A finiely preened group wih unbounded dead-end deph Sean Cleary and Tim R. Riley June 22, 2004 Revied Augu 25, 2004 and April 5, 2007 Abrac The dead-end deph of an elemen g of a group G, wih repec o a generaing e A, i he diance from g o he complemen of he radiu d A (1,g) cloed ball, in he word meric d A defined wih repec o A. We exhibi a finiely preened group G wih a finie generaing e wih repec o which here i no upper bound on he dead-end deph of elemen. 2000 Mahemaic Subjec Claificaion: 20F65 Key word and phrae: dead-end deph, lampligher The auhor regre ha he publihed verion of hi aricle (Proc. Amer. Mah. Soc., 134(2), page 343-349, 2006) conain a ignifican error concerning he model for G decribed in Secion 2. We are graeful o Jörg Lehner for poining ou our miake. In hi correced verion, ha model ha been overhauled, and ha ha neceiaed a number of change in he ubequen argumen. 1 Inroducion We explore he behavior of geodeic ray and he geomery of ball in Cayley graph of finiely generaed group. An elemen g in a group G i defined by Bogopol kiĭ [3] Suppor from PSC-CUNY gran #65752 i graefully acknowledged. Suppor from NSF gran 0404767 i graefully acknowledged. 1
o be a dead end wih repec o a generaing e A (which will alway be finie in hi aricle) if i i no adjacen o an elemen furher from he ideniy; ha i, if a geodeic ray in he Cayley graph of (G, A) from he ideniy o g canno be exended beyond g. Dead-end elemen occur in a variey of eing. Bogopol kiĭ howed ha SL 2 (Z), preened by x, y x 4, y 6, x 2 y 3, ha dead end wih repec o {x, y} and ha x, y x 3, y 3, (xy) k, wih k 3 ha dead end wih repec o {x, y} bu no {x, y, xy}. Fordham [8] found dead end in Thompon group F, preened by x 0, x 1 [x 0 x 1 1, x 0 1 x 1 x 0 ], [x 0 x 1 1, x 0 2 x 1 x 0 2 ], wih repec o he andard finie generaing e {x 0, x 1 }. In Lemma 4.19 of [4] Champeier how ha preenaion A R aifying he C (1/6) mall cancellaion condiion have no dead end wih repec o A. In general, generaing e can be conrived wih repec o which here are dead end; for example, Z ha dead end wih repec o he generaing e {a 2, a 3 }. Many of hee example are dicued in IV.A.13,14 of de la Harpe [7], where here i alo an exercie due o Valee concerning furher example. Dead end differ in heir everiy in he following ene. We define he deph, wih repec o a finie generaing e A, of an elemen g in an infinie group G o be he diance in he word meric d A beween g and he complemen in G of he cloed ball B g of radiu d A (1, g) cenered a 1. So g G i a dead end when i deph i a lea 2. Cleary and Taback [6] howed ha, wih repec o {x 0, x 1 }, he dead-end elemen in Thompon group F are all of deph 3. Cleary and Taback [5] exhibied wreah produc, uch a he lampligher group Z 2 Z, wih unbounded dead-end lengh wih repec o cerain andard generaing e. Independenly, Erchler oberved ha Z 2 Z provide an example reolving a cloely relaed queion of Bowdich (Queion 8.4 in Bevina problem li [2]): le Γ be he Cayley graph of an infinie finiely generaed group; doe here exi K >0 uch ha for all R > 0 and all verice v Γ B(R) here i an infinie ray from v o which doe no ener B(R K)? However, hee wreah produc example are no finiely preenable. So a naural queion, aked by Bogopol kiĭ [3], i wheher, given a finiely preenable group and a finie generaing e A, here i an upper bound on dead-end deph wih repec o A. Bogopol kiĭ [3] howed ha here i alway uch a bound in he cae of hyperbolic group. We anwer hi queion in he negaive wih he main reul of hi paper: Theorem 1. There i a finiely preenable group G ha ha a finie generaing e A wih repec o which G ha unbounded dead-end deph. We ake G o be he 2
group preened by P := a,, a 2 = 1, [a, a ] = 1, [, ] = 1, a = aa and A o be he generaing e {a,,, a, a, aa, a, a, aa}. We denoe he commuaor a 1 b 1 ab by [a, b] and he conjugae b 1 ab by a b. The lampligher group Z 2 Z, which ha preenaion a, a 2, [a, a i ], i Z, i a ubgroup of G. Removing he defining relaion a 2 from P give Baumlag remarkable example [1] of a finiely preened meabelian group conaining Z Z and hu a free abelian ubgroup of infinie rank. The reference o a pecific finie generaing e in he heorem i imporan a he following iue are unreolved. 1 I he propery of having unbounded deadend deph an invarian of finiely generaed group? Tha i, doe i depend on he choice of finie generaing e? Moreover, i hi propery a quai-iomery invarian? Indeed, doe he group G defined above have unbounded dead-end deph wih repec o all finie generaing e? Acknowledgmen. We began dicuing he idea in hi paper a he 2004 Cornell Topology Feival. We are graeful o he organizer for heir hopialiy. We would alo like o hank Oleg Bogopol kiĭ for an accoun of he hiory of dead-end deph and he anonymou referee for a careful reading. 2 The lampligher grid model for G The group Z 2 Z wa named he lampligher group by Cannon (ee Parry [9]) on accoun of he following faihful, raniive acion on P fin (Z) Z. Here, P fin (Z) denoe he e of finie ube of Z; ha i, finie configuraion illuminaed 1 Remark added Augu 2004. I ha ince been hown [10] ha a,, u a 2 = 1, [, u] = 1, a = a u ; i Z, [a, a i ] ha unbounded deph dead end wih repec o one finie generaing e, bu only a ingle dead end wih repec o anoher. 3
among a ring of lamp indexed by Z. The econd facor denoe he locaion of a lampligher (or a curor) in he ring of lamp. The acion of he generaor of a, a 2, [a, a i ], i Z i o incremen he locaion of he lampligher by one, and he acion of a i o oggle he lamp a he curren locaion of he lampligher beween on and off. We will develop a lampligher model for G and decribe a faihful, raniive lef acion of G on P fin (L) Z 2, where P fin (L) denoed he e of finie ube of a counable e L ha we define below. Thi will be be ueful for gaining inigh ino he meric properie of G. In our model, a lampligher move among he laice poin of he infinie rhombic grid illuraed in Figure 1. We will refer o he union of he -axi and he porion of he -axi ha i below he -axi a he lampand. Le L be he e of laice poin on he lampand; hee are he locaion of he lamp in our model. An elemen of P fin (L) denoe a finie configuraion of illuminaed lamp. The acion of and are o move he lampligher one uni in he - and - direcion, repecively, in he rhombic grid. The rhombic grid i ubdivided ino a riangular grid by inering a negaively loped diagonal ino each rhombu (he dahed line in Figure 1). A every laice poin here i a buon; he acion of a i o pre he buon a he locaion of he lampligher, and hi ha he following effec. If he buon i on he lampand, hen i oggle he lamp a i locaion. If he buon i off he lampand, a ignal i e off ha propagae and bifurcae in he riangular grid oward he lampand and oggle finiely many lamp a follow. When he buon i a a laice poin above he -axi, he ignal propagae downward in he riangular grid along he loped grid line. A each verex en roue i pli ino wo ignal, one advancing along he poiively loped diagonal below and one along he negaively loped diagonal below. The ignal op when hey hi he -axi, and each lamp on he -axi wiche beween on and off once for every ignal i receive. The manner in which hee ignal pli a hey propagae oward he lampand lead o a connecion wih Pacal riangle modulo 2. When he lampligher i above he -axi, we can underand he acion of a a follow. Suppoe he curren locaion of he lampligher ha, coordinae (p, q). Then for r {0, 1,..., p}, he lamp a poiion q+r i oggled beween on and off when here i a 1 in he r-h enry of row p of Pacal riangle mod 2. See he op row of diagram in Figure 1 for an example. 4
When he buon i a a laice poin below he -axi and o he lef of he -axi, he ignal propagae imilarly hrough he riangular grid oward he lampand, bu move in he horizonal direcion and in he -direcion. The ignal op when hey hi he lampand. There i again a connecion wih Pacal riangle modulo 2 bu hi ime i i roaed by 2π/3. The paern i inerruped a he - and -axe becaue ignal op when hey hi he lampand. The middle row of diagram in Figure 1 how an example. When he buon i a a laice poin below he -axi and o he righ of he - axi, he ignal propagae in he horizonal direcion and in he negaively loped diagonal direcion oward he lampand, where hey again op. Pacal riangle modulo 2 roaed by 4π/3 from i andard orienaion (and inerruped on he -axi) can be ued in decribing he acion, a illuraed in he boom row of diagram of Figure 1. To verify ha we have a well-defined acion of G on P fin (L) Z 2, i uffice o check ha a 2, [a, a ], and [, ] all ac rivially and ha he acion of aa and a agree. Thi and he proof ha he acion i raniive are lef o he reader. To how ha he acion i faihful we uppoe g G aifie g(, (0, 0) ) = (, (0, 0) ) and we will check ha g = 1 in G. Le w be a word repreening g. Conider he rhombic grid wih he ligh all off and he lampligher a (0, 0). Reading w from righ o lef deermine a pah for he lampligher aring and finihing a (0, 0), in he coure of which buon are preed: each leer ±1, ±1 in w move he lampligher, and each a ±1 pree a buon a he lampligher locaion, oggling ome of he lamp. Ue he relaion a 2 = 1 and aa = a o change w o a word w uch ha w = w in G, and here are no leer a 1 in w and no ubword a 2, and along he pah deermined by w, he only buon ha are preed are on he lampand. Then, by applying he relaion [, ] = 1 o w and removing invere pair, obain a freely reduced word w uch ha w = w in G and he pah deermined by w doe no leave he lampand. Then by inering invere pair of leer, w can be changed o a word w whoe pah reurn o he origin afer each buon-pre ha i, w i a concaenaion of word of he form a i (wih i < 0) and a j (wih j Z). Moreover, a w leave no lamp illuminaed, each a i and a j occur an even number of ime. Bu, a we how below, all a i and a j repreen commuing elemen of G. We deduce ha w, and herefore w, repreen 1 in he group. To ee ha [a i, a j ] = 1 in G for all i, j, i uffice o how ha [a, a i ] = 1 for 5
1 1 1 1 0 1 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 Figure 1: Example of a acing. The fir column of diagram how hg(, (0, 0)), where g = 3 a 1 a 2 4 a 2 a 1 a 3 a 1 a 1 a 3 and h i 6 4, 4 4, and 4 7, repecively. The hird column how he correponding ahg(, (0, 0)). (Off and on lamp are repreened by circle wih black and whie inerior, repecively.) all i 0. We will induc on i. Aume [a, a i ] = 1 for all 0 i k. Then [a, a k ] = 1 and o [aa, a k a k+1 ] = 1 a a = aa. Wih he excepion of a, a k+1, he inducion hypohei ell u ha any wo of a, a, a k, a k+1 commue. So [a, a k+1 ] = [aa, a k a k+1 ] = 1. To ee ha [a i, a j ] = 1 for all i, 0 and all j Z, noice ha [a i, a j ] i = [a, a j i ] = [a, (a i }) j ] which i 1 in G becaue (a i ) j i in he ubgroup S of G generaed by {a k. To ee ha [a i, a j ] = 1 for all i, j < 0, i uffice o how k Z ha [a, a l ] = 1 for all l > 0 and ha can be done imilarly. 6
3 Proof of he heorem Define map I : G P fin (L) and L : G Z 2 by ( I(g), L(g) ) = g(, (0, 0) ). So I give he lamp illuminaed and L give he locaion of he lampligher afer he acion of g on he configuraion in which no lamp are li and he lampligher i a he origin. (In fac, L i, in effec, he reracion G Z 2 =, ha kill a.) Define H n o be he ube of Z 2 of laice poin in (and on he boundary of) he hexagonal region of he grid wih corner a (±n, 0), (0, ±n), (n, n), ( n, n). (The haded region in Figure 2 i H 4.) In he following propoiion we deermine he diance of variou group elemen from he ideniy in he word meric wih repec o A = {a,,, a, a, aa, a, a, aa}. The crucial feaure of hi generaing e i ha he buon a a verex he lampligher i leaving or arriving a can be preed wih no addiional co o word lengh. Thu for g G {a}, he diance d A (g, 1) i he lengh of he hore pah in he rhombic grid ha ar a (0, 0), finihe a L(g), and uch ha preing ome of he buon a he verice viied produce he configuraion I(g) of illuminaed bulb. Propoiion 2. If g G i uch ha I(g) H n and L(g) H n, hen d A (1, g) 6n. Proof. Define (p, q) := L(g). So p and q are he and coordinae, repecively, of he poiion of he lampligher. We will decribe a pah from (0, 0) o (p, q) and pecify buon o puh en roue ha will illuminae I(g). We will fir addre he cae where p 0. Have he lampligher begin by ravelling a diance 2n fir along he -axi o ( n, 0) and hen back o (0, 0), illuminaing all bulb in I(g) ha are no on he -axi a i goe. How o proceed nex depend on wheher (p, q) i in he e T n of laice poin in, and on he boundary of, he riangular region wih corner (0, 0), (0, n), (n, n). 7
L 1 L 3 L 2 Figure 2: Region and line in he lamplighing grid. (In Figure 2 he region haded dark gray conain T 4.) Aume (p, q) T n. Have he lampligher ravel along he -axi o (0, n), hen along he -axi o (0, n), hen along he -axi o (0, q), and hen parallel o he -axi o (p, q). Thi involve ravering a mo 4n edge. Following hi pah, he lampligher vii all he lamp in I(g) ha are on he -axi and o can illuminae hem. So d A (1, g) 6n. Nex, we aume L(g) H n T n. Then L(g) i in, or on he boundary of, he rhombic region above he axi (haded medium gray in Figure 2). Have he lampligher follow a pah φ, fir along he -axi o (0, n), hen back along he -axi o (0, p), hen parallel o he -axi o (p, p), hen parallel o he -axi o (p, n p), and hen parallel o he -axi back o o (p, q). Thee five arc have lengh n, n p, p, n and n p q, repecively, oalling 4n p q, which i le han 4n becaue p > q. We now how ha preing ome combinaion of buon on φ achieve he configuraion of illuminaed lamp on he -axi required for I(g). The lamp in poiion n, n + 1,..., p 1 can be illuminaed a required when ravering he econd arc of φ. The lamp in poiion p, p + 1,..., n deermine a equence of n+p+1 zero and one which make up he bae of a rapezoid of zero and one a analyzed in Lemma 3, below. Overlaying hi rapezoid on he grid, we find ha i lef and op ide follow he hird and fourh arc of φ. Preing he buon a he locaion of he ummi (defined in Lemma 3) in he rapezoid illuminae he required lamp. Thi i becaue he rapezoid i conruced in uch a way ha each enry equal he oal number of ignal (ee Secion 2) mod 2 reaching 8
i locaion when he buon a he ummi are all preed. In oher word, he rapezoid i he overlay (mod 2) of a number of copie of Pacal riangle, one for each ummi; o, for example, he rapezoid in Figure 3 i he overlay of five riangle, wih heigh 1, 2, 4, 5 and 5. We now urn o he cae where p < 0 and q 0. The lampligher begin by ravelling along he -axi o (0, n) and hen back o (0, 0), illuminaing all he bulb in I(g) ha i ravere en roue. Nex i move along he -axi o (0, n) and hen parallel o he -axi o ( n, n). A in he argumen above involving Lemma 3, beween (0, n) and ( n, n) he lampligher can pre a combinaion of buon which pu he ligh on he -axi ino he configuraion of I(g). Thi will have ome effec on he ligh beween ( 1, 0) and ( n, 0). Bu, when ravelling from (0, 0) o (0, n), he lampligher can oggle ligh boh o counerac hi effec and o implemen he required configuraion for hoe ligh. By hi age he lampligher ha ravelled a diance of 4n, i locaed a ( n, n), and I(g) ha been achieved. All ha remain i for he lampligher o move o (p, q), which i wihin a furher diance of 2n a p < 0 and q 0. Finally we addre he cae where p < 0 and q > 0. Fir he lampligher ravel along he -axi o (0, n) and hen back o (0, 0), illuminaing all he bulb a per I(g) en roue. Then he lampligher move along he -axi o and hen parallel o he -axi o ( n, n). Beween (0, n) and ( n, n) buon are preed o illuminae he lamp on he -axi a per I(g), and beween (0, 0) and (0, n) buon are preed o counerac he effec on he bulb here and o achieve he configuraion I(g). The lampligher hen move o (p, q) which i wihin a diance 2n from ( n, n). Lemma 3. Suppoe S i a equence of m zero and one. Suppoe r {1, 2,...,m}. There i a rapezoid coniing of zero and one (enrie) aifying he following. The enrie are arranged in r row wih he boom row S and all he oher row conaining one fewer enry han ha below i (a in Figure 3). Every enry i he um of he wo enrie immediaely above i mod 2, wih poible excepion of ome enrie in he op row and a he lef end of he row. We call hee excepional enrie ummi. Proof. We ue inducion on r. When r = 1 here i nohing o prove. For he inducion ep, we aume he lowe r row have been conruced a per he hypohee. We add he nex row one enry a a ime aring from he righ-hand ide. We elec each enry o a o enure ha of he wo enrie immediaely 9
below i, ha o he righ ha he propery ha i i he um of he wo enrie immediaely above i mod 2. Thi enure ha only enrie in he lef ide he rapezoid and in he (r + 1)- row can fail o equal he um of he wo enrie immediaely above hem, mod 2. 0 0 1 0 1 1 0 1 1 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 Figure 3: A rapezoid of one and zero in which every enry oher han hoe circled (called ummi) i he um of he wo enrie above i mod 2. Propoiion 4. If g G aifie L(g) = (0, 0) and I(g) = {(0, n), (0, n), ( n, 0)}, hen d A (1, g) 6n. Proof. In order o ligh he lamp a (0, n), (0, n) and ( n, 0), he lampligher mu vii (or cro) he raigh line L 1 hrough (0, n) and (n, 0), L 2 hrough (0, n) and (n, n), and L 3 hrough ( n, 0) and ( n, n), and hen reurn o he origin. (See Figure 2.) We uppoe he lampligher follow a minimal lengh pah ha vii he hree line and reurn o he origin. We will how ha hi pah ha lengh a lea 6n. Drawing L 1, L 2, L 3 in andard R 2 wih he l 1 -meric, a ymmery become apparen on accoun of which we may a well aume ha he lampligher fir vii L 2. Bu hen becaue we are uing he l 1 -meric, he pah can be alered wihou increaing i lengh o ha i fir ravel from he origin o L 2 along he -axi. If he pah vii L 3 before L 1, hen i can be alered wihou increaing i lengh o ha i doe o by ravelling a diance n in he (negaive) -direcion one hen eaily check ha viiing L 1 and hen reurning o he origin co a lea 4n in addiional lengh. Suppoe, on he oher hand, he lampligher proceed nex o L 1 afer L 2. Then he final porion of he lampligher journey will be beween L 3 and he origin and he pah can be alered, wih no change in lengh, o ha i follow he -axi from ( n, 0) o (0, 0). One check ha a hore pah from (0, n) o ( n, 0) via L 1 ha lengh a lea 4n. So in each cae he lengh of he pah i a lea 6n and o d A (1, g) 6n, a claimed. 10
Proof of Theorem 1. Define g n := n a n n a 2n a n = n 1 (a) n n 1 (a) 2n (a) n 1. Then d A (1, g n ) 6n and o d A (1, g n ) = 6n by Propoiion 4 becaue g n (, (0, 0)) = ({( n, 0), (0, n), (0, n)},(0, 0)). By Propoiion 2, if w i a word repreening an elemen g wih d A (1, g) > 6n, hen he lampligher pah deermined by w (reading righ o lef and wih he lampligher iniially a he origin) mu leave H n. Bu L(g n ) = (0, 0) and H n conain he cloed ball of radiu n abou (0, 0) in he rhombic grid. So if v n i a word repreening g n and u n v n a word repreening an elemen h n wih d A (1, h n ) > 6n, hen he lengh of u n i more han n. So, wih repec o d A, he diance from g n o he complemen of he radiu d A (1, g n ) ball i more han n. Thu, wih repec o A, he dead-end deph of g n i a lea n and G ha unbounded dead-end deph. Reference [1] G. Baumlag. A finiely preened meabelian group wih a free abelian derived group of infinie rank. Proc. Amer. Mah. Soc., 35:61 62, 1972. [2] M. Bevina. Queion in geomeric group heory. hp://www.mah.uah.edu/ bevina/. [3] O. V. Bogopol kiĭ. Infinie commenurable hyperbolic group are bi-lipchiz equivalen. Algebra and Logic, 36(3):155 163, 1997. [4] C. Champeier. Propriéé aiique de group de préenaion finie. Adv. in Mah., 116:197 262, 1995. [5] S. Cleary and J. Taback. Dead end word in lampligher group and oher wreah produc. Quar. J. Mah. Oxford, 56(2):165 178, 2005. [6] S. Cleary and J. Taback. Combinaorial properie of Thompon group F. Tran. Amer. Mah. Soc., 356(7):2825 2849 (elecronic), 2004. [7] P. de la Harpe. Topic in geomeric group heory. Chicago Lecure in Mahemaic. Univeriy of Chicago Pre, 2000. [8] S. B. Fordham. Minimal lengh elemen in Thompon group F. PhD hei, Brigham Young Univeriy, 1995. 11
[9] W. Parry. Growh erie of ome wreah produc. Tran. Amer. Mah. Soc., 331(2):751 759, 1992. [10] T. R. Riley and A. Warhall. The unbounded dead-end deph propery i no a group invarian. In. J. Alg. Comp., 16(5): 969-984, 2006. Sean Cleary Deparmen of Mahemaic, The Ciy College of New York, Ciy Univeriy of New York, New York, NY 10031, USA cleary@ci.ccny.cuny.edu, hp://www.ci.ccny.cuny.edu/ cleary/ Tim R. Riley Deparmen of Mahemaic, 310 Malo Hall, Cornell Univeriy, Ihaca, NY 14853-4201, USA im.riley@mah.cornell.edu, hp://www.mah.cornell.edu/ riley/ 12