Diophantine Equations and the Freeness of Möbius Groups

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1 Alied Mthemtics, 04, 5, Published Olie Jue 04 i SciRes htt://wwwscirorg/ourl/m htt://ddoiorg/046/m0450 Diohtie Eutios d the Freeess of Möbius Grous Mri Gut Lbortoire de Mthémtiues, Uiversité Blise-Pscl, Clermot-Ferrd, Frce Emil: mrigut@mthuiv-bclermotfr Received 6 Mrch 04; revised 6 Aril 04; cceted Aril 04 Coyright 04 by uthor d Scietific Reserch Publishig Ic This work is licesed uder the Cretive Commos Attributio Itertiol Licese (CC BY htt://cretivecommosorg/liceses/by/40/ Abstrct Let d be two fied o zero itegers verifyig the coditio gcd (, = We check solutios i o zero itegers, b,, b d for the followig Diohtie eutios: (B ( b = 0 (B 4 4 b b b b bb = 0 The eutios (B d (B were cosidered by RC Lydo d JL Ullm i [] d AF Berdo i [] i coectio with the freeess of the Möbius grou G geerted by two mtrices of (, mely A = 0 0 d B = where = They roved tht if oe of the eutios (B or (B hs solutios i o zero itegers the the grou G is ot free We give lgorithms to decide if these eutios dmit solutios We obti rithmeticl criteri o d for which (B dmits solutios We show tht for ll d the eutios (B d (B hve oly fiite umber of solutios Keywords Diohtie Eutio, Möbius Grous, Free Grou Itroductio Let k d d be two ositive itegers with d d A,, Ak be mtrices of the grou d ( How to cite this er: Gut, M (04 Diohtie Eutios d the Freeess of Möbius Grous Alied Mthemtics, 5, htt://ddoiorg/046/m0450

2 Deote ( A A the grou, resectively g ( A A M Gut,, k,, k the semigrou, geerted by the mtrices A,, A k The followig roblem P hs bee studied i severl ers: Istce: A,, Ak d ( Questio: ( A,, Ak or g ( A,, Ak re they free with A,, Ak s geertors? Recll tht i 99 D Klrer, J-C Birget d W Stterfield i [] roved tht if d the the roblem P is ot decidble Moreover i 999 J Cssige, T Hru d J Krhümki i [4] roved tht the sme result is true if we suose tht ll the mtrices A,, Ak re lower trigulr The cse d = is oe d seems difficult I [5] d [6] results cocerig the freeess of the semigrous d grous geerted by two mtrices re estblished I this er we re studyig this roblem restricted to the cse of Möbius grous Let d = The Möbius grou G is the subgrou of ( geerted by A = 0 0 d B = The roblem of chrcteriztio of the set of comle vlues of or for which the grou G is free, ws studied i severl ers Thus i [] it is roved tht if is trscedetl or the G is free RC Lydo d JL Ullm i [] remrked tht G is ot free if d oly if there eists word w= b b b whose letters re o zero itegers so tht the roduct of the owers of b b b mtrices M ( w = A B A B A B A is lower trigulr mtri The elemet i the right uer corer of the mtri M ( w is of the form ( w where ( w = c 0 is olyomil i of degree with coefficiets c = Moreover c re olyomils with itegers coefficiets i the vribles, b,, b,, Results cocerig the set of lgebric vlues of or for which the grou G is ot free were obtied i [] [] [7]-[] Decidig if for ] 0, 4[ the grou G is ot free seems very difficult Let us recll some imortt results i this directio The grou G is ot free if belogs to oe of the followig sets: A=, m B= k,, C= m,, m > (see [] [] [7] [8] [0] [] k m I this er we check if for give ] 0, 4[ there eists o trivil word of o zero itegers w= b b b such tht Q ( w = 0 The mi results of our er cocer the freeess of Möbius grous: We rove tht if the legth of w is smll the the roblem is decidble (cses = d = (see Theorems, d We give lgorithms which solve the roblem for {,} (see Corollry d the roof of Theorem Moreover, we give rithmeticl criteri for this roblem whe = (see of Theorem We give lower boud umericl fuctio l defied from ],4[ to, icresig d ubouded, such tht for ech ],4 [, if M ( w is lower trigulr mtri the the legth of w is bigger th l ( (see Theorem 4 d Corollry As roved by AF Berdo ([] i these two cses {,} we hve to fid solutios for the eutios (B d (B I fct i our er we cosider d study two more geerl eutios: (B' ( b = 0 b b b b bb = (B' 0 Seueces of Polyomils Associted to Mtrices I this sectio, we study the roerties of some seueces of olyomils i fied mtrices of the grou G = ssocited to 40

3 M Gut ( 0, We cosider ( { } the free mooid of words o o zero itegers with the coctetio oertio We deote by the emty word of the free mooid ( { 0} d o emty word w ( { 0} by w= kkk k, where k,, k re o zero itegers The is clled the legth of w d is deoted by w The reversl of word w= kkk k is w = k kkk d the oosite of w is ( w= b b b of { } w= k k k k For every word roduct of legth w = we cosider the mtri 0 b b = M w A B A B A B A For istce, for, b, o zero itegers we hve: M We use the ottio: ( 0 = b d M ( b M ( w We remrk tht ( w, ( w, ( w d ( w lso hve ( ω = ( ω ( w = ( w d M ( ω = M ( ω b b = b b ( w ( w ( w ( w = re olyomils i with coefficiets i We If {, } the ( w = ( w d if {, } the ( w = ( w d if {, } the ( w = ( w We use the ottio M ( w ( w 0 = 0 = to idicte tht M ( w From ow o, i order to simlify the ottio we write: For istce, defied by: P Q w w = R S w w Also is lower trigulr mtri or tht P is bbrevitio for the olyomil i with rmeters, b,, b,,, b, Usig the fct tht AB, ( ( = ( P, b,, b,,, b, b b b we hve: PS Q R = ( The seueces of olyomils i, ( P, ( Q, ( R d ( S P Q = R S 0 P Q b b = R S b b ( ( P Q P bq P b Q = R S R bs R b S verify the followig reltios: ( ( (4 40

4 M Gut The reltios (4 d (5 follow from the eulity P Q P b Q P Q = R S R bs R S P Q P Q B A b = R S R S I the followig sectios, we lso use the followig two reltios: P = b b b bb (6 Q = Usig the revious reltios we obti Proositio The seueces ( Q d Proof From (5 we hve b b b b bb Q of olyomils i verify the followig idetities: Q b Q Q = 0 (8 bp b b bb P b P = 0 (9 P = Q Q d P = Q Q [ ] [ ] These idetities d the eutio P = P b Q give the eutio (8 The eutio (9 c be similrly obtied Let us suose tht = = where d re o zero itegers d gcd (, = If I the followig we cosider tht = the grou G is ot free becuse i this cse Q (,,( = 0 (see [] > The ( w 0, d ( w 0 Ideed, if ( w ( w = 0 the usig the fct tht det ( ( w = we deduce ( w ( w = fct tht gcd (, = This remrk llows us to defie ew seuece ( reltio: Thus we obti (5 (7 which is i cotrdictio with the by Q = This seuece stisfies the followig P = b = b b These reltios re similr with formuls for cotiued frctios The roerties of these seueces will be used i the et sectios of our er Let us lso cosider the seuece ( y defied by: (0 40

5 M Gut We remrk tht ( b b y = y(, b,,, b = = b,,,,, 0 if d oly if = ( = (, b,, y, b The followig lemm is the key elemet of Sectio 5 Lemm Let, b,,, b, be o zero itegers d suose tht > If (, b,,, b, = 0 the (, b,,, b,,, b, 0 we hve Proof If Let = (, b,, = y(, b = b b = such tht G is ot free We defie the followig umericl fuctio: { } = mi w ( { 0 }, w = d ( w = 0 The umber κ clibre of the grou G Hece κ will be clled the b 0 κ = if d oly if there re o zero itegers, b, such tht A B A = Also we b b 0 κ if d oly if there re o zero itegers, b,, b, such tht A B A B A = hve The Diohtie Eutio (B I the et three sectios, we cosider the followig roblem Istce: Two o zero itegers, with gcd (, = P, where, : Questio: Is there word of legth of o zero itegers w= b such tht M ( w 0 =, where =? So we check solutios i o zero itegers, b,, for the diohtie eutio Q, b,, = 0 ( The set of = for which the Möbius grou G is ot free coicides with the set of for which there eists such tht the Eutio ( dmits solutios I this sectio, we cosider the cse = d i the et sectio the cse = The reltio Q(, b, = 0 is euivlet to the Eutio (B' d the reltio Q(, b, = 0 is euivlet to the eutio (B' If d re erfect sures we obti the eutios (B d (B We will rove tht the roblem P is decidble The decidbility of the roblem P hs lredy bee estblished by AF Berdo (Theorem, [] for the cse whe d re erfect sures Our lgorithm is simler d llows us to give rithmeticl criteri for itegers d for which the roblem P hs solutios (see Theorem below First, we rove result cocerig the eutio (B' gcd, = d α Deote Proositio Let d be two itegers with The: { α} ( α ( L,, =, b, b = 404

6 M Gut If (, b, L( α,, d {,} The set (,, Proof L α is fiite i we hve b ( α i Let (, b, L( α,, d for {,} ( mod d di d i α As b = di i = we hve b results from ( ( α i ut di = ib The Becuse di bα d we deduce b ( α di i b ( α Usig the revious roositio we c obti the decidbility of the roblem Theorem Let d be two itegers with gcd (, = The eutio ( b = 0 hs solutios i o zero itegers There eists divisor d of, d such tht d ( mod d d = bα, P The followig seteces re euivlet: ( m, =, m 0 m Proof The euivlece betwee ( d ( results from the Proositio It is eough to cosider α = 0 i tht roositio The euivlece betwee ( d ( is obvious Remrk Let D( be the set of ll divisors of the iteger If d is like i ( of the revious Theorem the solutio (, b, to the eutio (B' c be obtied by tkig d d b D D d di i = for i {,} b where d = d d d d method We c write s i ( of the Theorem = Moreover y solutio (,, b of the eutio (B' c be obtied by this = = d d The results of AF Berdo ([], theorem cocerig the roblem P for the cse whe d re erfect sures (or euivletly whe result immeditely from the et corollry Corollry Let d be two o zero itegers with gcd (, = d = = The grou G is ot free with the clibre k ( = if d oly if there eists divisor d of d mod From the revious theorem it lso follows: 4, d such tht 405

7 M Gut The eutio (B' hs o solutio if = > k = i the followig cses: = ; b = ; c d = k ± with k Below we reset other form of the Theorem i which we use the decomositio of s roduct of rime umbers Theorem Let d be two itegers with > d gcd (, = Let us suose tht the α α αm decomositio of s roduct of owers of distict rime umbers π, π,, πm is = π π πm The k = if d oly if there eist: two disoit subsets I d J of {,,, m} with I J set of itegers ( δl l I with δ J l αl for every l I J ε, { } δi i such tht π ε π ( mod i I J δ β β βm Proof Let d = επ π π m be divisor of, d ± We c dro the cse d = becuse > β, β,, βm α, α,, αm d 0 βk αk for every k {,,, m} We ut I = i {,,, m} βi < αi d J = {,,, m} β > α The I J = d I J Let: Hece { } We hve δ α for every I J { } α β δ = β α if I if J The coditio d ( mod π ε π ( mod δi i i I J δ is euivlet to Corollry Let d α be two o zero itegers d π be rime umber Suose tht gcd (, π = The k α = if d oly if there eists iteger δ with δ α such tht π δ π ε( mod where ε {, } Proof We tke m = i the revious theorem Emle: Usig the revious results d emle from ([7] we hve k 7 = d k 7 = 4 The Berdo Diohtie Eutio (B Now we cosider the roblem P ( We metio tht the eutio Q, b,, b, = 0 hs bee cosidered i severl ers (see [] [8] [0] for the cse whe d re erfect sures From ow o, we suose tht Q b for every (,,,, 0 does ot belog to ] [ ( ϕ : 0, 4 A b ie followig Theorem, A= m,, m 0 Hece we c defie fuctio m by ϕ { αα A} ϕ = if ] [,4 ϕ mi, k k = if = We remrk tht 0 where ϕ > d b if ], [, k = Usig the reltios (8 for the seuece of olyomils ( Q we rove tht the roblem ( decidble Theorem Let Q ] 0, 4[ such tht does ot belog to the set ( m m, P is The the 406

8 M Gut eutio Q, b,, b, = 0 hs fiite umber of solutios 5, b,, b, Proof Usig the reltios (8 we deduce tht Q, b, Q, b, = Hece = Usig the fuctio ϕ we hve: b b bb bb ϕ We obti fiite umber of ossibilities for b, b d So d remi to be studied From the eutio it follows tht Q, b,, b, = 0 P, b,, b, S, b,, b, = Hece there eists ( d, d such tht 4 dd = ( b b b bb = d ( b b b bb = d Thus there eists fiite umber of ossibilities for d If Q(, b,, b, ( = 0 from the ieulity bb we obti ϕ If ],4[ the P ( hs o solutio b If, the b { }, b,, We lso remrk tht the eutio (B' is euivlet to the followig eutio b b = This ebles us to obti some elicit eressios for the rtiols such tht eutio (B' hs solutios i Z Proositio Let k, be two o zero itegers d k,k be two divisors of k If = k k k the the eutio (B' hs solutios i Z Proof Let b = k, b = k, = d b = b = k The (0 is euivlet to = k k k Note tht if i eutio (B' we hve b = b the is ectly give by the bove eressio Usig oce gi (0 we obti Proositio 4 Let α d α be i Z with α α If the the eutio (B' hs solutios i Z α α = αα α α ( ( 407

9 M Gut Proof Cosider (0 for b = b =, = α d = α The = α ρ, where b α α ρ = α α b = α ρ d It follows tht if we tke = α α the (0 is verified I the et roositio we give other method to obti solutios of Eutio (B' It is similr to those reseted i [8] d [0] Proositio 5 Let d be two itegers with gcd (, = Suose tht there eist, b d i Z such tht ( b = If = the the eutio (B' hs solutios i Z Proof Let A=, B= b, A=, B= b d A = The = A Hece B B A A the eutio (B' hs solutios We ed this sectio with the followig oe uestios: Questios: Fid ll the solutios of (B Fid rithmeticl chrcteriztios (similr to those give i Theorem for the ositive itegers d P hs solutios for which the roblem 5 Icresig Ubouded Lower Boud Fuctio for κ I this sectio, we rove tht i order to show tht the grou G is ot free for rtiol with = < 4 d close to 4, we hve to cosider loger d loger words i A d B Similr remrks (without y roof hve bee mde by AF Berdo i [] d SP Frbm i [7] Everywhere i this sectio, we cosider tht is rtiol umber i the oe itervl ],4[ From the Lemm, Sectio, if (,,, b, ( = 0 the (,, ( For this α of rtiol fuctios i the vrible, α = α, defied by: reso we cosider the seuece ( For emle α α 4 =, = α = α = α d α We lso defie the fuctio l : ], 4[ { 0,} l = 5 6 N by the formul: = if k N α k Thus oe hs l ( = if d oly if ],], d oly if l = if d oly if, 5 5, α α =,,,,,, For this reso we fid the mtri Now we will clculte Note tht l = if d 4 408

10 M Gut, where C = AB θ We suose ow tht = si with X = AB AB AB A = AB A = C A π θ, π, so = = ( cosθ As Usig this reltio we fid tht Hece α C trce C = = cosθ the mtri C verifies the eutio: cos = C θc I O θ si (( θ siθ si si ( θ = siθ θ θ si si ( θ si ( θ si ( si (( θ (( θ ( θ si ( θ si (( θ = = si si Lemm Let ( α,,, > be such tht > 0 b α α > The for every b, Z we hve Proof Sice < we obti tht < Hece b b α α The revious eressio for α ( d Lemm show tht l ( is well defied d l < κ, for every i the oe itervl ],4[ So l is lower boud umericl fuctio for the fuctio κ restricted to ], 4 [ Theorem 4 For y d ],4[ oe hs l( = if d oly if there eists θ π, π such tht cos = θ Proof Let l =, where > si (( k θ cosθ si kθ From the defiitio of the fuctio l this revious eulity holds if d oly if α d α >, for ll k {,, } But α Thus we obti the system of two ieulities Filly, l = if d oly if we hve k (( k θ si ( kθ (( θ (( θ si si si These ieulities give θ π, π Corollry The fuctio l is icresig d ubouded < d 0 d k (( k θ si ( kθ ( kθ (( k si > if d oly if < 0 si 0 si θ < for ll k {,, } 409

11 M Gut Therefore κ lim l = lim = 4, < 4 4, < 4 Emle: We cosider the seuece = 4, for N For = 0 we hve 0 follows tht κ( 0 = 7 For = we hve d sice we hve κ( = 5 α For = we hve ( 6 4 = < = 45 Questios: = So α ( 0 = = d α ( =, α ( =, α ( 5 4 Is it true tht for every ] 0, 4[ Is it true tht for every ] 0, 4[ decidble? Is it true tht for every ] 0, 4[ solutios? 4 Fid κ, for Ackowledgemets 5 0, hece l ( 0 = As ( 0 5 (,,,,,,,, = = < = 5,,,, = 0, it = d α ( =, α ( =, α ( =, Hece l ( = 6 d κ( From [7] we hve 7 κ 8 d, P is decidble? there eists, Hece l ( = 4 05 α5 =, 78, the roblem such tht the roblem there eists,, such tht the roblem P is P hs I thk Elis Thh (Uiversity S Bolivr, Crcs d Jerzy Tomsik (Uiversite d Auverge, Clermot- Ferrd for discussio cocerig some logicl sects of my er Refereces [] Lydo, RC d Ullm, JL (969 Grous Geerted by Two Lier Prbolic Trsformtios Cdi Jourl of Mthemtics,, htt://ddoiorg/045/cjm [] Berdo, AF (99 Pell s Eutio d Two Geertor Möbius Grous Bulleti of the Lodo Mthemticl Society, 5, 57-5 htt://ddoiorg/0/blms/5657 [] Klrer, D, Birget, J-C d Stterfield, W (99 O the Udecidbility of the Freeess of Iteger Mtri Semigrous Itertiol Jourl of Algebr d Comuttio,, -6 htt://ddoiorg/04/s [4] Cssige, J, Hru, T d Krhumki, J (999 O the Udecidbility of the Freeess of Mtri Semigrous Itertiol Jourl of Algebr d Comuttio, 9, htt://ddoiorg/04/s [5] Cssige, J d Nicols, F (0 O the Decidbility of Semigrou Freeess RAIRO Theoreticl Iformtics d Alictios, 46, htt://ddoiorg/005/it/000 [6] Gwrychowski, P, Gut, M d Kisielewicz, A (00 O the Problem of Freess of Multlictive Mtri Semigrous Theoreticl Comuter Sciece, 4, 5-0 htt://ddoiorg/006/tcs [7] Frbm, SP (995 No-Free Two-Geertor Subgrous of SL ( Q Publiccios Mthemàtiues, 9, 79-9 htt://ddoiorg/05565/publmat_995_ [8] T, E-C d T, S-P (996 Qudrtic Diohtie Eutios d Two Geertors Möbius Grous Jourl of the Austrli Mthemticl Society, 6, htt://ddoiorg/007/s [9] de l Hre, P (000 Toics i Geometric Grou Theory Chicgo Lectures i Mthemtics Uiversity of Chicgo Press, Chicgo 40

12 M Gut [0] Grytczuk, A d Wotowicz, M (000 Berdo s Diohtie Eutios d No-Free Möbius Grous Bulleti of the Lodo Mthemticl Society,, 05-0 htt://ddoiorg/007/s [] Bmberg, J (000 No-Free Poits for Grous Geerted by Pir of Mtrices Jourl of the Lodo Mthemticl Society, 6, htt://ddoiorg/0/s