A Short Introduction to Self-similar Groups

Transcription

1 A Short Introution to Self-similr Groups Murry Eler* Asi Pifi Mthemtis Newsletter Astrt. Self-similr groups re fsinting re of urrent reserh. Here we give short, n hopefully essile, introution to them. 1. Introution 0 1 The figure ove shows (prt of) the infinite e inry tree, T. The is the noe (or vertex) t the top, n eh noe hs extly two noes elow it joine y n ege. It goes on forever own the pge. We hve lelle eh noe with inry numer in systemti wy if w is inry string lelling noe, then the two noes elow it joine y n ege re lelle w0 n w1. An utomorphism of T is ijetive mp from the noes to the noes whih preserves jenies, mening if two noes re joine y n ege, then the noes they mp to re gin joine y n ege. This efinition works for ny grph, ut we ll stik with T for now. Before we give n exmple, here is onvention whih will help esrie utomorphisms of T. Drwing t noe lelle y w (some inry string) mens exhnge the sutree with w0 n the sutree with w1, s inite here: n utomorphism hs severl s then it oesn t mtter in whih orer they re performe. Now for n exmple. Define to e the utomorphism of T whih fixes the, sens noes lelle 0w to 1w, n noes lelle 1w to 0w, where w is ny inry string (possily empty). The tree on the right shows wht T looks like fter is pplie Applying twie puts T k s it strte, so we sy tht is the sme s the mp tht oes nothing to the tree. We ll the mp whih leves the tree unhnge the ientity, n enote it y the letter e. Note tht if n re utomorphisms of T, the nottion mens pply first then. For exmple, if is the utomorphism given y 0 then the reer n hek tht sens the noe lelle 00 to position 11 while sens it to 10. w0 w1 w00 w01 w10 w11 w w1 w0 w10 w11 w00 w01 The representtion of n utomorphism of T y T eorte y s is lle portrit of the utomorphism. Note tht every utomorphism of T n e expresse using this nottion (possily with infinitely mny s). One the move is performe we remove the, n we n verify tht with this efinition, if Invite tehnil pper, ommunite y Jon Borwein. The uthor s reserh is supporte y the Austrlin Reserh Counil. The inverse of n utomorphism x is n utomorphism y suh tht xy = e( = yx). Sine utomorphisms re ijetive mps, they hve inverses. If G is set of utomorphisms of T n their inverses, suh tht for eh x, y G the prouts Jnury 2013, Volume 3 No 1 17

2 Asi Pifi Mthemtis Newsletter xy n yx re lso in G, then the lgeri ojet we otin is lle group. 1 A goo wy to ensure the property tht prouts of G sty in G is to tke set of utomorphisms, sy n, their inverses (whih in this se re the sme), n let G e the set of ll finite prouts of these utomorphisms. In this se we sy G is generte y the set {, }, n n re the genertors. Whenever group is generte y finite set of elements (utomorphisms), we ll it finitely generte group. Here is nother wy to efine utomorphisms. Let w e inry string. The mp sens the noe lelle 0w to position 1w, n the noe lelle 1w to 0w, so we n esrie it using the following rules: (0w) = 1.e(w), (1w) = 0.e(w), where e(w) mens pply the ientity mp (o nothing) to the suffix w. More interesting re the rules esriing : (0w) = 0.e(w), (1w) = 0.(w). The first rule just sys tht noes on the left sutree of the re not hnge, ut the seon rule sys if noe lel strts with 1, pply to the suffix of the lel. Note tht the rules for uses n e, n the rules for only uses e, so the set {,, e} of utomorphisms n e esrie y self-referening or selfsimilr set of rules. Prouts of n n lso e expresse with rules of this form, for exmple (0w) = (1.e(w)) = 1.(e(w)) (i.e. pply e first to w then ), n (0w) = (0.e(w)) = 1.ee(w). Definition 1.1. Let G e group of utomorphisms of T. Then G is self-similr group if for eh g G, eh x {0, 1}, n eh inry string w, there is y {0, 1} n h G suh tht g(xw) = y.h(w). See [6] for more etils. Note tht the efinition esily extens to groups of utomorphisms of e n-ry trees, ut gin we will stik with inry trees for this pper. 2. Automt Another wy to esrie utomorphisms of T is y n utomton. Here is n exmple: 1 Groups re not just sets of utomorphisms of T they n e the onfigurtions of Ruik s ue, utomorphisms of grphs other thn T, ris, n mps of the rel line to itself. A group is just set with multiplition efine on it, so tht prouts of things in the set re lso in the set, suh tht the multiplition is ssoitive, it hs some ientity (like e) n eh element hs n inverse (eh x hs y so tht xy = e). 1/0 x 0/1 e This utomton hs two sttes lelle e n x. If we strt t the stte x n re the string 010, we follow the ege lelle 0/1, repling the first letter of the string y 1, then from the stte e we follow the ege, repling the seon letter 1 y 1, then (sine we re still in the stte e) we follow n keep the thir letter s 0. So the ege lel tells us how to rewrite the next letter of the string, n the stte tells us wht to o with the suffix of the string. If we strt t the stte e n re string, the string stys the sme. Exerise 2.1. Drw the portrit of the utomorphism x in this exmple. 2 Exerise 2.2. Drw the utomton enoing the rules for n in the previous setion. Exerise 2.3. If you know some si group theory, o you reognise the group generte y {x} 3 n the group generte y {, }? 3. Grigorhuk s Group Here is n exmple whih relly kike off the theory of self-similr (or utomton) groups. We strt with the utomton esriing the selfsimilr rules for five tions. 0/1 1/0 e If we strt t stte, we swith the first letter of the string, then move to stte e for the rest of the string. So is the sme tion s esrie t the strt (while isn t, sine it sens 00 to 01). Exerise 3.1. Write the self-similr rules for,,, y reing off the utomton. Here re portrits of, n (the perio 3 pttern keeps going ll the wy own). 2 Solutions for ll the exerises n e foun t 3 Hint: think of x n ting on w s oing something to inry numer (written kwrs). 18 Jnury 2013, Volume 3 No 1

3 Asi Pifi Mthemtis Newsletter Exerise 3.2. Using the portrits, or otherwise, work out wht,, n o. Whih utomorphisms re they the sme s? The Grigorhuk group is the self-similr group generte y,,,. If you i the exerises right, you woul hve foun some reltions etween letters. Eh genertor one twie is e, n is the sme s. It follows from these fts tht ny prout n e written more effiiently y never writing n never putting two letters,, next to eh other (sine =, =, = ). Tht is, every prout in the group n e reue to something of the form x 1 x 2... or x 1 x 2... where x i =, or. Exerise 3.3. Show tht is the sme utomorphism s. A finitely generte group G is si to e finitely presente if finite numer of reltions, sy u 1 = v 1,..., u n = v n where u i n v i re finite prouts of genertors (or inverses of genertors), suffie to show equlity etween ritrry prouts of genertors. Even though we hve foun few reltions tht the genertors,,, for Grigorhuk s group stisfy, like = = = = e, =, =, it is known tht Grigorhuk s group is not finitely presente. 4. Wor Prolem The wor prolem for finitely generte group is the following question: given wor (or finite prout) of genertors, is the prout equl to the ientity element or not? In the se of groups of utomorphisms of T, this is the sme s sking if prout of genertors puts T k in its originl onfigurtion. For exmple, oes nothing to the tree, so the nswer on this input is yes. 4 Here is n lgorithm (given y Grigorhuk) to solve the wor prolem for his group. Write the input wor in the form x 1 x 2... or x 1 x 2... where x i {,, }. Count the numer of letters. If it is o, we know tht the utomorphism it represents swithes the noes 0 n 1, so this wor is not the ientity. So suppose the numer of letters is even. If the wor strts with, write it s (x 1 )x 2 (x 3 )x 4..., n otherwise write it s x 1 (x 2 )x 3 (x 4 ).... Now 4 This follows from Exerise 3.3 sking if u = v in group is the sme s sking if uv 1 = e. we know the wor oes not swith the two noes t level 1. Wht oes it o to the sutree hnging from the noe 0? The suwor () hs the effet of oing wht oes to the sutree (hek this moves the sutree over to the right sie, then ts y swithing own the right sie of the sutree, then puts it k). Similrly we n work out tht () ts like on the sutree, n () ts like. In very similr wy, we n work out wht, n o to the sutree n flip it (so t like ) n oes nothing to it. So to work out wht the input wor oes to the sutree hnging from 0, we rewrite (x 1 )x 2 (x 3 )x 4... or x 1 (x 2 )x 3 (x 4 )... y repling n y, n y e, n,, y,, respetively. It s similr story for the sutree hnging from noe 1. Exerise 4.1. Work out the replement rules for,,,,, for the right sutree. We shoul now e le to see how this will turn into reursive lgorithm given sutree n wor in,,, ting on it, ount the numer of s, n if it is even, see wht hppens to the two sutrees (uner two rewritten (n shorter) wors). A goo exerise is to figure is out the worst-se time (n spe) omplexity of the lgorithm. A vrition of this lgorithm works for lrge lss of self-similr groups; see [6]. In generl, the wor prolem for n ritrry finitely generte group is n uneile prolem there re even finitely presente groups for whih, if we oul eie if given wor is the sme s the ientity, then we oul solve the Hlting prolem for Turing mhines, whih is unsolvle (see for exmple [5] for more etils). 5. Growth Grigorhuk s group is fmous euse it ws the first exmple of group hving intermeite growth. For group G generte y finite set of elements, efine the growth funtion f : N N y f (n) where f (n) is the numer of elements of G tht re equl to some prout of genertors of length t most n; the mximum vlue this funtion oul ttin is exponentil in the numer of genertors, sine there re k n strings of k letters of length n. Milnor ske if group oul hve growth funtion tht is superpolynomil, ut suexponentil (like f (n) = e n ), whih is lle intermeite, n Grigorhuk showe tht his group hs n intermeite growth funtion. Two exellent soures in whih to re ounts of this re [3] n [4]. Jnury 2013, Volume 3 No 1 19

4 Asi Pifi Mthemtis Newsletter Open Question 5.1. Is there finitely presente group of intermeite growth? 6. Shreier Grphs Whenever we hve group G generte y finite set G, ting on set X, n M is some suset of X, there is useful evie lle the Shreier grph, whih is efine s follows. A goo exmple to keep in min while reing this efinition is to tke X s the noes of T, M s the set of noes t some fixe level k of T, n G s self-similr group ting on T. For eh element of M rw noe lelle y this element. Connet noes m i, m j y irete ege lelle s G whenever m j = sm i. Note tht this grph is onnete if for ny two points in M there is lwys some group element (whih n e expresse s finite prout of genertors from G) whih tkes the point m i to m j. If this is stisfie, we sy tht G ts trnsitively on M. For exmple, if G is Grigorhuk s group, X = T, n M is the set of noes t level k, then the tion of G is trnsitive on M sine we n fin omintions of,,, whih move ny point to ny other in this level. Here re the Shreier grphs for Grigorhuk s group ting on levels 2 n 3: ontining ll elements whih fix noe t level k. Then the Shreier grph for G ting on G/H is the sme s the grph for G ting on T when M is the set of noes t level k. 7. Geoesis Eh element of finitely generte group is the prout of some numer of genertors. A prout of genertors is geoesi if there is no shorter prout tht equls the sme element. For exmple is not geoesi in the Grigorhuk group, ut is 5 If G is group with finite generting set G, the geoesi growth funtion for G with respet to G ounts the numer of geoesis of length (t most) n. It is ler tht this funtion is oune elow y the usul growth rte (sine eh element hs t lest one geoesi representing it), n oune ove y n exponentil funtion (sine there re k n strings of k letters of length n). In [1], groups with polynomil geoesi growth funtions re onsiere, n nturl extension to Milnor s question rises: Open Question 7.1. Is there group of intermeite geoesi growth? The first exmple to try is Grigorhuk s group, sine the numer of geoesis of length n is t lest the numer of elements, so its geoesi growth is superpolynomil We n use the Shreier grphs of the Grigorhuk group to fin geoesis s follows. A wor of the form u = x 1 x 2... x n/2 lelling pth strting from the noe lelle n moving right (ening t noe we will ll k) enoes n To get the grph for the next level, we mke two opies of the grph for the previous level, ppen 1 to the noes in one opy n 0 to the other, flip the 0 opy then glue them together. The she lines inite where gluing ours to get level 3. In this wy we see some more self-similrity. Exerise 6.1. Drw the Shreier grph for level 4. More generlly, given ny group G with generting set G, n ny sugroup H, the set of left osets G/H is set on whih G ts trnsitively, so we n form Shreier grphs for G ting on G/H. If H is the trivil sugroup, then the Shreier grph oinies with nother stnr onstrution in group theory: the Cyley grph. If G is self-similr group ting trnsitively on eh level of T, let H e the sugroup of G utomorphism tht sens to k in T. Suppose v is prout of,,, tht lso sens to k. Then v lels pth in the Shreier grph, strting t n ening t k. Sine the Shreier grph esries ll possile wys of moving etween noes t fixe level of the tree, if v oes not trvel in stright line in this grph (like u oes) it hs no hope of sening to k. In other wors, no wor shorter thn u n e the sme group element s u. For eh x i we hve two hoies for,,, so there re 2 n/2 = ( 2) n ifferent wors of length n like this, so the geoesi growth funtion is exponentil. More etils n e foun in [2]. 5 Don t elieve me? Run the wor prolem lgorithm on u 1 for ll wors u of length t most three. 20 Jnury 2013, Volume 3 No 1

5 Asi Pifi Mthemtis Newsletter 8. Exmple: The Bsili Group Here is one more self-similr group. Let B e the group ting on T, generte y two utomorphisms, esrie y these self-similr rules: (0w) = 1(w), (1w) = 0e(w), (0w) = 0(w), (1w) = 1e(w). Exerise 8.1. Drw n utomton (with sttes lelle,, e) whih enoes these rules. Exerise 8.2. Drw the Shreier grph of the tion of B on level 2 of T. Here is the Shreier grph for level 3, whih shoul look like two opies of the Shreier grph for level 2, stuk together y reking some eges n reonneting Exerise 8.3. Drw the Shreier grph of the tion of B for levels 4, 5,... of T. Wht o you see? I hope this short introution might inspire some reers to explore the topi further. Some exellent strting points re [3], [4] n [6]. Aknowlegements I m extremely grteful to Nik Dvis, Slv Grigorhuk, Dim Krvhek, Zorn Šunić n Mike Whittker who hve tught me muh out the sujet. Referenes [1] Brison, M., Burillo, J., Eler, M. n Šunić, Z. (2012). On groups whose geoesi growth is polynomil. Internt. J. Alger Comput. 22 (to pper). [2] Eler, M., Gutiérrez, M. n Šunić, Z. Geoesis in the first Grigorhuk group. In preprtion. [3] Grigorhuk, R. n Pk, I. (2008). Groups of intermeite growth: n introution. Enseign. Mth. 54, [4] Mnn, A. (2012). How Groups Grow (Lonon Mthemtil Soiety Leture Note Series 395). Cmrige University Press. [5] Miller III, C.F. (1992). Deision prolems for groups survey n refletions. In Algorithms n Clssifition in Comintoril Group Theory, es Bumslg, G. n Miller III, C.F. (Mth. Si. Res. Inst. Pul. 23), Springer, New York, pp [6] Nekrshevyh, V. (2005). Self-Similr Groups (Mthemtil Surveys n Monogrphs 117). Amerin Mthemtil Soiety, Proviene, RI. Murry Eler Murry.Eler@newstle.eu.u Murry Eler is n ARC Future Fellow t The University of Newstle. He i PhD t The University of Melourne working with Wlter Neumnn on utomti groups n geometri group theory, then took severl postos n leturing positions in the US n Sotln, efore returning to Austrli in 2008 when he seure temporry position t The University of Queensln. His reserh interests inlue forml lnguges n utomt, geometri group theory, omputtion n experimentl mthemtis, omplexity n omputility, n enumertive omintoris. Currently he hols permnent position t the Shool of Mthemtil n Physil Sienes, University of Newstle, University Drive, Cllghn NSW 2308, Austrli. Jnury 2013, Volume 3 No 1 21