Rewriting Systems and Geometric 3-Manifolds *

Transcription

1 Rewriting Systems and Geometric 3-Manifods * Susan Hermier New Mexico State University Las Cruces, NM 88003, USA Michae Shapiro University of Mebourne Parkvie, Vic 3052, Austraia Abstract: The fundamenta groups of most (conjecturay, a) cosed 3- manifods with uniform geometries have finite compete rewriting systems. The fundamenta groups of a arge cass of amagams of circe bundes aso have finite compete rewriting systems. The genera case remains open. 1. Introduction A finite compete rewriting system for a group is a finite presentation which soves the word probem by giving a procedure for reducing each word down to a norma form. For cosed irreducibe 3-manifods, resuts of [6] show that if the fundamenta group is infinite and has a finite compete rewriting system, then the group has a tame combing, so resuts of [12] show that the manifod has universa cover homeomorphic to R 3. In this paper we point out that we-known properties of finite compete rewriting systems and we-known facts about geometric 3-manifods combine to give the foowing. (See beow for definitions.) Theorem 1. Suppose that M is a cosed 3-manifod bearing one of Thurston s eight geometries. Suppose further that if M is hyperboic, that M virtuay fibers over a circe. Then π 1 (M) has a finite compete rewriting system. According to a conjecture of Thurston ([15], question 18), every cosed hyperboic 3-manifod obeys the ast hypothesis. We aso exhibit a cass of non-uniform geometric 3-manifods whose fundamenta groups have finite compete rewriting systems. In particuar, suppose that M is a graph of circe bundes based on a graph Γ. We wi ca an edge of this graph a oop if it has the * AMS Cassifications 20F32, 68Q42, 57M05. The first author wishes to thank the Nationa Science Foundation for partia support from grant DMS The second author wishes to thank the Austraian Research Counci. 1

2 same initia and termina vertex. We suppose that when a oops are removed, the resuting graph is a tree. Under certain conditions on the way the vertex manifods are gued aong their boundary tori, the fundamenta group π 1 (M) has a finite compete rewriting system. We woud ike to thank Mark Brittenham for many hepfu discussions. 2. Background and Definitions Let G be a group with finite generating set A. We write A for the free monoid on A. Each eement of A evauates into G under the identity map and this extends to a unique monoid homomorphism of A onto G which we denote by w w. A rewriting system R over the set A is a subset of A A. We write a pair (u, v) R as u v and ca this a rewriting rue or repacement rue. If u v is a rewriting rue, then for any xuy A, we write xuy xvy. We say a finite set R = {u i v i } is a finite compete rewriting system for (G, A) if 1) The monoid presentation A u i = v i is a presentation of the underying monoid of G. 2) There is no word w 0 A spawning an infinite sequence of rewritings, w 0 w 1 w 2. Such a system is caed Noetherian. 3) For each eement g G there is exacty one word w A so that g = w and w contains no u i as a substring (that is, w is irreducibe). We wi say that G has a finite compete rewriting system if there is a generating set A for which there is a finite compete rewriting system for (G, A). Given a proposed rewriting system for (G, A), we woud ike to be abe to check that it performs as advertised. Suppose that we know that 1) is satisfied. It turns out that we can often proceed in a mechanica manner. Property 2) can often be checked by giving an ordering on A. It then suffices to show that is a we-founded ordering and that is -decreasing. That is, we woud ike to know that any non-empty subset of A has a -minimum eement and that whenever u v, v u. This can often be done by means of recursive path ordering. Definition 2.1. [1] Let > be a partia we-founded ordering on a set S. The recursive path ordering > rpo on S is defined recursivey from the ordering on S as foows. Given s 1,..., s m, t 1,..., t n S, s 1...s m > rpo t 1...t n if and ony if one of the foowing hods. 1) s 1 = t 1 and s 2...s m > rpo t 2...t n. 2) s 1 > t 1 and s 1...s m > rpo t 2...t n. 3) s 2...s m rpo t 1...t n. The recursion is started from the ordering > on S. Recursive path ordering is a we-founded ordering which is compatibe with concatenation of words [1]. Thus, in order to check that a rewriting system R = {u i v i } is Noetherian, it suffices to order the generators so that u i > rpo v i for each u i v i R. The critica pair anaysis of the Knuth-Bendix agorithm [10] is a computationa procedure for checking that a Noetherian rewriting system is compete. This agorithm checks for overapping pairs of rues either of the form r 1 r 2 s, r 2 r 3 t R with r 2 1, or of the form r 1 r 2 r 3 s, r 2 t R, where each r i A ; these are caed critica pairs. In 2

3 the first case, the word r 1 r 2 r 3 rewrites to both sr 3 and r 1 t; in the second, it rewrites to both s and r 1 tr 3. If there is a word z A so that sr 3 and r 1 t both rewrite to z in a finite number of steps in the first case, or so that s and r 1 tr 3 both rewrite to z in the second case, then the critica pair is said to be resoved. The Knuth-Bendix agorithm checks that a of the critica pairs of the system are resoved; if this is the case, then the rewriting system is compete. The existence of software packages impementing these procedures makes experimentation and exporation much ess painfu. In the course of our work we have used Rewrite Rue Laboratory [9], a software package for performing the Knuth-Bendix agorithm, as an aid in exporation. 3. Proof of Theorem 1 We wi need the foowing facts about finite compete rewriting systems. Proposition 3.0. The trivia group has a finite compete rewriting system. Proposition 3.1. Z has a finite compete rewriting system. Proposition 3.2. [5],[11] If G is a surface group, then G has a finite compete rewriting system. Proposition 3.3. [3] If H is finite index in G and H has a finite compete rewriting system, then G has a finite compete rewriting system. Proposition 3.4. [2] If 1 K G Q 1 is a short exact sequence, and K and Q have finite compete rewriting systems, then G has a finite compete rewriting system. A thorough account of Thurston s eight geometries is given in [14]. If M is a Riemannian manifod, then the Riemannian metric on M ifts to a Riemannian metric on the universa cover, M. Suppose now that M is a cosed 3-manifod with a uniform Riemannian metric. (This means that the isometry group of M acts transitivey.) Thurston has shown that up to scaing, there are ony eight possibiities for the Riemannian manifod M and that π 1 (M) is constrained in the foowing way. (We say that G is virtuay H if G contains a finite index copy of H.) Proposition 3.5. Suppose M is a cosed Riemannian 3-manifod with a uniform metric. Then one of the foowing hods: 1) M is the 3-sphere and π 1 (M) is finite, i.e., virtuay trivia. 2) M is Eucidean 3-space and π 1 (M) is virtuay Z 3. 3) M is S 2 R and π 1 (M) is virtuay Z. 4) M is H 2 R and π 1 (M) is virtuay H Z, where H is the fundamenta group of a cosed hyperboic surface. 5) M is Ni, the Lie group consisting of upper trianguar rea 3 3 matrices with one s on the diagona, and π 1 (M) contains a finite index subgroup G which sits in a short exact sequence 1 Z G Z

4 6) M is PSL2 (R), the universa cover of the unit tangent bunde of the hyperboic pane, and π 1 (M) contains a finite index subgroup G which sits in a short exact sequence 1 Z G H 1, where H is the fundamenta group of a cosed hyperboic surface. 7) M is So, a Lie group which is a semi-direct product of R 2 with R, and π 1 (M) has a finite index group G which sits in a short exact sequence 1 Z 2 G Z 1. 8) M is hyperboic space. Under the further assumption that M virtuay fibers over a circe, π 1 (M) has a finite index group G which sits in a short exact sequence 1 H G Z 1, where H is the fundamenta group of a cosed hyperboic surface. The proof of Theorem 1 now consists of appying Propositions to the cases of Proposition Non-uniform geometric 3-manifods For an arbitrary cosed 3-manifod M satisfying Thurston s geometrization conjecture (see [14] for detais), but not necessariy admitting a uniform Riemannian metric, finding rewriting systems becomes much more compicated. If M is not orientabe, then M has an orientabe doube cover; Proposition 3.3 then says that if the fundamenta group of the cover has a finite compete rewriting system, then so does M. So we may assume M is orientabe. Any cosed orientabe 3-manifod M can be decomposed as a connected sum M = M 1 #M 2 # #M n in which each M i is either a cosed irreducibe 3-manifod, or is homeomorphic to S 2 S 1 [4]. The fundamenta group of M, then, can be written as the free product π 1 (M) = π 1 (M 1 ) π 1 (M 2 ) π 1 (M n ). Another resut of [2] says that the cass of groups with finite compete rewriting systems is cosed under free products; therefore, if π 1 (M i ) has a finite compete rewriting system for each i, then so does π 1 (M). So we may assume that our cosed orientabe 3-manifod is aso irreducibe. Resuts of [7] and [8] state that a cosed irreducibe 3-manifod M can aso be decomposed in a canonica way. (Much of this has been greaty simpified by [13].) There is a finite graph Γ associated to M. For each vertex of v of Γ, there is a compact 3-manifod M v M. For each edge e of Γ there is an incompressibe torus Te 2 M. The boundary of M v consists of v e T e 2 and M is the union aong these boundary tori of the pieces M v. Consequenty, the fundamenta group of M can aso be reaized as the group of the graph of groups given by pacing the fundamenta groups of the vertex manifods at the corresponding vertices of Γ, and the fundamenta group of a torus on each edge, together with the appropriate injections. If M satisfies Thurston s geometrization conjecture, then the interior of each one of these vertex manifods admits a uniform Riemannian metric. The simpest case of this type of decomposition occurs when the graph Γ consists of a singe vertex with no edges; this is deat with in Theorem 1. In the case where Γ has edges, the vertex manifods are either cusped hyperboic 3-manifods or Seifert fibered manifods with boundaries. In Section 5 we begin our pursuit of rewriting systems for non-uniform 3-manifods. In this case our resuts depend on assuming that the vertex manifods are circe bundes, that the graph obeys some moderate assumptions and that the guings are of a particuar 4

5 type. In spite of these assumptions, the cass of manifods for which we can produce finite compete rewriting systems seems quite arge. Its simpest cases are those in which Γ consists of a singe edge with two vertices or one vertex. In these cases, π 1 (M) is a free product with amagamation or an HNN extension. We begin with the case of a free product with amagamation since this case iustrates the assumptions in our more genera resut. 5. Guing two circe bundes In this section we dea with the case in which M can be constructed by guing two circe bundes together aong a torus boundary. Suppose Γ is a graph with two vertices v and w and a singe edge between them. The manifod M v attached to the vertex v is formed by taking the product of a surface of genus g that has a singe boundary component, with a circe. The fundamenta group of the surface can be presented as a 1,..., a g, b 1,..., b g, p p = g [a i, b i ] = a 1,..., a g, b 1,..., b g, and the fundamenta group of the circe as x. Then the fundamenta group A of M v is A = π 1 (M v ) = a 1,..., a g, b 1,..., b g x. The manifod M w is aso the product of a surface of genus h that has a singe boundary component, with a circe, so we can write C = π 1 (M w ) = c 1,..., c h, d 1,..., d h y. Fig. 1. The manifods M v and M w In each of these manifods the fiber is represented schematicay by a oop. The product of these oops with the surface boundaries gives each of these manifods a torus boundary. These torus boundaries are gued together producing a singe torus in the amagam. 5

6 M v and M w are circe bundes, each with a singe torus boundary component, and M is formed by guing aong these tori. The fundamenta group G of M is then the free product with amagamation of the fundamenta groups of M v and M w, where the amagamating subgroup is the fundamenta group of the torus. So G = π 1 (M) = A X C, where the subgroup X = Z 2 is identified with the subgroups g [a i, b i ] x and h [c i, d i ] y. With this notation in hand, we see that each guing is described by h x = [c i, d i ] k 1 y n 1 and g h [a i, b i ] = [c i, d i ] k 2 y n 2, where the matrix ( ) k1 n φ = 1 SL k 2 n 2 (Z). 2 If the guing aong X gues fibers of M v to fibers of M w, then M is itsef a Seifert fibered manifod and consequenty has a uniform geometry. The simpest nontrivia guing is given by guing the fiber of M v to the surface boundary component of M w and vice versa, giving h g x = [c i, d i ] and [a i, b i ] = y. In this case φ is the identity matrix. We have been abe to find finite compete rewriting systems for a one parameter famiy of guings, namey those where so that x = φ = ( ) 1 n, 0 1 h [c i, d i ]y n and g [a i, b i ] = y. We wi ca guings of this form shear guings. For the case in which the guing is a shear guing, the fundamenta group of the resuting manifod M has a presentation given by G = π 1 (M) = a 1,..., a g, b 1,..., b g, c 1,..., c h, d 1,..., d h, x, y [a i, x] = [b i, x] = [c j, y] = [d j, y] = 1 (for a i, j) h g x = [c i, d i ]y n, [a i, b i ] = y. 6

7 Theorem 2. If the manifod M can be decomposed into two vertex manifods, such that each vertex manifod is the product of a surface with boundary with a circe, and such that these are connected with a shear guing, then π 1 (M) has a finite compete rewriting system. Proof. We wi utiize the presentation deveoped above for the fundamenta group of M. The generating set for G = π 1 (M) wi be S S 1 where S = {a 1,..., a g, b 1,..., b g, x, c 1,..., c h, d 1,..., d h, y}. The set R of rewriting rues is given by: inverse canceation reators: {ss 1 1 s 1 s 1 s S} bue vertex reators: {x ±1 a ±1 i a ±1 i x ±1, x ±1 b ±1 i b ±1 i x ±1 }, i }, red vertex reators: {c ±1 i y ±1 y ±1 c ±1 i, d ±1 i y ±1 y ±1 d ±1 edge reators: {x ±1 y ±1 y ±1 x ±1 } v amagam reators: {a 1 b 1 yp 1 b 1 a 1, a 1 b 1 1 b 1 1 py 1 a 1, a 1 1 yp 1 b 1 b 1 a 1 1, a 1 1 b 1 1 b 1 1 a 1 w amagam reators: 1 yp 1 }, {c 1 d 1 y n xq 1 d 1 c 1, c 1 d 1 1 y n d 1 1 qx 1 c 1, c 1 1 xq 1 d 1 y n d 1 c 1 1, c 1 1 d 1 1 y n d 1 1 c 1 Here we use the etter p to denote the string of etters so p 1 denotes Simiary the etter q denotes the string and q 1 denotes 1 xq 1 } a 2 b 2 a 1 2 b a gb g a 1 g b 1 g, b g a g b 1 g a 1 g...b 2a 2 b 1 2 a 1 2. c 2 d 2 c 1 2 d c hd h c 1 h d 1 h, d h c h d 1 h c 1 h...d 2c 2 d 1 2 c 1 2. The discussion above Theorem 2 shows that these rewriting rues are a presentation of G, so this is a rewriting system for G. In order to show that this system is Noetherian, we wi show that there is a wefounded ordering on the words in (S S 1 ) so that whenever a word is rewritten, the resuting word is smaer with respect to this order. It suffices to take the appropriate order on S and use recursive path ordering. The foowing emma is proved by inspection of the rues in R. Lemma 5.1. Let > be the recursive path ordering on (S S 1 ) induced by a 1 1 > a 1 > b 1 1 > b 1 > > a 1 g > a g > b 1 g > b g > x > c 1 1 > c 1 > d 1 1 > d 1 > > c 1 h > c h > d 1 h > d h > y. Then for each of the rues u v in R, we have u > v. It foows from Lemma 5.1 that this system is Noetherian. In order to show that the system is aso compete, it suffices to show that in the monoid presented by R, for each 7

8 eement m in this monoid, there is exacty one word in (S S 1 ) representing m that cannot be rewritten. We have used the critica pair anaysis of the Knuth-Bendix procedure to check that the rewriting system R is compete. Rather than give the detais of this computation, we wi give a description of the norma forms that these rewriting rues produce. To understand these norma forms, we consider severa subanguages. Let L(A/X) be the set of irreducibe words on {a i, b i, y} ±1 which do not end in y ±1. Simiary, we et L(X\C) be the set of irreducibe words on {c i, d i, x} ±1 which do not begin in x ±1. We take L(X) = {y m x n m, n Z}. Lemma ) L(A/X) bijects to A/X. 2) L(X\C) bijects to X\C. 3) L(X) bijects to X. Proof. Ceary L(A/X) surjects to A/X, for the set of reduced words on these etters surjects to A, and deeting any traiing y ±1 does not change the coset. Thus, to prove 1) we must show that if u, u L(A/X) with ux = u X then u = u. Here u and u both evauate into the free group on {a i, b i }, so in this case we have uy m = u for some m. Observe that no rewriting rue has a eft hand side consisting of etters of {a i, b i, y} ±1 and ending in y ±1 (other than free reduction). Thus, if u is irreducibe, then so is uy m for any m. If m 0 we have two distinct irreducibe words representing the same group eement. However, it is not hard to carry out the Knuth-Bendix procedure on the set of rues evauating into A. This ensures that for any eement of A there is a unique irreducibe word and thus u and u are identica as required. The proof of 2) is simiar. Once again it is easy to see that L(X\C) surjects to X\C. Now we suppose that Xw = Xw with w and w in L(X\C). We then have y m x n w = w for some m and n. Once again, these are both irreducibe as there is no rewriting rue beginning in y m x n that can be appied. Appeaing to the Knuth-Bendix procedure in C forces m = n = 0 and thus w = w as required. The proof of 3) is immediate. Now observe that any irreducibe word θ has the form θ = u 1 v 1 w 1...u k v k w k where For each i, v i = y m i x n i L(X). For each i, u i is a maxima subword ying in L(A/X); that is, u i is a subword of θ ying in L(A/X), and there is no arger subword of θ containing u i that is aso contained in L(A/X). For each i, w i is a maxima subword ying in L(X\C). The maximaity of each u i ensures that no v i consists soey of y ±1 s directy preceding v i+1. Likewise the maximaity of each w i ensures that no v i consists soey of x ±1 s directy foowing v i 1. We ca k the ength of θ. Let L k be the set of a irreducibe words of ength k. For each g G the AC-ength of g is the minima k such that g (AC) k. Let G k be 8

9 the set of a group eements of AC-ength k. That is, g G k if k is minima such that g = A 1 C 1...A k C k with A i A, C i C. Lemma 5.3. L k bijects to G k. Proof. This is an induction on k. When k = 0, the ony irreducibe word of ength 0 is the empty word, and the ony eement of G with AC-ength 0 is the trivia eement, so there is a bijection between L 0 and G 0. We next check the case k = 1. The set L 1 surjects to G 1, since any eement of A has the form u 1 v 1 and each eement of C has the form v 1 w 1. Mutipying these together and appying the repacement rues produces a word of the form u 1 v 1 w 1 as required. We now check that the map from L 1 to G 1 is injective. Suppose g G 1 and g = u 1 v 1 w 1. Notice that A/X bijects to AC/C. Thus g determines a coset gc in AC/C and thus a unique eement of A/X. Consequenty, g determines u 1. On the other hand, having determined the coset representative u 1 of gc in AC/C, there is a unique c C so that g = u 1 c and this, in turn, determines v 1 w 1. We now assume by induction that L k bijects to G k and check that L k+1 bijects to G k+1. First check that L k+1 surjects to G k+1. Suppose g G k+1. Then g has the form g k h with g k G k, h G 1. We represent g k by a word of L k and h by a word of L 1. We concatenate these words and appy our rewriting rues. The resuting word θ ies in k+1 L i. Since k L i misses G k+1, it foows that θ L k+1. We must show that L k+1 injects to G k+1. Notice that g k and h are determined by g up to an eement of X. Thus, if g = g k h = g k h then there is z X so that g k = g kz and h = z 1 h. Suppose then that g is represented by two irreducibe words θ = u 1 v 1 w 1...u k v k w k u k+1 v k+1 w k+1 θ = u 1v 1w 1...u kv kw ku k+1v k+1w k+1 We take g k and g k to be the group eements represented by the L k portions of θ and θ. Thus h and h are represented by the remaining portions of these two words. Notice that if w k ends in x ±1 then u k+1 and v k+1 are both empty, for otherwise, the fina x ±1 s of w k woud have moved right through u k+1 and any y ±1 s of v k+1. This cannot happen, since each w i was chosen to be maxima. In the same manner, we do not have w k empty and v k ending in x ±1. The same argument appies to θ. Suppose the eement z X is represented by the word y m x n L(X). Then if the word u 1 v 1 w 1...u k v k w k y m x n is reduced using the rues of the rewriting system, for the resuting irreducibe word u 1 v 1 w 1...u k 1 v k 1 w k 1 ũ k ṽ k w k, we have that either w k = w k x n or w k = w k is empty and ṽ k = v k x n. Now this irreducibe word represents the same eement of G k as u 1 v 1 w 1...u k v k w k. Therefore our induction hypothesis says that ṽ k = v k and w k = w k. So either w k = w k x n = w k or ese w k and w k are both empty and ṽ k = v k x n = v k. Since w k cannot end with x±1 and if w k is empty then v k cannot end with x ±1, this shows that n must be zero. Consequenty, u 1 v 1 w 1...u k v k w k and u 1 v 1 w 1...u k v k w k differ by at most a power of y in G; that is, we have z = ym. On the other hand, if u k+1 begins in y ±1, we must have v k and w k empty, for any eading y ±1 s of w k+1 woud have had to move eft through w k and any x ±1 s of v k. Maximaity of the words u i does not aow this to happen. Aso, we cannot have u k+1 9

10 is empty and v k+1 beginning with y ±1. It foows by a simiar argument, then, that u k+1 v k+1 w k+1 and u k+1 v k+1 w k+1 differ in G by at most a power of x; that is, z = xn. Since z is now both a power of x and a power of y, that power is painy 0, so g k = g k and h = h. By induction and so θ = θ as required. u 1 v 1 w 1...u k v k w k = u 1v 1w 1...u kv kw k u k+1 v k+1 w k+1 = u k+1 v k+1 w k+1, Since G = G k and the anguage of irreducibe words is L k it foows that the anguage of irreducibe words is a norma form which bijects to G. This competes the proof of Theorem A graph of circe bundes It is natura to ask whether the rewriting system in Section 5 can be modified to give rewriting systems for HNN extensions and arger graphs of circe bundes. In this Section we offer such rewriting systems. More specificay, suppose that Γ is a finite graph, and suppose that at any vertex v the vertex manifod M v is a circe bunde over a surface with boundary of genus g v. Then M v wi have a torus boundary component for each boundary component in the base surface; the edges of the graph Γ determine how these boundary components wi be gued together. We impose the foowing restriction on the graph Γ. Reca that a oop is an edge with the same initia and termina vertex. We assume that if a of the oops in the graph Γ are removed, the resuting graph is a tree. It now foows the vertices of Γ can be coored aternatey red and bue, so that each edge that is not a oop joins a bue vertex to a red vertex. We use this to impose restrictions on the guings. We orient a of the nonoop edges in Γ by taking the bue vertex to be the initia vertex, so the red vertex is the termina vertex. Let V be the set of vertices of Γ, et E be the set of edges in Γ that are not oops, and et L be the set of oops in Γ. For each edge e E, ι(e) wi denote the initia (bue) vertex of e, and τ(e) wi denote the termina (red) vertex. For each edge in L, ι() = τ() and this may be either red or bue. Suppose that v is a bue vertex. The vertex manifod M v is the product of a surface with boundary of genus g v with a circe, as before. For each edge e E with ι(e) = v, there is one boundary component in the surface. Since v is bue, there are no edges ending at v. For each oop with ι() = v, there are two corresponding boundary components in the surface. The fundamenta group of the surface can therefore be written as a v1,..., a vgv, b v1,..., b vgv, {p e e E, ι(e) = v}, {r, s L, ι() = v} 10 ι(e)=v p e ι()=v r s = g v j=1 [a vj, b vj ].

11 Fig. 2. A graph Γ satisfying our restrictions The vertices are of two types, red and bue. After deeting oops, we are eft with a tree. Each edge of this tree is directed from a bue vertex to a red vertex. Then the fundamenta group of the manifod is π 1 (M v ) = a v1,..., a vgv, b v1,..., b vgv, {p e e E, ι(e) = v}, {r, s L, ι() = v} g v p e r s = [a vj, b vj ] x v. ι(e)=v ι()=v Suppose w is a red vertex; the vertex manifod M w is constructed simiary as the product of a punctured surface of genus g w with a circe. There are no edges that start at w, but each edge e E with τ(e) = w corresponds to a singe puncture in the surface, and each oop at w contributes two punctures. In this case we can present the fundamenta group of M w by π 1 (M w ) = a w1,..., a wgw, b w1,..., b wgw, {q e e E, τ(e) = w}, {r, s L, ι() = w} τ(e)=w q e ι()=w r s = g w j=1 j=1 [a wj, b wj ] x w. Suppose the edge e E has initia vertex v (so v is bue) and termina vertex w (so w is red). Then the manifods M v and M w are gued aong the tori given by the product of corresponding punctures with the circe. On the eve of fundamenta groups, this gives an amagamation with reations x v = q k e e xn e w and p e = q k e e x n e w 11

12 Fig. 3. A graph manifod with a singe vertex manifod M v The surface has two boundary components. The fiber is represented schematicay by a oop. The product of the oop with the surface s two boundary components gives M v two torus boundary components. These torus boundary components are gued together producing a singe torus in the graph manifod. where the matrix ( ) ke n φ = e SL 2 (Z). As before, we wi ca this a shear guing if the matrix is of the form ( ) 1 ne φ = ; 0 1 our reations in this case are k e n e x v = q e x n e w and p e = x w. For each red vertex w, define the number n w to be the sum over a the edges e E, with target τ(e) = w, of the numbers n e. If the oop has initia and termina vertex z (z can be either red or bue), then a generator t is added to form the presentation for M, aong with reations t x z t 1 = s k xm z and t r t 1 = s k x m z where, again, the matrix ( ) k m φ = SL 2 (Z). k m 12

13 As before, in order to find a finite compete rewriting system, we assume this matrix is ( ) 1 m φ = ; 0 1 our reations in this case are t x z t 1 = s x m z and t r t 1 = x z, and we aso refer to these as shear guings. The fundamenta group for the manifod M, then, is the free product of the fundamenta groups of the vertex manifods, with the amagamations for each edge, and the HNN extensions for each oop, described above. In order to simpify this presentation for M, for each edge e E, repace the generator p e with the generator x τ(e) in this presentation, and repace the generator q e with the word x ι(e) x n e τ(e). Then the presentation for M, assuming the restriction on Γ and assuming that a of the guings are shear, is given by π 1 (M) = a zj, b zj, x z,r, s, t [a zj, x z ] = [b zj, x z ] = [r, x ι() ] = [s, x ι() ] = [x ι(e), x τ(e) ] = 1, ι(e)=v τ(e)=w t x ι() t 1 x τ(e) ι()=v x ι(e) x n e w r s = τ()=w g v j=1 r s = [a vj, b vj ], g w j=1 = s x m ι(), t r t 1 = x ι(), [a wj, b wj ], where the generators and reations range over a vertices z, a 1 j g z, a bue vertices v, a red vertices w, a edges e in E, and a oops. Theorem 3. Suppose that Γ is a graph for which, when a of the oops in Γ are removed, the resuting graph is a tree. If the manifod M can be decomposed into a graph of circe bundes with graph Γ, such that each vertex manifod is the product of a surface with boundary with a circe, and such that the guing corresponding to each edge and oop is a shear guing, then π 1 (M) has a finite compete rewriting system. Proof. Using the presentation above, the foowing is a rewriting system for the graph of circe bundes M, with aphabet A = S S 1, where S = {a zj, b zj, x z, r, s, t z V, 1 j g z, L}. inverse canceation reators: {ss 1 1, s 1 s 1 s S} bue vertex reators: {x ±1 v a±1 vi a ±1 vi x±1 v, x±1 v b±1 vi b ±1 vi x±1 v, x ±1 ι(k) r±1 k r ±1 k x±1 ι(k), x±1 ι(k) s±1 k s ±1 k x±1 13 ι(k) }

14 red vertex reators: {a ±1 w x ±1 wi x±1 w x ±1 r ±1 x ±1 ι() x±1 ι() r±1, s ±1 x ±1 ι() x±1 ι() s±1 w a ±1 wi, b±1 wi x±1 edge reators: {x ±1 ι(e) x±1 τ(e) x±1 τ(e) x±1 ι(e) } bue amagam reators: 2 {a v1 b v1 Λ v j=g v [b vj, a vj ]b v1 a v1, a v1 b 1 a 1 v1 Λ v v1 b 1 v1 gv j=2 [a vj, b vj ]Λ 1 v a v1, 2 j=g v [b vj, a vj ]b v1 b v1 a 1 v1, a 1 v1 b 1 v1 b 1 v1 a 1 v1 Λ 2 v j=g v [b vj, a vj ]} red amagam reators: w b ±1 wi, } {a w1 b w1 x n w 2 w Ω w j=g w [b wj, a wj ]b w1 a w1, a w1 b 1 w1 xn w w b 1 gw w1 j=2 [a wj, b wj ]Ω 1 w a w1, a 1 w1 Ω 2 w j=g w [b wj, a wj ]b w1 x n w w b w1 a 1 a 1 w1 b 1 w1 x n w w w1, b 1 w1 a 1 w1 Ω w 2 j=g w [b wj, a wj ]} bue HNN reators: {x ι(k) t k t k r k, x 1 ι(k) t k t k r 1 k, r k t 1 k t 1 k x ι(k), r 1 k t 1 k t 1 k x 1 ι(k) s k t k t k r m k k x ι(k), s 1 x ι(k) t 1 k t 1 k s kx m k ι(k), k t k t k r m k k x 1 ι(k) t 1 k and red HNN reators: {t r x ι() t, t r 1 x 1 ι() t, t 1 x ι() r t 1, t 1 x 1 t x ι() x m ι() s t, t 1 s x ι() r m x 1 ι(k), t 1 k s 1 k x m k ι(k) }, ι() r 1 t 1, t x 1 ι() x m ι() s 1 t, x 1 t 1, t 1 s 1 ι() rm t 1 } These rues range over a bue vertices v, red vertices w, and edges e E, as we as a oops k at bue vertices and a oops at red vertices. The etter Λ v denotes the string of etters Λ v = x τ(e) r k s k. ι(e)=v ι(k)=v Λ 1 v, then, denotes the forma inverse of Λ v, taking the etters in the string Λ v in the opposite order with their signs changed. The etter Ω w denotes the string of etters and Ω 1 w is its forma inverse. Ω w = τ(e)=w x ι(e) ι()=w Denote this set of rues to be R; the discussion preceding Theorem 3 shows that the generators A together with our rewriting rues R give a presentation for the fundamenta group of the graph of circe bundes. 14 r s,

15 In order to show that this rewriting system is compete, we wi first show that a subset of the rues give rise to a compete rewriting system. Let A = A {t ±1 k k L, ι(k) is bue}, and define R to be the rewriting system consisting of a of the rues above except the bue HNN reators and the inverse canceation reators invoving the etters of A A. In order to show that this system R is Noetherian, we wi again show that these rues decrease the we-founded recursive path ordering. The foowing emma is proved by inspection of the rues in the set R. Lemma 6.1. Let > be the recursive path ordering induced by t 1 > t > a 1 w1 > a w1 > b 1 w1 > b w1 > a 1 w2 > > b wg w > x 1 v > r 1 > r > s 1 > s > x v > a 1 v1 > a v1 > b 1 v1 > b v1 > a 1 v2 > > b vg v > r 1 k > r k > s 1 k > s k > x 1 w > x w, where v is any bue vertex, w is any red vertex, k is any oop at a bue vertex, and is any oop at a red vertex. Then for each of the rues u v in R, we have u > v. It foows from the Lemma that this system R is Noetherian. In order to show that R is aso compete, the remaining property to check is that in the monoid presented by (A, R ), for each eement m in this monoid, there is exacty one word in A representing m that cannot be rewritten. This proof has been done in Section 5 when Γis a graph with a singe edge; for more compicated graphs, this becomes much more difficut. For the rewriting system R, we have checked that R is compete using the Knuth-Bendix agorithm [10]. Since the rewriting system R is compete, for each word u A, there is a bound on the engths of a sequences of rewritings u w 1 w n (where the ength of this sequence is defined to be n). The maximum of the engths of a of the possibe rewritings of u is caed the disorder of u, denoted d R (u). We wi use these numbers in order to show that the arger rewriting system R is Noetherian. In order to define a we-founded ordering on the set A, note that every word w A can be written uniquey in the form w = u 1 t 1 u 2 t 2 u j t j u j+1, where each u i is a (possiby empty) word in A and each t i is a etter in (A A ) (A A ) 1. Define functions ψ i from A to the nonnegative integers by ψ 0 (w) =j, ψ 2i (w) =d R (u i ), and ψ 2i+1 (w) =ength(u i ), where i ranges from 1 to j + 1, and ength denotes the word ength over A. In order to compare words of different ength, define ψ i (w) = 0 if i > 2j + 3. For two words w 1 and 15

16 w 2 in A, define w 1 > w 2 if ψ 0 (w 1 ) > ψ 0 (w 2 ) or if ψ i (w 1 ) = ψ i (w 2 ) for a i < k and ψ k (w 1 ) > ψ k (w 2 ). We caim this defines a we-founded ordering on A. To check the caim, suppose w A. If a rue in R is appied to w, the rue must to be appied to one of the subwords u i, so the vaue of ψ 2i is reduced without atering the vaues of ψ k for any 0 k 2i 1. Suppose an inverse canceation reator invoving the etters of A A is appied to w; in this case, the vaue of ψ 0 is reduced. Finay, if a bue HNN reator is appied to w, the rue must be appied to a subword u i t i of w. Then the vaues of ψ k for any 0 k 2i 1 are not atered; the vaue of ψ 2i either decreases or remains unchanged; and the vaue of ψ 2i+1 is reduced. So each time a word is rewritten, the resuting word is smaer with respect to this ordering. Therefore the rewriting system R is aso Noetherian. Since R is Noetherian, we have again appied the Knuth-Bendix procedure to check that the rewriting system R is compete. The dedicated reader may wish to check this; our computation resoved 84 critica pairs. 7. A question When the guings of the circe bundes at the vertices of Γ are more compicated, or when the circe bundes themseves are repaced by more genera Seifert-fibered spaces, we were unabe to find finite compete rewriting systems. So we end with the foowing. Question. Does every fundamenta group of a cosed 3-manifod satisfying Thurston s geometrization conjecture have a finite compete rewriting system? References [1] N. Dershowitz, Orderings for term rewriting systems, Theoret. Comput. Sci. 17 (1982), [2] J.R.J. Groves and G.C. Smith, Rewriting systems and soube groups, preprint, University of Mebourne, [3] J.R.J. Groves and G.C. Smith, Soube groups with a finite rewriting system, Proc. Edinburgh Math. Soc. 36 (1993), [4] J. Hempe, 3-manifods, Ann. Math. Studies 86, Princeton University Press, Princeton, [5] S. Hermier, Rewriting systems for Coxeter groups, J. Pure App. Agebra 92 (1994), [6] S. Hermier and J. Meier, Tame combings, amost convexity, and rewriting systems for groups, Math. Z., 225 (1997) [7] W. Jaco and P.B. Shaen, Seifert fibered spaces in 3-manifods, Mem. Amer. Math. Soc. 220, American Mathematica Society, Providence, [8] K. Johannson, Homotopy equivaence of 3-manifods with boundary, Lecture Notes Math. 761, Springer, Berin, [9] D. Kapur and H. Zhang, An overview of Rewrite Rue Laboratory (RRL), in Rewriting Techniques and Appications (N. Dershowitz, Ed.), pp , Lecture Notes in Computer Science 355, Springer, Berin,

17 [10] D.E. Knuth and P. Bendix, Simpe word probems in universa agebras, in Computationa Probems in Abstract Agebra (J. Leech, Ed.), pp , Pergamon Press, New York, [11] P. Le Chenadec, Canonica forms in finitey presented agebras, Pitman, London; John Wiey & Sons, New York, [12] M.L. Mihaik and S.T. Tschantz, Tame combings of groups, Trans. Amer. Math. Soc., 39 (1997) [13] W.D. Neumann and G.A. Swarup, Canonica decompositions of 3-manifods, Geom. Topo. 1 (1997) [14] P. Scott, The geometries of three-manifods, Bu. London Math. Soc. 15 (1983), [15] W.P. Thurston, Three dimensiona manifods, Keinian groups and hyperboic geometry, Bu. Amer. Math. Soc. 6 (1982),