Theorem 5.1 If the (cid:28)nitely generated group G is free-by- Z then G is largeif the free group F is in(cid:28)nitely generated, or if Z × Z ≤ G and F has rankat least 2.Proof. If F is in(cid:28)nitely generated then applying Theorem 4.2 with H = G tells us that G has de(cid:28) ien y at least 2. This is be ause if l = 0 then thekernel of the asso iated homomorphism χ is ∞ [ n =0 s − n As n where A = h a , . . . , a k i but then β ( ker χ ; Q ) ≤ β ( A ; Q ) whereas we have F = ker χ .If F has (cid:28)nite rank then this is Corollary 4.6. Note that a semidire tprodu t A ⋊ B is residually (cid:28)nite if both A and B are residually (cid:28)nite and A is (cid:28)nitely generated, so we do not need to use [13℄ when applying this orollary to G . ✷ In ontrast G. Baumslag gives in [5℄ an example of an in(cid:28)nitely generatedfree-by- Z group with every (cid:28)nite quotient y li so that this group is notresidually (cid:28)nite. Indeed its (cid:28)nite residual R G must ontain G ′ and hen eit is not large be ause every (cid:28)nite index subgroup F of G has the propertythat all of the (cid:28)nite quotients of F are abelian, as F ′ ≤ G ′ ≤ R G = R F .This is a striking demonstration of how largeness and residual (cid:28)niteness arebest suited to (cid:28)nitely generated groups. He then proves in [4℄ that (cid:28)nitelygenerated groups whi h are F -by- Z for F an in(cid:28)nitely generated free groupare residually (cid:28)nite. As for the residual (cid:28)niteness of (cid:28)nitely generated groupsthat are as ending HNN extensions of in(cid:28)nitely generated free groups, thisappears to be open (it orresponds to the ase l > in Theorem 4.2); indeedthat they are Hop(cid:28)an is Conje ture 1.4 in [22℄.Corollary 5.2 If G is F n -by- Z for F n the free group of rank n then either G is large or G is word hyperboli or G = BS (1 , ± .Proof. It is known by [8℄, [9℄ and [14℄ that su h a G being word hyperboli is equivalent to G ontaining no subgroups isomorphi to Z × Z and also tothe automorphism α having no periodi onjuga y lasses. ✷ FREE BY CYCLIC GROUPS 22In fa t the equivalen e of the last two notions an be proved dire tly byquoting the lassi al result of Higman [26℄ whi h says that an automorphismof a free group that maps a (cid:28)nitely generated subgroup into itself maps itonto itself. So if α has a periodi onjuga y lass we an assume there is i > and w ∈ F n \{ } su h that θ i ( w ) is onjugate to w ± .This leaves us with an important question:Question 5.3 If G is F n -by- Z for n ≥ and G is a word hyperboli groupthen is G large?As for whether the six onsequen es of largeness given in the introdu tionhold for these groups G , the (cid:28)rst is obvious whereas it is unknown if G hassuperexponential subgroup growth or has in(cid:28)nite virtual (cid:28)rst Betti number:Question 12.16 by Casson in [7℄ is equivalent to asking whether there exists H ≤ f G with β ( H ) ≥ . However being word hyperboli means that theother properties are known to hold, giving us a de(cid:28)nitive result for thesethree ases.Theorem 5.4 If G is (cid:28)nitely generated and is virtually free-by- Z then forall (cid:28)nitely generated subgroups H of G we have:(i) H has word problem solvable strongly generi ally in linear time(ii) H has uniformly exponential growth(iii) H is SQ-universalunless H is virtually S for S = Z or Z × Z .Proof. We have shown by Theorem 5.1 and Corollary 5.2 that any free-by- Z group G whi h is (cid:28)nitely generated (ex epting BS (1 , ± and Z ) is largeor is non-elementary word hyperboli . This implies SQ-universality (by [44℄for the hyperboli ase and [45℄ when G is large) and uniformly exponentialgrowth (by [33℄ for the hyperboli ase and [25℄ when we have largeness).Then Corollary 4.1 in [32℄ (where we also have the relevant de(cid:28)nitions) proves(i) if G has a (cid:28)nite index subgroup with a non-elementary word hyperboli quotient. Moreover if G ≤ f Γ then the properties (i), (ii) and (iii) hold for Γ as well, by [32℄ for (i), [25℄ for (ii) and [42℄ for (iii).Finally if H ≤ G where G is free-by- Z then H/ ( H ∩ F ) ∼ = HF/F ≤ G/F = Z so either this is trivial with H ≤ F hen e H is free, or it isisomorphi to Z and so H is an extension of the free group H ∩ F by Z (although if F is (cid:28)nitely generated then H ∩ F is not ne essarily (cid:28)nitelygenerated if H has in(cid:28)nite index in F ). Thus if H is (cid:28)nitely generated then 1 RELATOR GROUPS 23(i), (ii), (iii) or the ex eptions hold for H too, and if L is a (cid:28)nitely generatedsubgroup of the virtually free-by- Z group Γ with G ≤ f Γ then L ∩ G ≤ f L so L ∩ G is a (cid:28)nitely generated subgroup of G , hen e we gain our propertiesfor L ∩ G and then also for L . ✷ BS (1 , m ) or is y li , see [39℄ II Proposition 5.27 and [49℄.Indeed if the presentation has at least 3 generators then we know by [2℄ thatthe group is large so we need only on ern ourselves here with 2 generator 1relator presentations h a, b | r i . Largeness is also known by [23℄ and [48℄ when r is a proper power, whi h is exa tly when the group has torsion, but for l, m oprime we have by [20℄ that BS ( l, m ) is not large as it has virtual (cid:28)rst Bettinumber equal to 1, and similarly Example 3.5 (iii) is not large. Thus anotherdire tion in whi h to go when looking for large 2 generator 1 relator groupsis if r is in the ommutator subgroup of F , as at least that gives (cid:28)rst Bettinumber equal to 2. The starting ase to be onsidered here should be to take r a tually equal to a ommutator and appli ation of Theorem 3.6 gives us anear de(cid:28)nitive result.Corollary 6.1 If G = h a, b | uvU V i where u and v are any elements of F = h a, b i with uvU V not equal to abAB , baBA or their y li onjugates whenredu ed and y li ally redu ed then G is large or is NARA.Proof. It is well known that G = Z × Z if and only if the relator is of theabove form (equivalently if and only if u, v form a free basis for F ); see forinstan e [41℄ Theorem 4.11. Otherwise we are in Theorem 3.6 ase (ii) or(iii). ✷ Question 6.2 If G = h a, b | uvU V i then an G be NARA?No examples are known to us. This is an important question be ause ayes answer gives us a non residually (cid:28)nite 1 relator group with the relatora ommutator, the existen e of whi h is Problem (OR8) in the problem listat [43℄ (however there it is shown that non residually (cid:28)nite examples exist if 1 RELATOR GROUPS 24the relator is merely in the ommutator subgroup) and a no answer gives uslargeness. We an prove that we do not have NARA groups in a whole rangeof ases.Proposition 6.3 If G = h F | uvU V i then G an only be NARA if u, v / ∈ F ′ with the images of u and v generating the abelianisation Z × Z of F andsu h that u is a free basis element for F or G u = h F | u i is NARA, alongwith the same ondition for v .Proof. If the images of u and v do not generate the homology of F up to(cid:28)nite index then we are done by Theorem 3.1. Now suppose that G u is notNARA or Z (the latter happening if and only if u is an element of a freebasis for F ) then as G surje ts to G u we see that a non-abelian (cid:28)nite imageof G u is also an image of G . Then swap u and v .Otherwise we an take a free basis α, β for F su h that there are k, l ≥ with u equivalent to α k in homology and v to β l . If k > then onsider thehomomorphism of F into the linear (hen e residually (cid:28)nite) group SL (2 , C ) given by α (cid:18) e πi/k e − πi/k (cid:19) , β (cid:18) (cid:19) . This is non-abelian but does make u and v ommute so we are done byProposition 3.4 (iii). ✷ In fa t there are other ways to on lude that G = h F | uvU V i is notNARA and hen e large, for instan e the powerful algorithm of K. S. Brownin [15℄, whi h determines whether a 2 generator 1 relator group is a mappingtorus of an inje tive endomorphism of a (cid:28)nitely generated group (whi h mustne essarily be free), an be used (along with [13℄ proving that su h groupsare residually (cid:28)nite). If all else fails then there is the option of using the omputer to (cid:28)nd the abelianisation of some low index subgroups of G andlook for one whi h is not Z × Z in order to obtain largeness. For instan e in[18℄ it is shown that G = h a, b | a k b l a k b l a k b l i for k k k , l l l = 0 and k + k + k = l + l + l = 0 is large for all possibilitiesex ept for one parti ular relator (on whi h we use the omputer to (cid:28)nd a(cid:28)nite index subgroup H of the form in Theorem 3.6 and with β ( H ) > )and two in(cid:28)nite families of relators (these are of the required form so we an RESIDUAL FINITENESS 25use Brown's algorithm) thus we have shown that the remaining ases are alllarge.We (cid:28)nish this se tion by mentioning a onje ture of P. M. Neumann in[42℄ from 1973: that a 1 relator group is either SQ-universal or is isomorphi to BS (1 , m ) or y li (the next omment that (cid:16)a proof of this by G. Sa redoteseems to be almost omplete now(cid:17) turns out with hindsight to be somewhatover optimisti ). In addition to presentations with 3 or more generators orwith the relator a proper power, at least this an be seen to be true forthose 2 generator 1 relator groups G whi h are free by y li , and possiblyas ending HNN extensions of free groups if the two questions at the end ofSe tion 4 are true, as well as if G is virtually a group of this type. We makeprogress on this question from a di(cid:27)erent dire tion in the next se tion.7 Residual FinitenessIt appears that often when we have a ounterexample to statements aboutlargeness, this is a hieved by taking a group whi h is not residually (cid:28)nite.The following straightforward observation suggests why:Proposition 7.1 A group G is large if and only if the residually (cid:28)nite group G/R G is large, where R G is the interse tion of all the (cid:28)nite index subgroupsof G .Proof. A group is large if any quotient is large, whereas any homomorphismfrom G to a residually (cid:28)nite group fa tors through G/R G and if H ≤ f G then R H = R G . ✷ Thus perhaps we should take the same approa h as those who ount (cid:28)niteindex subgroups of (cid:28)nitely generated groups by only onsidering residually(cid:28)nite groups. However the example of G = BS (2 , , where G/R G is solublebut not (cid:28)nitely presented, means we an lose good properties of our originalgroup. This assumption removes the obvious ounterexamples whi h areSQ-universal but not large, for instan e taking free produ ts of groups withno (cid:28)nite index subgroups, and then the two properties begin to look moresimilar. A re ent result of [1℄ shows that (cid:28)nitely generated groups within(cid:28)nitely many ends are SQ-universal. Whilst they annot all be large, aseviden ed by these free produ ts, it is straightforward to establish this in theresidually (cid:28)nite ase by adapting an argument of Lubotzky from [37℄. RESIDUAL FINITENESS 26Theorem 7.2 A residually (cid:28)nite group with in(cid:28)nitely many ends is large.Proof. If Γ = G ∗ φ G where φ is an isomorphism from A a (cid:28)nite subgroupof G to B a (cid:28)nite subgroup of G then, as G will be residually (cid:28)nite, we an take M ✂ f G with M ∩ A = I and [ G : M ] > | A | , meaning thatthe subgroup AM/M of G /M has index greater than 2 and is isomorphi to A . We an also get N ✂ f G with N ∩ B = I and [ G : N ] > | B | .Now we an form ( G /M ) ∗ φ ( G /N ) , where φ ( aM ) = φ ( a ) N provides anisomorphism from AM/M to BN/N . This is a quotient of Γ and is virtuallyfree by [47℄ II Proposition 11, with the index onditions ensuring that it isvirtually non-abelian free (see [47℄ 2.6 Exer ise 3).As for HNN extensions Γ = G ∗ φ , where φ is an isomorphism with domaina (cid:28)nite proper subgroup A of G , and φ ( A ) ≤ G is onjugate to A in Γ viathe stable letter t , we now take N ✂ f Γ su h that there exists g ∈ G with ag / ∈ N for all a ∈ A , whi h implies that AN = GN . Thus AN/N and φ ( A ) N/N are subgroups of
GN/N whi h are onjugate in Γ /N via tN , thusisomorphi , with both of these proper subgroups. Hen e the HNN extension h GN/N, s | s ( aN ) s − = φ ( a ) N i an be formed and this is a non-as endingHNN extension of a (cid:28)nite group, thus it is virtually non-abelian free. ✷ A group G is alled LERF (equivalently subgroup separable) if every(cid:28)nitely generated subgroup is an interse tion of (cid:28)nite index subgroups. Anobservation in [12℄ is that G annot be LERF if there is a (cid:28)nitely generatedsubgroup H and t ∈ G with tHt − ⊂ H , thus proper as ending HNN ex-tensions of (cid:28)nitely generated groups are never LERF. If G is F n -by- Z and β ( G ) ≥ then it is possible for G to be simultaneously free-by- y li andalso to be a proper as ending HNN extension of a (cid:28)nitely generated freegroup with respe t to another asso iated homomorphism onto Z , so beingLERF is quite a lot stronger than merely being residually (cid:28)nite. However ifwe assume this we an gain some very spe i(cid:28) on lusions.Theorem 7.3 If G is (cid:28)nitely presented and LERF then either G has virtual(cid:28)rst Betti number equal to 0, or G is large, or G is virtually L ⋊ Z for L (cid:28)nitely generated.Proof. Any group with positive (cid:28)rst Betti number is an HNN extension (itis a semidire t produ t ( ker χ ) ⋊ Z for χ a homomorphism onto Z ) but if itis (cid:28)nitely presented then [11℄ tells us that it is an HNN extension L ∗ φ with RESIDUAL FINITENESS 27stable letter t and with L and the domain A of φ both (cid:28)nitely generated.Thus on taking H ≤ f G with β ( H ) > we have H = L ∗ φ with presentation h L, t | ta i t − = φ ( a i ) for ≤ i ≤ m i where a , . . . , a m is a generating set for A . Now if A = L then, as subgroupsof LERF groups are also LERF, we have F ≤ f H whi h ontains A but not L . We an take N ≤ F whi h is normal in H and of (cid:28)nite index. Thisgives us AN ≤ F < LN and so we an argue as in Theorem 7.2 to get anon-as ending HNN extension: We have that
LN/N is (cid:28)nite and AN = LN implies that AN/N is a proper subgroup of
LN/N . Now the isomorphism φ from A to φ ( A ) is indu ed by onjugation by t , so that in H/N we have that
AN/N and φ ( A ) N/N are onjugate by the element tN . Hen e the indu edmap φ ( aN ) = φ ( a ) N is well de(cid:28)ned and is an isomorphism between thesetwo subgroups. Therefore we an form ( LN/N ) ∗ φ with the domain of φ equalto AN/N , and this has presentation h LN/N, s | s ( a i N ) s − = φ ( a i N ) for ≤ i ≤ m i whi h is an image of H under the homomorphism L LN/N, t s and islarge by [47℄ II Proposition 11 so H is large too.Otherwise the HNN extension is as ending but the LERF ondition meansthat H is in fa t a semidire t produ t. ✷ Going ba k to de(cid:28) ien y 1 groups, we have [28℄ Theorem 6 whi h statesthat if G has de(cid:28) ien y 1 and is an as ending HNN extension with base the(cid:28)nitely generated subgroup L then the geometri dimension of G (thus the ohomologi al dimension) is at most two. But on ombining this with [22℄Corollary 2.5, whi h states that if L is of type F P and has ohomologi aldimension 2 then G has ohomologi al dimension 3, we see that if G hasde(cid:28) ien y 1 then L (cid:28)nitely presented (or even F P ) implies that L is in fa tfree. This means that the only way a de(cid:28) ien y 1 group G ould fail theTits alternative of not being virtually soluble and not ontaining F is for G to be an as ending HNN extension L ∗ φ where L is (cid:28)nitely generated butnot (cid:28)nitely presented and where L fails the Tits alternative. It is unknownwhether this an o ur but at least we an on lude the Tits alternative holdsfor oherent de(cid:28) ien y 1 groups. In [49℄ it is shown that a soluble de(cid:28) ien y1 group is BS (1 , m ) or Z . As these groups are oherent, this result is easilyextended to virtually soluble de(cid:28) ien y 1 groups by the above and the resultEFERENCES 28in [11℄ that a (cid:28)nitely presented group G with β ( G ) > whi h does not ontain F is an as ending HNN extension of a (cid:28)nitely generated group.We an use this to obtain some results about LERF groups in parti ular ases.Corollary 7.4 If G is LERF and has de(cid:28) ien y 1 then either G is SQ-universal or G is BS (1 , ± or Z or G = L ⋊ Z for L (cid:28)nitely generated butnot (cid:28)nitely presented.Proof. Certainly β ( G ) > so by the proof of Theorem 7.3 G is large orequals L ⋊ Z with L (cid:28)nitely generated. If L is (cid:28)nitely presented then theabove omment shows that L is free and Theorem 5.4 (iii) applies if L isnon-abelian free. ✷ We an (cid:28)nish by making some progress on P. M. Neumann's onje turegiven in the last se tion.Corollary 7.5 If G is a 1 relator group whi h is LERF then either G isSQ-universal or G is y li or G = BS (1 , ± .Proof. We know that G is (cid:28)nite y li or has de(cid:28) ien y at least two or hasde(cid:28) ien y one whereupon Corollary 7.4 applies. But if G = L ⋊ Z for L (cid:28)nitely generated then L must be free by [15℄ Se tion 4. ✷✷