Typical elements in free groups are in different doubly-twisted conjugacy classes

T ypical elemen ts in free groups are in differen t doubly-t wisted conjugacy classes P . Christopher Staec k er ∗ †‡ No v em b er 4, 2018 Abstract W e giv e an easily chec k able alge braic condition whic h implies that tw o elemen ts of a finitely generated free group are members of distinct doubly- tw isted conjugacy cla sses with respect t o a pair of homomorphisms. W e further sh ow that this criterion is satisfied with p ro bability 1 when the homomorphisms and elements are chosen at random. 1 In tro duction Let G a nd H b e finitely g enerated fr e e gro ups, and ϕ, ψ : G → H be homo- morphisms. The gr oup H is partitioned into the set of doubly-twiste d c onjugacy classes as follows: u, v ∈ H are in the same class (w e write [ u ] = [ v ]) if and only if there is some g ∈ G with u = ϕ ( g ) v ψ ( g ) − 1 . Our principal motiv ation for studying do ubly-t wisted co nju ga cy is Nielsen coincidence theory (see [4] for a survey), the study of the coincidence set o f a pair of mappings and the minimization of this set while the mappings are changed by homotopies. Our fo cus on free groups is motiv ated sp ecifically by the pr oblem of computing Nielsen classes of coincidence po ints for pairs of mappings f , g : X → Y , where X and Y are compac t surfaces with bo undary . A nece ssary condition for tw o coincidence points to be combined by a homo - topy (thus reducing the tota l n umber of co incidence p oin ts) is that they b elong to the same Nielsen class. (Much o f this theory is a dir ect generaliz ation of sim- ilar tec hniques in fixed p oint theory , see [5 ].) The n umber of “essential” Nielsen classes is ca lled the Nielsen num b er, and is a lo wer b ound fo r the minimal n um- ber o f co incidence p oin ts when f and g are allow ed to v a ry by homo topies. ∗ Email: cstaec ker@fairfield.edu † Keyw ords: Nielsen theory , coincidence theory , twisted conjugacy , doubly t wisted conju- gacy , asymptotic densit y ‡ MSC2000: 54H25, 20F10 1 In our setting, deciding when t wo coincidence p oin ts ar e in the sa me Nielsen class is equiv alent to so lv ing a natural doubly-t wisted conjugacy problem in the fundamen tal gro ups, using the induced homomorphisms giv en by the pair of mappings. Th us the Nielsen classes o f coincidence po ints corr espond to twisted conjugacy classes in π 1 ( Y ). The problem of co mput ing doubly-twisted co nj uga cy classes in free g r oups is nontrivial, even in the singly-twisted case which ar is es in fixed p oin t theory , where ϕ is a n endomorphism and ψ is the identit y . Existing techniques for computing doubly-twisted conjugacy are adapted fro m singly-twisted metho ds using abelia n and nilpotent quotients [10]. Our main result is no t adapted from singly-twisted methods: it is suited sp ecifically for doubly-twisted conjugacy and in fact can never apply in the c a se where ψ = id. In Section 2 w e will present a r e m nant condition which can b e used to show that t wo w or ds are in different doubly- twisted co njuga cy classes. In Section 3 we show in fact that this remnant condition is very common for “most” homomo r- phisms. In the sense of as y mp totic densit y , w e sho w that if the homomorphisms ϕ, ψ , and elements u , v are all c hosen at random, then [ u ] 6 = [ v ] with probability 1. The author would like to thank Rob ert F. Brown, Benjamin Fine, Arma ndo Martino and Enric V entura for many helpful comments. 2 A remnan t condition for doubly-t wisted con- jugacy Given homomo rphisms ϕ, ψ : G → H , the e qualizer sub gr oup Eq( ϕ, ψ ) ≤ G is the subgroup Eq( ϕ, ψ ) = { g ∈ G | ϕ ( g ) = ψ ( g ) } . Our first lemma is an equalizer version o f a r esult for singly-twisted conjugacy which app ears in the pro of o f Theorem 1.5 of [3 ]. Let h z i b e the free g roup generated b y z , a nd let ˆ G = G ∗ h z i and ˆ H = H ∗ h z i , with ∗ the fr e e pro duct. Let h v = v − 1 hv , a nd ϕ v ( g ) = v − 1 ϕ ( g ) v . The following lemma holds when G a nd H are a n y gr oups (not necessarily free): Lemma 1. Given u ∈ H , let ˆ ϕ u : ˆ G → ˆ H b e the ex tension of ϕ given by ˆ ϕ u ( z ) = uz u − 1 , and let ˆ ψ : ˆ G → ˆ H b e the extension of ψ given by ˆ ψ ( z ) = z . Then [ v ] = [ u ] if and only if ther e is some g ∈ G with g z g − 1 ∈ Eq( ˆ ϕ v u , ˆ ψ ) . Pr o of. First, assume that [ v ] = [ u ], and let g ∈ G b e some elemen t with v = ϕ ( g ) u ψ ( g ) − 1 . Then we hav e ˆ ϕ v u ( g z g − 1 ) = v − 1 ϕ ( g ) uz u − 1 ϕ ( g ) − 1 v = ψ ( g ) u − 1 ϕ ( g ) − 1 ϕ ( g ) uz u − 1 ϕ ( g ) − 1 ϕ ( g ) uψ ( g ) − 1 = ˆ ψ ( g z g − 1 ) 2 as desired. Now assume that there is s ome element g z g − 1 ∈ E q( ˆ ϕ v u , ˆ ψ ) for g ∈ G . Consider the c omm utator: [ u − 1 ϕ ( g ) − 1 v ψ ( g ) , z ] = u − 1 ϕ ( g ) − 1 v ψ ( g ) z ψ ( g ) − 1 v − 1 ϕ ( g ) uz − 1 = u − 1 ϕ ( g ) − 1 v ˆ ψ ( gz g − 1 ) v − 1 ϕ ( g ) uz − 1 = u − 1 ϕ ( g ) − 1 v ˆ ϕ v u ( g z g − 1 ) v − 1 ϕ ( g ) uz − 1 = u − 1 ϕ ( g ) − 1 v v − 1 ϕ ( g ) uz u − 1 ϕ ( g ) − 1 v v − 1 ϕ ( g ) uz − 1 = 1 . Thu s z commutes with u − 1 ϕ ( g ) − 1 v ψ ( g ), and so u − 1 ϕ ( g ) − 1 v ψ ( g ) = 1, since this word do es not contain the letter z . Thus [ u ] = [ v ]. The ab o ve lemma is difficult to apply for the purpo se of computing t wisted conjugacy classes , since the problem of computing the equalizer subgro up of homomorphisms is difficult. In fixed point theor y (where ψ = id), if ϕ is a n automorphism, an a lgorithm of [8] is given to compute the fixed point subg roup Fix( ϕ ). This a lgorithm r elies fundamentally on the methods o f Be s t vina and Handel [2] for r epresen ting automorphisms of free groups using train tracks, and these tec hniques do not extend in an obvious wa y to coincidence theory . Though the equalizer subgroup is in general difficult to compute, we will show that a certain r emnan t pr operty will for ce the equaliz e r subg roup to b e trivial. Remnant prop erties were first used by W agner in [11]. Definition 2. Let G b e a finitely genera ted free g roup with a s pecified set of generator s G = { g 1 , . . . , g n } . The homomor ph ism ϕ has r emnant if for each i , the w or d ϕ ( g i ) has a no n tr ivial subw o rd which has no cancellation in an y of the pro ducts ϕ ( g j ) ± 1 ϕ ( g i ) , ϕ ( g i ) ϕ ( g j ) ± 1 , except for j = i with exp onen t − 1 . The maximal such noncance lling sub word of ϕ ( g i ) is called the r emnant of g i , written Rem ϕ ( g i ). W e will o ccasionally discuss the length of the remnant subwords, in one of t wo wa ys. If, for s ome natura l num b er l , we hav e | Rem ϕ ( g i ) | ≥ l for a ll g i , we will say that ϕ has r emnant length l . F or some r ∈ (0 , 1), we say that ϕ has r emn a nt r atio r when | Rem ϕ ( g i ) | ≥ r | ϕ ( g i ) | for each i . The condition that ϕ has remna n t is slightly weaker than saying that ϕ ( G ) is Nielsen reduced (see e.g. [6]), which w ould make some additional assumptions on the w ord length of the remnant subw ords. Throughout the rest of the pap er, we will fix a particula r g enerating set G = { g 1 , . . . , g n } for G . Given tw o homomorphisms ϕ, ψ : G → H , there is a homomorphism ϕ ∗ ψ : G ∗ G → H , defined as follows: Denoting G = h g 1 , . . . , g n i , we write G ∗ G = h g 1 , . . . , g n , g ′ 1 , . . . , g ′ n i . Then we define ϕ ∗ ψ on the generators of G ∗ G by ϕ ∗ ψ ( g i ) = ϕ ( g i ) and ϕ ∗ ψ ( g ′ i ) = ψ ( g i ). W e hav e: 3 Lemma 3. If ϕ ∗ ψ : G ∗ G → H has r emnant, then ϕ ( G ) ∩ ψ ( G ) = { 1 } . In p articular this me ans t hat Eq( ϕ, ψ ) = { 1 } . Pr o of. If ϕ ( G ) ∩ ψ ( G ) contains so me nontrivial ele m ent, then we hav e x, y ∈ G , bo th nontrivial, with ϕ ( x ) ψ ( y ) − 1 = 1 , which is not p ossible when ϕ ∗ ψ has remnant: writing x and y in terms of generator s will s how that the left s ide ab o ve cannot fully canc e l. The tw o lemmas ca n be used to compute do ubly - t wis t ed conjugacy classes as in the following example: Example 4. W e will examine do ubly-t wisted conjuga cy cla s ses o f the homor- phisms ϕ, ψ : G → G with G = h a, b i defined by: ϕ : a 7→ aba b 7→ b − 1 a ψ : a 7→ b 2 a − 1 b 7→ a 3 Given w ords u, v ∈ G , let η ( u,v ) = ˆ ϕ v u ∗ ˆ ψ . Our tw o lemmas tog ether sho w that for u , v ∈ G , we will hav e [ u ] 6 = [ v ] whenever η ( u,v ) : ˆ G ∗ ˆ G → ˆ G ha s remnant. This remnant condition can be easily chec ked b y hand. W e will apply this strategy for all words u , v of length 0 or 1. The homomorphism η = η (1 ,b ) is as follows: η = a 7→ b − 1 abab b 7→ b − 2 ab z 7→ b − 1 z b a ′ 7→ b 2 a − 1 b ′ 7→ a 3 z ′ 7→ z W e can see that η ha s remnant, and thus by Lemma 3 that Eq( ˆ ϕ b 1 , ˆ ψ ) = 1, and th us by Lemma 1 that [1] 6 = [ b ]. Checking the appropr iate homomorphisms (due to the asymmetry in the role of u a nd v , an unsucc e s sful check for the pair ( u, v ) might actually succeed for the pair ( v , u )) we obtain the following additional inequalities: [ a ] 6 = [1 ] , [ a ] 6 = [ b ] , [ a − 1 ] 6 = [1] , [ a − 1 ] 6 = [ b ] . This metho d is somewha t tedious to p e rform by hand. A web-based computer implemen tation of the pro cess is av aila ble for testing at the a ut hor ’s website. 1 The c hecks for re mna n t in the abov e example are equiv alent to a related noncancellation condition: 1 The tec hnique is implemente d in GAP , with a web-base d frontend. The fr on t-end and GAP source code are a v ai l ab le at http://faculty. fairfield.edu/cstaecker . 4 Theorem 5. L et u, v ∈ H b e distinct wor ds. If ϕ v ∗ ψ has r emnant, and if, for e ach gener ator g of G ∗ G , the r emnant wor ds Rem ϕ v ∗ ψ ( g ) do not ful ly c anc el in any pr o du ct of the form ( ϕ v ∗ ψ ( g )) v − 1 u, u − 1 v ( ϕ v ∗ ψ ( g )) , (1) then [ u ] 6 = [ v ] . Pr o of. Let ˆ G = G ∗ h z i and ˆ H = H ∗ h z i , and let ˆ ϕ u , ˆ ψ : ˆ G → ˆ H be defined as in Lemma 1. W e will show that ˆ ϕ v u ∗ ˆ ψ has remnant. F or brevity , let η = ˆ ϕ v u ∗ ˆ ψ , a nd let us denote the genera to rs of the free pro duct so that ˆ G ∗ ˆ G = h g 1 , . . . , g n , z , g ′ 1 , . . . , g ′ n , z ′ i . Then the homo morphism η is given by: η = ˆ ϕ v u ∗ ˆ ψ : g i 7→ ϕ v ( g i ) z 7→ v − 1 uz u − 1 v g ′ i 7→ ψ ( g i ) z ′ 7→ z T o s ho w tha t η has remnant, we must show that the words η ( g i ) hav e non- cancelling s ub words in v ar ious pr oducts of the form in Definition 2. F or each i , let w i be the subw or d of the remnant of Rem ϕ v ∗ ψ ( g i ) which has no cancellatio n in any of the pro ducts in (1). Similar ly let w ′ i be the subw ord of the remnant of Rem ϕ v ∗ ψ ( g ′ i ) with no cancellation in an y of the pro ducts in (1). Let us first examine sub words o f η ( g i ) in pro ducts of the form η ( g i ) η ( g j ) ± 1 = ϕ v ( g i ) ϕ v ( g j ) ± 1 or η ( g i ) η ( g ′ j ) ± 1 = ϕ v ( g i ) ψ ( g j ) ± 1 . In these pro ducts, w i will remain unca ncelled (unless j = i with exp onent − 1) bec ause it is a subw ord o f Rem ϕ v ∗ ψ ( g i ). Now consider η ( g i ) η ( z ) ± 1 = ϕ v ( g i ) v − 1 uz ± 1 u − 1 v . Here w i will r emain uncancelled by the hypothese s o n pro du cts of the form in (1), to gether with the fact that no cancellatio n can o ccur with the z ± 1 bec ause u and v do no t use the letter z . Finally we mu st consider pro ducts o f the form η ( g i ) η ( z ′ ) ± 1 = ϕ v ( g i ) z , in which clearly w i do es not cancel. W e hav e shown that w i has no cancella t ion in pr oducts of the for m in Defi- nition 2 inv olving η ( g i ) on the left. Similar arguments will show that w i has no cancellation in pr oducts inv olving η ( g i ) on the right. Identical arguments will show that the words w ′ i are uncancelled in v a rious pro ducts inv olving η ( g ′ i ), and th us ˆ ϕ v u ∗ ˆ ψ has remnant. Since ˆ ϕ v u ∗ ˆ ψ has remnant, we hav e E q( ˆ ϕ v u , ˆ ψ ) = { 1 } by Lemma 3, and th us [ u ] 6 = [ v ] by Lemma 1. Note that Theorem 5 cannot b e used in fixed point theory to distinguish singly-twisted conjugacy classes , since ϕ v ∗ id can nev er hav e remnant. 5 3 Generic p rop e rties A theorem of Rober t F. Brown in [11] shows that “most” homomor phisms hav e remnant. Theo rem 3.7 of that paper is: Lemma 6. L et G b e a fr e e gr oup with gener ators g 1 , . . . , g n with n > 1 . Given any ǫ > 0 , ther e exists some M > 0 such that, if ϕ : G → G is an endomorphi sm chosen at r andom with | ϕ ( g i ) | ≤ M for al l gener ators g i ∈ G , then the pr ob ability that ϕ has r emnant is gr e ater than 1 − ǫ . The ab ov e is the only result of its k ind typically referenced in the Nielsen theory literature, but it is in the spirit of a well established theory of gener ic group prop erties. (See [9] for a survey .) F or a free gr o up G and a natural n umber p , let G p be the subset of all w or ds of length at most p . F or a subset S ⊂ G , let S p = S ∩ G p . The asymptotic density (or simply density ) of S is defined a s D ( S ) = lim p →∞ | S p | | G p | , where | · | denotes the cardinality . The set S is said to be generic if D ( S ) = 1. Similarly , if S ⊂ G l is a set of l -tuples of elemen ts of G , the asy mptotic density of S is defined as D ( S ) = lim p →∞ | S p | | ( G p ) l | , where S p = S ∩ ( G p ) l , and S is ca lled generic if D ( S ) = 1. A homo morphism on the free group G = h g 1 , . . . , g n i is equiv alent combina- torially to an n -tuple of elements of G (the n elements are the words ϕ ( g i ) for each g enerator g i ). Thus the a symptotic densit y of a set of homorphisms can be defined in the same sense a s ab ov e, viewing the s et of homo mo rphisms as a collection of n -tuples. The statement of Lemma 6 , then, is s imply that the set of endomor phisms G → G with re mna nt is gener ic. Similarly we can define the density of a set of pair s o f homorphisms by view ing it as a collection of 2 n -tuples (a pair of homomorphisms is equiv alent to a pair of n -tuples). The s tatement of Lemma 6 can b e strengthened and extended easily to general homomo rphisms (p ossibly no n-endomorphisms) using a generic prop er ty from [1]. Consider the setting of homo morphisms G → H , where G and H are finitely generated free and H has more than one generator. Lemma 3 of [1] implies tha t for any l , the co llection o f l - tuples o f H which a re Nielsen reduced (when viewed as sets of elemen ts of H ) is generic. This directly gives Lemma 7 . If t he r ank of H is gr e ater than 1, then the set of homomorphisms G → H with r emnant is generic. Applying the ab ov e to homomorphisms G ∗ G → H and applying Lemma 3 gives an in teresting cor ollary: 6 Corollary 8. If the r ank of H is gr e ater than 1, then the set of p airs of homo- morphisms ϕ, ψ : G → H with E q( ϕ, ψ ) = { 1 } is generic. This gives a s omewhat co unterin tuitive result: If F i is the free gr oup o f rank i , a pair of ho momorphisms from F 1000 to F 2 will generically hav e imag es whose intersection is trivial. (See results of a similar spir it in [7], e.g . that homomorphisms of fr e e groups are generically injective but not surjective.) The cited result fr o m [1] is quite a bit stronger: it is shown that the s et of subsets of G having sma ll cancellation prop erty C ′ ( λ ) is g eneric fo r a ny λ > 0. This gives stronger results concer ning remnant properties of gener ic homomorphisms: Lemma 9. L et the r ank of H b e gr e ater t han 1. Then: • F or any natur al numb er l , the set of homomorphi sms ϕ : G → H with r emn ant length l is generic. • F or any r ∈ (0 , 1) , t he set of h omomorphisms ϕ : G → H with r emnant r atio r is generic. W e include the ab ov e lemma for the sake of completeness, but we will not ac- tually r e quire its full strength in order to prove o ur gener ic pro p erty for do ubly- t wisted conjugacy . Theorem 10. L et G and H b e fr e e gr oups, with the r ank of H gr e ater than 1. Then the set S = { ( ϕ, ψ , u, v ) | [ u ] 6 = [ v ] } is generic. Pr o of. W e will s lightly extend our free-pro duct no tation for homomorphisms as follows: for a homo morphism ϕ : G → H and a w ord w ∈ H , let h x i be the free gr oup generated by some new letter x . Then define ϕ ∗ w : G ∗ h x i → H by ϕ ∗ w ( g i ) = ϕ ( g i ) for g i a genera tor of G , and ϕ ∗ w ( x ) = w . By Theorem 5, S contains all tuples ( ϕ, ψ , u, v ) such that ϕ v ∗ ψ ∗ uv − 1 has remnant. This remnant condition will be satisfied when the brack eted w ords b e- low hav e sub words which do no t ca ncel in any of the following products (exce pt those whic h ar e trivia l): [ ϕ v ( g i )]( ϕ v ( g j )) ± 1 , [ ϕ v ( g i )] ψ ( g j ) ± 1 , [ ϕ v ( g i )]( uv − 1 ) ± 1 , ( ϕ v ( g j )) ± 1 [ ϕ v ( g i )] , ψ ( g j ) ± 1 [ ϕ v ( g i )] , ( uv − 1 ) ± 1 [ ϕ v ( g i )] , [ ψ ( g i )]( ϕ v ( g j )) ± 1 , [ ψ ( g i )] ψ ( g j ) ± 1 , [ ψ ( g i )]( uv − 1 ) ± 1 , ( ϕ v ( g j )) ± 1 [ ψ ( g i )] , ψ ( g j ) ± 1 [ ψ ( g i )] , ( uv − 1 ) ± 1 [ ψ ( g i )] , [ uv − 1 ]( ϕ v ( g j )) ± 1 , [ uv − 1 ] ψ ( g j ) ± 1 . It ca n b e verified that there will be nonca nceling subw or ds in the bracketed parts a b ov e when ϕ ∗ ψ ∗ u ∗ v has remna nt . W e will verify the firs t and last of these: 7 In the pro duct [ ϕ v ( g i )]( ϕ v ( g j )) ± 1 = [ v − 1 ϕ ( g i ) v ] v − 1 ϕ ( g j ) ± 1 v , if ϕ ∗ ψ ∗ u ∗ v has remnant, then a portio n of ϕ ( g i ) will remain uncanceled. No w consider the pro duct: [ uv − 1 ] ψ ( g j ) ± 1 . Again, if ϕ ∗ ψ ∗ u ∗ v ha s remnant, then a p ortion of u will remain uncanceled. It is easy to check that v ar ious remnant words o f ϕ ∗ ψ ∗ u ∗ v are similarly uncanceled in the other of the 14 pro ducts a b ov e. Thu s S contains the set of tuples ( ϕ, ψ , u, v ) such that ϕ ∗ ψ ∗ u ∗ v has remnant. But this set of tuples is generic b y Lemma 9, since a c hoice o f a tuple with ϕ ∗ ψ ∗ u ∗ v having r e mnant is combinatorially equiv alent to choosing a single homomorphism η : F 2 n +2 → H , wher e F 2 n +2 is the free g roup o n 2 n + 2 generator s. Since S contains a gener ic set, it is itself g eneric. References [1] G. Arzhantsev a a nd A. O l’shanskii. The class of groups all of whose s ub- groups with lesser num b er of gener ators a re fr ee is gener ic. Mathematic al Notes , 59:35 0–35 5 , 19 96. [2] M. Bestvina and M. Handel. T ra in tracks a nd automor phisms of free groups. Annals of Mathematics , 135:1– 51, 1992. 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