Tame combings, almost convexity and rewriting systems for groups

Transcription

1 Tame combins, almost convexity and rewritin systems for roups Susan M. Hermiller* Department of Matematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA Jon Meier Department of Matematics, Lafayette Collee, Easton, PA 18042, USA Abstract: A finite complete rewritin system for a roup is a finite presentation wic ives an aloritmic solution to te word problem. Finite complete rewritin systems ave proven to be useful objects in eometric roup teory, yet little is known about te eometry of roups admittin suc rewritin systems. We sow tat a roup G wit a finite complete rewritin system admits a tame 1-combin; it follows (by work of Mialik and Tscantz) tat if G is an infinite fundamental roup of a closed irreducible 3-manifold M, ten te universal cover of M is R 3. We also establis tat a roup admittin a eodesic rewritin system is almost convex in te sense of Cannon, and tat almost convex roups are tame 1-combable. Keywords: Rewritin systems, tame combins, coverin conjecture, almost convex roups, isoperimetric inequalities. * Researc supported in part by NSF rants DMS at MSRI and INT at te University of Melbourne 1

2 Tame combins, almost convexity and rewritin systems for roups 1 Introduction Several properties for finitely presented roups ave been defined wic can be used to sow tat a closed P 2 -irreducible tree-manifold as universal cover omeomorpic to R 3. For example, work of Poénaru [P] sows tat if te fundamental roup is infinite and satisfies Cannon s almost convexity property, ten te universal cover is simply connected at infinity, and ence is R 3 (see [B-T]). Casson later discovered te property C 2, wic rerettably is presentation dependent, but wic also implies tat te universal cover is R 3 [S-G]. Brick and Mialik eneralized te condition C 2 to te quasi-simply-filtered condition [B- M], wic is independent of presentation and also implies te coverin property. Later Mialik and Tscantz [M-T] defined te notion of a tame 1-combin for a finitely presented roup, wic implies te quasi-simply-filtered condition and sowed tat asyncronously automatic roups and semiyperbolic roups are tame 1-combable. Usin a fairly eometric arument we sow tat roups wit finite complete rewritin systems ave tame 1-combins. Usin similar tecniques, we obtain oter eometric properties for roups wit finite complete rewritin systems satisfyin more restrictive conditions. In particular, a roup wit a eodesic finite complete rewritin system is almost convex. To finis tis circle of ideas we also sow tat a roup wic is almost convex for some eneratin set admits a tame 1-combin. Tese results establis te followin implications: GF = F AC = T were GF and F denote te cateory of roups admittin eodesic finite complete rewritin systems and roups admittin finite complete rewritin systems, respectively; AC is te cateory of almost convex roups; and T is te cateory of tame 1-combable roups. General results about finite complete rewritin systems for fundamental roups of tree-manifolds are quite scarce. However, Hermiller and Sapiro [HS] ave observed te followin teorem concernin tree-manifolds wic admit a eometric structure. (See [Sc] for more information on tese manifolds.) 2

3 Teorem. (Hermiller, Sapiro) Suppose M is a closed tree-manifold carryin one of Turston s eit eometries. In te case tat M is yperbolic, make te furter assumption tat M is finitely covered by a manifold wic fibers over te circle. Ten π 1 (M) as a finite complete rewritin system. (Accordin to folklore, Turston as conjectured tat all closed yperbolic 3- manifolds obey tis additional assumption.) Tis result follows by combinin results in [Sc] wit teorems on closure properties for te class of roups admittin finite complete rewritin systems in [G-S1], [G-S2], alon wit a finite complete rewritin system for surface roups ([L] or [H]). Any closed irreducible tree-manifold satisfyin Turston s eometrization conjecture wit infinite fundamental roup as universal cover R 3. So in tese cases te rewritin systems do not actually ive additional information about te coverin conjecture. However, since tere are computer prorams available wic searc for finite complete rewritin systems for roups, tis tecnique of findin rewritin systems provides a new metod for ceckin te coverin property for any tree-manifold. Neiter orizontal implication in te diaram above can be reversed. For example, solvroups admit finite complete rewritin systems but tey are not almost convex on any set of enerators ([C+]). Terefore solvroups are examples of roups wic ave finite complete rewritin systems, but wic do not ave a eodesic finite complete rewritin system on any eneratin set. (We note tat solvroups are not contained in any of te cateories studied in [M-T]; owever, since tey do ave rewritin systems tey are tame 1-combable.) Many rewritin systems tat ave been constructed for roups are eodesic because te rules are compatible wit a sortlex orderin on words in te enerators of te roup. Rewritin systems ave also been constructed wit rules compatible wit a weitlex orderin. Gersten [G2] as sown tat a roup wic is almost convex wit respect to a finite eneratin set satisfies a linear isodiametric inequality and an exponential isoperimetric inequality. We sow tat tese inequalities are also satisfied by roups wit rewritin systems compatible wit a weitlex orderin. Tus we are able to build anoter diaram of implications. Below SF and WF are te classes of roups admittin sortlex rewritin systems and weitlex rewritin systems, respectively. We use LE to denote te class of roups wit linear isodiametric and exponential isoperimetric inequalities. 3

4 SF = WF AC = LE Gersten [G2] as sown tat solvroups satisfy a linear isodiametric inequality, and results in [B] and [E+] sow tat solvroups satisfy an exponential isoperimetric inequality. So solvroups are examples of roups wic lie in LE but not in AC. Rerettably, it is currently unclear wic oter implications cannot be reversed in tis diaram. Te connections between te two diarams of implications is also unclear. We note tat tere are roups admittin finite complete rewritin systems wic do not lie in LE. Gersten as sown tat te roup x, y, z x y = x 2, y z = y 2 does not satisfy a linear isodiametric inequality nor an exponential isoperimetric inequality. (See [G1] and [G3] for more details.) Gersten also created te followin rewritin system for tis roup. (A capital letter denotes te inverse of te enerator.) xx 1 Xx 1 yy 1 Y y 1 zz 1 Zz 1 xy Xyx xy Y xx XXy yx XY Y XX ZY Y Zy zy Y Y z Zyy yz zy yyz Buildin on Gersten s work, wit elp from Brazil, we ave establised tat tis rewritin system is complete. Consequently, tis roup is an example of a roup wic admits a finite complete rewritin system but does not admit a weitlex rewritin system. In te next section we ive te central definitions and formally state te main teorems. We ten proceed to te proofs of eac teorem. 2 Definitions and statements of teorems Given a finite eneratin set S (wic we will assume is closed under inverses, or symmetric) for a roup G, let C S denote te correspondin Cayley rap. Tere is a natural map from te free monoid on S, S, onto G. A set of normal forms is a section of tis map, tat is, it is a coice of ow to express eac roup element in terms of te enerators. Te correspondin collection of pats in te Cayley rap from te identity out to eac vertex described by tis set of normal forms is often referred to as a combin. 4

5 Suppose G is a finitely presented roup, and let X be te universal cover of te standard 2-complex associated to some finite presentation. Coose a base point ɛ X 0. Te followin more eneral definitions of combins are due to Mialik and Tscantz [M-T]. A 0-combin is essentially a coice of a pat in te 1-skeleton X 1 from ɛ to eac point of X 0. Viewin X 1 as te Cayley rap of te roup G, tis can be tout of as te standard notion of a combin for G. Formally, a 0-combin of X is a omotopy Ψ : X 0 [0, 1] X 1 suc tat Ψ(x, 0) = ɛ and Ψ(x, 1) = x for all x X 0. Te map Ψ is a tame 0-combin if for eac compact set C X tere is a compact set D X suc tat for all x X 0, Ψ 1 (C) ({x} [0, 1]) is contained in a sinle pat component of Ψ 1 (D) ({x} [0, 1]). To define a 1-combin, tese definitions are extended to one dimension ier, ivin a coice of pat in X from ɛ to eac point of X 1, in a continuous fasion. A 1-combin of X is a omotopy Ψ : X 1 [0, 1] X suc tat Ψ(x, 0) = ɛ and Ψ(x, 1) = x for all x X 1, and suc tat Ψ(X 0 [0, 1]) X 1. It follows tat te restriction of Ψ to X 0 [0, 1] is a 0-combin. Te map Ψ is a tame 1-combin if tis restriction is a tame 0-combin, and if for eac compact set C X tere is a compact set D X suc tat for all edes e X 1, Ψ 1 (C) (e [0, 1]) is contained in a sinle pat component of Ψ 1 (D) (e [0, 1]). A finite complete rewritin system for a finitely presented roup G also includes a coice of normal forms for te elements of G. A rewritin system consists of a finite set Σ and a subset R Σ Σ, were Σ is te free monoid on te set Σ. An element (u, v) R, called a rule, is also written u v. In eneral if u v ten for any x, y Σ, we write xuy xvy and say tat te word xuy is rewritten (or reduced) to te word xvy. for some An element x Σ is irreducible if it cannot be rewritten. Te ordered pair (Σ, R) is a rewritin system for a monoid M if Σ u = v if (u, v) R is a presentation for M. A rewritin system for a roup is a rewritin system for te underlyin monoid; tat is, te elements of Σ must be monoid enerators for te roup. A rewritin system (Σ, R) is complete if te followin conditions old. C1) Tere is no infinite cain x x 1 x 2 of rewritins. (In tis case te rewritin system is called Noeterian.) C2) Tere is exactly one irreducible word representin eac element of te monoid presented by te rewritin system. 5

6 Te rewritin system is finite if R is a finite set. Rerettably, unlike tame combins, te existence of a finite complete rewritin system is dependent upon te coice of eneratin system. Because we want to move between te Cayley rap of a roup for a iven set of enerators and rewritin systems based on tose enerators, all of te rewritin systems in tis paper will ave eneratin sets wic are closed under takin formal inverses. Tis often occurs naturally since Σ must be a set of monoid enerators, and in most cases tis is a armless restriction since every roup wic as a finite complete rewritin system on some set of enerators Σ, also as one on Σ = Σ Σ 1, wit extra rules takin enerators in Σ 1 to teir irreducible representatives in Σ from te old rewritin system. For more information on rewritin systems, consult [Co], [H], [L], and te references cited tere. Terminoloy: In order to improve te exposition we often drop te adjectives finite and complete wen referrin to rewritin systems. None te less we are always assumin te rewritin systems are finite complete rewritin systems, and tat tey ave a symmetric set of enerators. Teorem A. A roup wic admits a finite complete rewritin system as a tame 1-combin. In addition to tinkin of te Cayley rap as a topoloical space, we also tink of C S as a metric space wit te word metric, tat is, were eac ede is iven te metric structure of te unit interval. If n is any positive inteer, ten we can define te spere of radius n to be S(n) = { x C S d(ɛ, x) = n } were ɛ is te identity element; similarly, te ball of radius n is B(n) = { x C S d(ɛ, x) n }. A roup G is almost convex [Ca] wit respect to a finite presentation on a symmetric set of enerators if tere is a constant A suc tat for any inteer n and any pair of roup elements, G wit, S(n) and d(, ) 2, tere is a pat in B(n) of lent at most A joinin and. Similarly we can discuss te eometry of pats; in particular, a pat in C S is eodesic if it is a minimal lent pat connectin its endpoints. Te collection of irreducible words of a finite complete rewritin system is a set of eodesic normal forms if and only if none of te rules of te system increase lent; suc a rewritin system is a eodesic rewritin system. Geodesic rewritin systems, and te special case of sortlex rewritin systems discussed below, are te only instances in tis paper were te assumption tat te eneratin set is symmetric mit be restrictive. It is conceivable tat a roup could admit a eodesic (or 6

7 sortlex) rewritin system, but tat no suc rewritin system can be created for a symmetric set of enerators. No examples of suc patoloy are known. In practice one establises te Noeterian condition C1) for a rewritin system by usin a well-founded orderin on te words in Σ and ceckin tat if u v R ten u > v in te orderin. For example, one can establis te Noeterian condition by sowin tat te rewritin rules are compatible wit a sortlex orderin. In te sortlex orderin a total order is placed on te enerators of G; ten a word w is defined to be less tan a word v if w is sorter tan v or tey are of te same lent but w is lexicorapically prior to v. Rewritin systems tat are compatible wit a sortlex orderin are eodesic. Teorem B. If G as a eodesic finite complete rewritin system, ten G is almost convex. In order to complete te first circle of implications mentioned in te introduction we establis te followin result. Teorem C. If G is almost convex, ten G as a tame 1-combin. An alternative metod of establisin te Noeterian property is to use a weitlex orderin. Here eac enerator is assined a positive inteer weit and w is less tan v if te total weit of w is less tan te total weit of v, or if te weits are te same, w is lexicorapically prior to v. Any roup wit a finite complete rewritin system compatible wit a weitlex orderin, as one on a symmetric set of enerators, so aain in tis case our assumption is not restrictive. A finitely presented roup G satisfies a linear isodiametric inequality if tere is a linear function f : N N suc tat for any freely reduced word w representin te trivial element of G, tere is a van Kampen diaram D wit boundary label w and wit te (word metric) distance from any vertex of D to te basepoint at most f(l(w)), were l(w) is te word lent of w. Te roup G satisfies an exponential isoperimetric inequality if tere is an exponential function : N N suc tat for any freely reduced word w representin te trivial element, tere is a van Kampen diaram D for w containin at most (l(w)) 2-cells. Te reader is referred to [L-S] for information on van Kampen diarams and to [G4] for backround on te isodiametric and isoperimetric inequalities. Teorem D. If G admits a finite complete rewritin system compatible wit a weitlex orderin, ten G satisfies linear isodiametric and exponential isoperimetric inequalities. 7

8 3 Proof of Teorem A We bein by sowin tat a rewritin system defines a tame 0-combin for a roup. Lemma 1. Te irreducible words of a finite complete rewritin system for G ive a tame 0-combin for G. Proof. Given a finite complete rewritin system (Σ, R) for G, let X denote te universal cover of te 2-complex associated to te presentation iven by te rewritin system; define a 0-combin usin te rewritin system by takin te pat Ψ(x, t) : [0, 1] X 1 for a iven x X 0 to be te pat followin te irreducible word representin x. A prefix u of an irreducible word w = uv Σ cannot contain te left and side of a rule, so te prefix must also be irreducible. Tis means tat te 0-combin Ψ is prefix closed; tat is, for any x X 0, Ψ(x, t) = Ψ(x, t ) implies t = t, and for any y Ψ({x} (0, 1)) X 0, te combin pat Ψ({y} [0, 1]) is a simply a reparametrization followin te pat Ψ({x} [0, t]) from ɛ to y. In [M-T] it is sown tat te compact sets in te definitions of tame 0- and 1-combins can be replaced wit finite subcomplexes. Given a finite subcomplex C X, let D be te subcomplex D = C ( x C 0Ψ({x} [0, 1])). Ten D is also a finite subcomplex. Prefix closure implies tat for any point x X 0, Ψ 1 (D) ({x} [0, 1]) must be an interval [0, t] for some t. It is ten immediate tat Ψ 1 (C) ({x} [0, 1]) is contained in a sinle pat component of te pullback of te set D. Hence Ψ is tame. In order to be able to extend te 0-combin from Lemma 1 to a 1-combin, we first replace te rewritin system for G by an equivalent minimal one. For any rewritin system (Σ, R), write x y wenever x = y or tere is a finite cain of arrows x x 1 x 2 y. Two complete rewritin systems (Σ, R) and (Σ, R ) are equivalent if Σ = Σ, teir irreducible words are te same, and wenever x z in R, wit z irreducible, ten x z in R, also. Te rewritin system (Σ, R) is minimal if for every (u, v) R, every proper subword of u is irreducible and v is irreducible. Te followin lemma was proven in [Sq]. Lemma 2. (Squier) Given a finite complete rewritin system (Σ, R) for G, tere is an equivalent minimal finite complete rewritin system. Now assume te rewritin system for G is minimal, and let X be te universal cover of te 2-complex associated to tis rewritin system. Te 1-combin Ψ on X 1 [0, 1] will be defined inductively. 8

9 Given an ede e X 1, te endpoints of e correspond to elements, G, were we can write s = for some enerator s Σ. Write te irreducible word representin a roup element as. Put any total orderin on X 0, for example a sortlex orderin on te irreducible words, and suppose > in tis orderin. We can consider e to be te union of two oriented edes f 1 and f 2, were te initial vertices are i(f 1 ) =, i(f 2 ) = and te terminal vertices are t(f 1 ) =, t(f 2 ) =. A omotopy Θ fi : f i [0, 1] X will be defined for bot orientations i = 1, 2. Tese will aree on te endpoints of te edes and will be compatible wit te zero combin previously defined. Tat is, if e is and ede and f 1 and f 2 are te orientations of e, ten Θ fi (x, t) = Ψ(x, t) wenever x is one of te endpoints, X 0 and t [0, 1]. (We need combins on bot orientations in order to mimic te idea of prefix closure used in te proof of Lemma 1.) Te 1-combin Ψ restricted to te ede e is defined to be Θ f1 on te correspondin oriented ede f 1. Te prefix-size ps(f) of an oriented ede f wit i(f) =, t(f) =, and s = in G, is defined to be te number of rewritins needed to rewrite s to, were at eac step te sortest reducible prefix is rewritten. Minimality of R implies tat tere is only one way to rewrite tat prefix in eac case, and completeness of R implies tat te prefix-size of an oriented ede must always be finite. We define Θ f on an oriented ede f by inductin on te prefix-size of f. If ps(f) = 0, ten s =, so f is in te imae of te 0-combin line to ; tat is, (abusin notation, and tinkin of f as representin te correspondin unoriented ede also) f Ψ({} [0, 1]). For any point x f, define te pat Θ f ({x} [0, 1]) to be a reparametrization of te pat from ɛ to, stoppin at x (see Fi. 1). ε s Fi. 1. Te prefix size is zero Now suppose tat ps(f) = 1. Te word s Σ contains a suffix wic is a left and side of a rule; tis rule is uniquely determined because R is minimal and is irreducible. So we can write = wu were us v is a rule in R. If te rule us v is of te form s 1 s 1, ten te unoriented ede correspondin to f is in te imae of te 0-combin, and as before we can define Θ f to be a reparametrization of te 0-combin (see Fi. 2). Oterwise, te relation iven by us = v defines te boundary of a 2-cell in X and f is a face of tis 2-cell. 9

10 Te combin pats and from ɛ to te endpoints of f follow a common pat alon w from ɛ to te 2-cell, and ten follow te boundary of te 2-cell. All of te edes in te boundary of tis 2-cell except f are in te imae of te 0-combin, since tey are on te pats iven by eiter or. Θ f is defined on f [0, 1] to follow w and ten fill te interior of tis cell (see Fi. 3). ε s Fi. 2. Te prefix size is one, first case ε w u v s Fi. 3. Te prefix size is one, second case Finally, suppose tat ps(f) = n, and tat for any oriented ede f wit ps(f ) < n, we ave defined Θ f. As before, te word s Σ contains a suffix wic is a left and side of a rule, and tis rule is uniquely determined. Te map Θ f is defined on f [ 1 2, 1] to fill te 2-cell defined by te application of tis rule. If we write = wu in Σ, were us v is a rule in R, ten te imae Θ f (x, 1 ) for any point x f lies on te pats based at te end of w defined by 2 u or v. Since wu = defines te 0-combin pat Ψ({} [0, 1]), if Θ f (x, 1 2 ) lies on te pat iven by u, ten Θ f ({x} [0, 1 ]) can be defined to follow wu from ɛ 2 to Θ f (x, 1 2 ), by reparametrization (see Fi. 4). On te oter and, suppose Θ f (x, 1 2 ) lies on te pat defined by v based at te end of w. If Θ f (x, 1 2 ) Ψ(X0 [0, 1]), ten aain define Θ f ({x} [0, 1 2 ]) to follow te 0-combin pat from ɛ to Θ f(x, 1 2 ) wit reparametrization. If Θ f (x, 1 2 ) / Ψ(X0 [0, 1]), ten Θ f (x, 1 2 ) must lie on an ede e. Tere is an orientation f of e wit i(f ) =, t(f ) =, and s = for an s Σ, suc tat at some stae in te rewritin of s by rewritin sortest prefixes, te process must rewrite, in some finite number of steps, a word of te form s z to z, for some z Σ. In particular, tis procedure must rewrite te prefix s to by always rewritin sortest possible prefixes. Tis implies tat ps(f ) < ps(f). By induction, ten, te omotopy Θ f on f [0, 1] as already been defined; define Θ f ({x} [0, 1 2 ]) to follow (wit reparametrization) te pat defined by te omotopy Θ f, from ɛ to Θ f (x, 1 2 ) (see Fi. 5). 10

11 s u v w Fi. 4. Definin Θ f on [ 1 2, 2] ε x ' Θ ( ) f x, 1 2 ' s ' ' Θ f ' ' ( f [0,1]) ' Fi. 5. Completin te definition of Θ f by induction ε We now ave omotopies defined for every vertex and for every orientation of every ede in X. Te 1-combin Ψ on X 1 [0, 1] is defined to be te union of 0-combin of Lemma 1 on X 0 [0, 1] and te ede combins, were te combin for an ede e wit endpoints > is te omotopy Θ f on te oriented ede f wit i(f) = and t(f) =. Te ede combins are identified alon te combins of teir boundin vertices. Proposition 1. Te 1-combin Ψ is a tame 1-combin. 11

12 Proof. Suppose C is a finite subcomplex of X. Let D be te finite subcomplex D = C ( x Ψ({x} [0, 1])) ( f Θ f (f [0, 1])), were te union is taken over all points x C 0 and bot orientations f of all edes in C 1. To establis tat Ψ is tame it is sufficient to sow tat once te pat Ψ(y, t) leaves te subcomplex D it does not return to C; tat is, for any y X 1, Ψ 1 (C) ({y} [0, 1]) is contained in te component of Ψ 1 (D) ({y} [0, 1]) containin (y, 0). Assume to te contrary tat tere is some t suc tat Ψ(y, t) C wile tere is some t < t wit Ψ(y, t ) / D. By possibly takin a larer t we can assume furter tat Ψ(y, t) C 1. If Ψ(y, t) is actually a vertex ten, by te construction, Ψ({y} [0, t]) is te combin pat for tat vertex, ence Ψ({y} [0, t]) is contained in D, contradictin te existence of t. Te only oter possibility is tat Ψ(y, t) e for some ede e in C. But in tis case Ψ(y [0, t]) is contained in Θ f1 (f 1 [0, 1]) or Θ f2 (f 2 [0, 1]) were te f i are te two orientations of e. But bot of tese sets are contained in D ence once aain Ψ({y} [0, t]) D. 4 Proof of Teorem B Suppose (Σ, R) is a eodesic finite complete rewritin system for a roup G wit symmetric eneratin set Σ, and suppose tat, G wit, S(n) and d(, ) 2. We will sow tat tere is a pat in B(n) of lent at most A = 2σρ joinin and, were σ is te cardinality of Σ, and ρ is te lent of te lonest relator in te rewritin system. Write for te irreducible word representin, as in te proof of Teorem A. Suppose tat d(, ) = 2. Ten tere is an element G wit d(, ) = 1 = d(, ). If S(n 1), ten tere is a pat of lent 2 in B(n) from to. If S(n), we ave te same situation as te case wen d(, ) = 1, and we will find a pat in B(n) from to of lent at most σρ; ence tere is a pat in B(n) from to of lent at most A. If S(n + 1), let t be te last letter in ; since is a eodesic, t 1 S(n). A simple modification of te proof for te case wen d(, ) = 1 sows tat tere is a pat in B(n) from to t 1 of lent at most σρ (we leave te details to te reader). Repeatin tis construction to find a pat from t 1 to ives te desired bound. Finally, suppose tat d(, ) = 1, and let s 0 Σ be te enerator suc tat s 0 =. Te word s 0 can be expressed as w 0 u 0 s 0 were u 0 s 0 v 0 s 1 R and 12

13 s 1 Σ. Let 1 be te roup element iven by te word w 0 v 0 so tat = 1 s 1 ; since rewritin cannot increase lent, 1 B(n). If 1 represents a vertex in te pat iven by, ten we are done. Oterwise we can repeat tis process by expressin te word 1 s 1 as w 1 u 1 s 1 wit u 1 s 1 v 1 s 2 R and lettin 2 B(n) be te roup element iven by te word w 1 v 1 so tat = 2 s 2 (see Fi. 6). s 0 u 0 w 0 v 0 1 w 1 u 1 2 s v w s u 2 2 Fi. 6. Rewritin to find te almost convexity constant If, in tis process, it ever occurs tat s i = s j for i j ten te process above would outline ow to construct a rewritin from i s i to j s j, ence from a word back to itself. By te Noeterian property tis is not possible. Tus tere can be at most σ steps in tis process, and so i s i = for some i < σ. Te word u 1 0 v 0u 1 1 v 1...u 1 i v i s i ten defines a pat in B(n) from to of lent less tan σρ. ε 5 Proof of Teorem C Given Poénaru s work [P] and te central motivation for te definition of a tame 1-combin, it surprises us tat tis teorem as not yet appeared in te literature. Let G be a finitely presented roup wit finite presentation P wic is almost convex. Let A be te constant suc tat between any two vertices and in S(n) wit d(, ) 2 tere is a pat in B(n) connectin to of lent less tan or equal to A. We will work wit an extended presentation for G, wit te same enerators, but wit extra relations; we add to te set of relations every word of lent A+2 wic represents te identity. Te condition AC is known to be dependent on 13

14 te coice of eneratin set [T], but for a fixed eneratin set it is independent of coice of relations. We denote te universal cover of te 2-complex associated to tis padded presentation by X and define B(n) to be te subcomplex of X consistin of B(n) X 1 toeter wit all of te 2-cells in X wose boundaries are completely contained in B(n). Fix a prefix closed eodesic 0-combin; for example, te combin iven by takin te sortlex minimal representative of eac element. Assume tat every ede in te ball B(n) X 1 as a combin wic is consistent wit te 0-combin at its endpoints and satisfies te followin monotone increasin property: (M) For any point y X 1 for wic te 1-combin Ψ is already defined, and any t [0, 1], if Ψ(y, t) B(k + 1)\ B(k), ten Ψ(y, t ) X\ B(k) for all t > t. We sow ow to extend tis combin to B(n + 1). Let e be an ede in B(n + 1)\B(n) correspondin to te enerator s and wit vertices and. Let and be te eodesic combins to te end points of e. If e is part of te 0-combin ten simply reparametrize tis combin to ive te combin for te ede e. Assume ten tat e is not part of te 0-combin. Tere are two possibilities, eiter (1) bot end points of e are in B(n), or (witout loss of enerality) (2) te last ede t in is also not in B(n). We assume we are in case (2) and leave te easier case (1) to te reader. Denote te ede from t 1 to = s by f. Since te presentation is almost convex, tere is a pat p in B(n) of lent at most A between and t 1. Tus te pat pts 1 is a loop based at te end of te word of lent A + 2, and terefore in X it is filled by a 2-cell. By assumption, te 1-combin as already been defined for all of te edes in te pat p, and f is in te imae of te 0- combin. Te combin for te ede e consists of te union of te combins for te edes in te pat pt (identified alon te 0-combins for te vertices in pt) plus a mappin from te pat pt to e trou te 2-cell bounded by pts 1. Tis extension still satisfies property (M); we define te 1-combin Ψ to be te union of te ede combins identified alon te cosen 0-combin. Proposition 2. Te combin just described is a tame 1-combin. Proof. Suppose C is some finite subcomplex of X, and suppose C B(n) for some n. Let D = B(n). Because te 1-combin is monotone, once a combin line leaves B(n), it never returns, and tus it could not possibly intersect C. In terms of te definition of a tame 1-combin, for all y X 1, Ψ 1 (C) ({y} [0, 1]) is contained in te component of Ψ 1 (D) ({y} [0, 1]) wic contains (y, 0). 14

15 6 Proof of Teorem D Let (Σ, R) be a finite complete rewritin system for G suc tat te rules of te rewritin system are compatible wit a weitlex orderin. Define constants λ = max{ wt(s i ) s i Σ } and λ l = min{ wt(s i ) s i Σ }, and let λ = λ λ 1 l. Ten for any w Σ, te lent of w is less tan or equal to λ 1 l wt(w) and te weit of w is at most λ l(w). So for any G, l( ) λd(ɛ, ). (Since any subword of an irreducible word is irreducible it follows tat te 0-combin defined by te irreducible words is quasi-eodesic.) Let w Σ be any word evaluatin to te identity in G. Ten because of te Noeterian property of te rewritin system, w rewrites to te identity. Tus a strin of rewritins defines a van Kampen diaram for w wit a (possibly deenerate) 2-cell for eac application of a rule. Let te rewritin be w = z 0 z 1 z 2 1 come from rewritin te sortest possible reducible prefix at eac stae. Since te rewritin rules are compatible wit a weitlex orderin, eac rewritin preserves or reduces te weit of te previous word. For a iven k Z te total number of words in Σ of weit k or less is bounded by te number of words of lent at most kλ 1 l, wic in turn is at most (σ + 1) kλ 1 l. Here σ is te cardinality of Σ and te +1 indicates an empty letter, tus allowin words of lent less tan kλ 1 l. Tus te total number of rewritin steps is bounded by (σ + 1) wt(w)λ 1 l (σ + 1) λl(w). Proposition 3. If G admits a rewritin system compatible wit a weitlex orderin, ten G satisfies an exponential isoperimetric inequality. Let be any vertex in tis van Kampen diaram for w; by construction corresponds to a subword of some word z i in te rewritin. Because te rewritin of w rewrites te sortest irreducible prefixes, te irreducible word connectin ɛ to is contained in te diaram for w. Since rewritin can only preserve or reduce weit, wt( ) wt(z i ) wt(w) Combinin tis wit te bounds iven above on lents, we et l( ) λ 1 l wt( ) λ 1 l wt(w) λl(w). So te distance from ɛ to a vertex in te van Kampen diaram for w is bounded by te linear function λl(w). Proposition 4. If G admits a rewritin system compatible wit a weitlex orderin, ten G satisfies a linear isodiametric inequality. 15

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