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JOURNAL OF ALGEBRA 89, 349-374 (1984)
Gromov’s Theorem on Groups of
Polynomial Growth and Elementary Logic
L. VAN DEN DRIES
Department of Mathematics, StaGford University, Stanford, California 94305, U.S.A.
AND A. J. WILKIE*
Department of Mathematics, The University, Manchester MI3 9PL, England Communicated by J. Tits Received July 20, 1981
INTRODUCTION In the fall of 1980 the authors attended Professor Tits’ course at Yale University in which he gave an account of Gromov’s beautiful proof that every finitely generated group of polynomial growth has a nilpotent subgroup of finite index. An essentialpart of Gromov’s argument consists of constructing for each group of polynomial growth a locally compact metric space and an action of a subgroup of finite index on that space.The intuitive motivation underlying this construction is fairly clear but it required an elaborate theory of “limits” of metric spacesto be carried out. It occurred to us to give a simple nonstandard definition of a space which has all the nice properties needed in the rest of Gromov’s argument. Besides shortening proofs our construction works for arbitrary finitely generated groups, not only for those of polynomial growth, and it has functorial properties. This enables us to state some of Gromov’s lemmas without the restriction of polynomial growth, e.g., (4.2) and (5.5). We also found a new proof of local compactness of the space, see Section 6, under an a priori weaker hypothesis than polynomial growth, and this led to a slight extension of Gromov’s theorem: If‘ the group r with finite generating set X has growth function G, with G,(n) < c. nd for infinitely many n and positive constants c, d, then r has a nilpotent subgroup of jinite index. (Gromov’s hypothesis is that G,(n) < c. nd for afl n > 0.)
- The authors were supported by NSF grants. 349 0021.8693184 $3. Copyright D 1984 by Academic Press, Inc. All rights of reproduction in any form reservrd.
350 VAN DEN DRIES AND WILKIE
We have tried to make this paper reasonably self-contained: For the reader’s convenience we give all the basic definitions and repeat arguments which occur in the literature. Sections 1 and 2 contain a proof of the main theorem just quoted, Theorem (1. lo), module a demonstration of the basic properties of the space attached to any finitely generated group. (These properties are only summarized in Section 2.) In Section 3 we define nonstandard extensions and describe its properties, concentrating on those we need later. (This section may seem a bit long, but we are confident that together with the rest of the paper, it will help readers not versed in the subject to acquire an understanding of how nonstandard extensions are actually used in various situations.) In Section 4 we give our (nonstandard) space construction, and in Sections 5 and 6 we derive the properties of the space we had used before in Section 2 in the proof of the main theorem. In Section 7 we show how another simple application of logic gives an algorithm, based on trial and error, to compute bounds related to Gromov’s theorem, where previously only the existence of bounds was known. For other accounts of Gromov’s theorem and geometric applications we refer the reader to the original paper [5] and to Tits’ Bourbaki seminar lecture [ 141. The authors would like to thank Professors Macintyre, Mostow and Tits for stimulating discussions, and the referee and Professor Kreisel for their suggestions on the presentation of the material.
1. PRELIMINARIES AND PREVIOUS RESULTS
(1.1) Let r be a group generated by a finite subset X. The length function 1 ( = ( Ix: T-r N is defined as follows:
Igl=lengthof s or esh t t^ word^ in X^ U^ X-^ ’ representing^ g.
Properties
(i) I g( = 0 o g = e (the empty word represents the identity e). (ii) Ig/=lg-‘I. (iii> I ghl <I gl + IhI.
The norm-like properties of I I give rise to a metric d = dx : r X r+ N, defined by d(g, h) = 1g-‘hi. Note that d is invariant under left multiplication: d(ag, ah) = d( g, h). (1.2) We define the growth function
G=G,:N+N of (r,X)
352 VAN DEN DRIES AND WILKIE
such that G,(n) < #(X) - G,(bn) for n = 1, 2, 3,.... Hence, H has (near) growth degree <d (resp. exponential growth) if and only if r has.
(1.7) The easy fact that f.g. abelian groups are of polynomial growth was generalized by Wolf as follows:
A Jg. nilpotent group has polynomial growth.
(The precise growth degree was given by Bass; see [ 141.)
(1.8) Milnor and Wolf also proved [ II] :
If r is solvable, then r is either of exponential growth or has a nilpotnent subgroup of finite index.
(1.9) These theorems characterize the groups of polynomial growth among the f.g. solvable groups. Gromov managed to remove the hypothesis of solvability, cf. [5].
If r is of polynomial growth, it has a nilpotent subgroup ofjinite index.
We will slightly weaken the hypothesis of Gromov’s theorem and prove:
(1.10) If r is of near polynomial growth, it has a nilpotent subgroup of finite index.
(1.11) A rough sketch of Gromov’s remarkable proof is as follows: Consider the sequence of discrete metric spaces (r,(l/n) d). (As n increases one moves, so to speak, away from the space (r, d) so that its points seem to get closer together.) In case r is of polynomial growth Gromov shows that some subsequence (r, (l/n,) d) “converges” to a metric space Y with the following properties: (i) Y is homogeneous (for any two points there is an isometry carrying one to the other). (ii) Y is connected and locally connected. (iii) Y is complete. (iv) Y is locally compact and finite dimensional. From the solution of Hilbert’s fifth problem, it then follows that the isometry group of Y is a Lie group. Now one can let a subgroup of finite index of r act on Y in such a way that, using that Isom(Y) is a Lie group and theorems of Jordan and Tits on linear groups, one obtains a homomorphism of this subgroup onto Z (assuming r is infinite). It then follows that the kernel is of polynomial growth of lower degree. An inductive
GROUPS OF POLYNOMIALGROWTH 353
assumption allows us to conclude that r has a solvable subgroup of finite index so that an application of the theorem of Milnor-Wolf finishes the proof.
(1.12) Our proof of (1.10) follows the same lines. The difference is mainly in the construction of the space Y, which we obtain in Section 4 by a very simple and general nonstandard argument. The next section, Section 2, just assembles the relevant properties of the space Y and shows how (1.10) follows.
- PROOF OF GROMOV'S THEOREM ASSUMING PROPERTIES OF THE SPACE Y
r continues to denote a finitely generated group (with finite generating set X). The following algebraic lemma is essentially due to Milnor.
(2.1) LEMMA. Let 1 -+ K + T-+h Z -t 0 be exact and r not of exponential growth. Then K isJinitely generated. Moreover: (1) If r has near growth degree <d + 1, then K has near growth degree<d; (2) if K has a solvable subgroup of finite index, then T has one, too. Proof: Take y E r with h(y) = 1, and take e,,..., ek E K such that r = (Y, e, ,..., ek).^ Define^ ym,i = y”eiy-”^ for^ m^ E Z,^ i^ =^ l,...,^ k.^ Then^ one easily checks that K is generated by the Y~,~. Fix an i in (I,..., k}. For m > 0 consider the elements of r of the form yzi .a. yz,,, si = 0 or 1. There are 2mt’ words on {y,e 1,..., ek} here, each of length <2m. The assumption of nonexponential growth implies that for some m > 0 two of those words represent the same element, say y& ..a y,“,i = yipi .*. ~2~ and E, # 6,. Then Ym,i E (Y o,i,..., y,,- I,i).^ Conjugating^ this^ relation^ by^ y we see that^ ym+ I,i E (Y~,~,..., y,,,) c (yO,i ,..., Y,,-~,~) and by induction we obtain
Yp,i E (YO,i~ae.~Ym-1.i) for all^ p > 0.
A similar argument for negative m gives us that K is generated by a finite set
{Ym,i: 1 <i<k,ImIGW, ME^ fN.
To prove (l), let c > 0 and S c N infinite such that G,(n) < c. ndfl for all n E S. Without loss of generality, see (1.6)(i), we may assume that X = YU (y), where Y generates K. Let n E S and let g,, i = l,..., GY([n/2]) be the distinct elements in K of Y-length <[n/2]. Then the n. G,([n/2]) elements giy’, i = l,..., GY( [n/2]), -[n/2] <j < [n/2] are distinct and of X-
GROUPS OF POLYNOMIALGROWTH 355
(2.5) THEOREM. Suppose Y = Y(r) is locally compact and jinite dimen- sional, and T is infinite. Then P has a subgroup ofjkite index which has Z as a homomorphic image.
Prooj If r has an abelian subgroup of finite index, the conclusion is immediate and from now on we assumethat we are not in this case. The hypothesis of the theorem, together with (I), (II), (III) above, allows us to use the deep results of Gleason-Mongomery-Zippin on Hilbert’s fifth problem. In fact, we use [ 12, 6.31 and [ 1, p. 6061 to conclude: ’
Isom(Y) is a Lie group with finitely many connected components. (^) (*)
Let L be the connected component of the identity. So L is a connected Lie group of finite index in Isom(Y). We claim:
r contains a subgroup A offinite index which has arbitrarily large homomorphic images in L. (^) (**>
(“arbitrarily large”: for each n E N there is one of cardinality >n). The claim holds trivially if l(r) c Isom(Y) is infinite. (Take A = Z-‘(L) nr.) So from now on we supposethat I(T) is finite. In particular, r’ = kernel(Z) is of finite index in K As r has no abelian subgroup of finite index, we can use property (IV), which implies that r’ has homomorphic images in Isom(Y) containing elements fly arbitrarily close to 1y. Now, as a Lie group, Isom(Y) has the property that for each n > 0 a suitable neighborhood of 1, contains no elements# 1,, of order <n. (The “no small subgroups” property.) It follows that r’ has arbitrarily large homomorphic images in Isom(Y). Now there are only finitely many subgroups of r’ of any given index, so at least one of the subgroups of r’ of index < [Isom(Y): L], say A, has arbitrarily large homomorphic images in L. Claim (**) is proved. Let C be the center of L. So L/C embedsinto GL,(C), where n = dim(L) (by a fundamental property of connected Lie groups). Consider the morphisms A -+ L/C obtained by composing the morphisms A -+ L with the natural map L + L/C. If all of these have images of order bounded by q, say, then their kernels are subgroups of A of index <q which have arbitrarily large images in C, and so the intersection of those kernels is a subgroup A’ of finite index in A with arbitrarily large abelian homomorphic images. Hence the commutator subgroup of A’ has infinite index in A’, and it follows that A’ which is of finite index in I- and therefore finitely generated, has L as a homomorphic image, so the conclusion of the theorem holds.
’ It is useful to have some familiarity with the subject treated in [ 121 to see that the theorems we refer to apply.
356 VAN^ DEN^ DRIES^ AND^ WILKIE
So from now on we assume that the morphisms d + L/Cc GL,(C) referred to above have arbitrarily large images. We distinguish two cases:
(a) the morphisms A + GL,(C) have arbitrarily large finite images. (b) there is a morphism A -P GL,(C) with an infinite’ image d. In case (a), Theorem (2.5) follows by very similar arguments as above using the following theorem of Jordan [3, 36.131:
There is an integer q = q(n) such that each finite subgroup of GL,((c) has an abelian subgroup of index <q.
Case (b) is handled by a deep result of J. Tits, cf. [ 131:
A finitely generated subgroup of GL,(C) has either a free subgroup of rank 2 or has a solvable subgroup offinite index.
If d has a free subgroup of rank 2, then 2, hence A and r are of exponential growth, which is excluded by the hypothesis of the theorem and property (VI) of (2.4). So d has a solvable subgroup of finite index, and replacing, if necessary, A by a suitable subgroup of finite index, we may as well assumethat d is solvable, and that its commutator subgroup has infinite index. Then d, hence A, has Z as a homomorphic image. The proof of the theorem is finished. (2.6) Proof of (1.10). (^) Given that r has near growth degree <d for some d E R\l,we have to show that r has a nilpotent subgroup of finite index. The proof is by induction on d. If d = 0, then r is finite, and we are done. Supposer is of near growth degree <d + 1, and r is infinite. Now we use property (V) of (2.4), and apply Theorem (2.5) and (1.6)(iii) to reduce to the case that there is a surjective morphism h: r+ Z. Let K = kernel(h). By (2.1)( 1) and the induction hypothesis K has a nilpotent, hence solvable, subgroup of finite index. By (2.1)(2) r has a solvable subgroup of finite index. An application of the Milnor-Wolf theorem, cf. (1.8), to this subgroup complete the proof. 1
- SOME INTRODUCTORY REMARKS ON NONSTANDARD EXTENSIONS'
(3.1) As already remarked in the introduction we are going to use the theory of nonstandard extensions to construct a space Y having the
2 The reader already familiar with nonstandard methods can skip this section, although we shall occasionally refer to results described here, and use notation introduced here, in the sequel.
358 VAN^ DEN^ DRIES^ AND^ WILKIE
called nonstandard elements. We also sometimes refer to the elements of S as standard elements in this context. We leave the reader to verify:
(3.5) S=S*^ iff S is finite.
It is not literally true that
(3.6) TcS*T”cS”
since if h E T’ (so h E Sr) then h/D evaluated in T* is in general a proper subset of h/D evaluated in S*. However, identifying these two equivalence classes is completely harmless (since any fuction in the first class is equal, almost everywhere, to any function in the second class) and we shall do it, so that (3.6) holds. This also implies (together with the identification of S and v(S)) that
(3.7) TcSTnS=T.
(3.8) We can generalize (3.7) as follows. Given sets S,,..., S, and vc s, x -. (^) X S, define V = {(f,/D ,..., f,/D) E Sf X ..+ x Sz : (h(i) ,...,
f,(i)) E K p.p.i.). Then V*cSf x... X Sz and it is easy to check that
(3.9) Vn(S,X~~XS,)=V.
Further, if V happens to be a function S, x .e. x S,,-, + S, (so we write W 1,***,x,-^1 ) =x,^ for (xi ,..., x,)^ E V), then we also have
(3.10) V* isafunctionS:XV..~S~-,+S~ and V*rS,X...XS,-,=V.
In fact, V(f,/D ,...,f,-,/D) = V(fi ,...,f,-,)/D, where V(fi ,..., f,) E S: is of course defined by V(fi ,..., f,_ I)(i) = V(fi(i) ,...,f,- I(i)). Note that (3.9) and (3.10) tell us that S is a sub-structure of S (more precisely, v is an embedding) with respect to all functions and relations defined on S. This partially justifies remark(a) of (3.2) but we need something much stronger. (For example, we shall need to know that if 0 is a group operation on S, then (S, 0) is also a group.) To this end we define a subset W of S:x... x S,* to be internal if membership to W can be computed co-ordinatewise almost everywhere, i.e., if there exists for each i E I, a subset Wi of S, x ... X S, such that for all f, E Si ,...,f, E SL:
(3.11) (^) (f,/D ,..., f,lD) E Wu (fi(i) ,..., f,(i)) E Wi p.p.i.
We refer to ( Wi)iel as a family of components for W. We leave the reader to
GROUPSOFPOLYNOMIALGROWTH 359
check that (3.11) is a well-defined equivalence, and that if ( Wl)i,, is another family of components for W, then Wi = Wi p.p.i.
(3.12) It is also immediate that if we are given any family ( Wi)ir, of subsets of S, x ... x S,, then (3.11) uniquely defines a (necessarily internal) subset of ST x ... x S,$ with components Wi. Note that if V c S, x ... X S,, then V* is an internal subset of ST x ‘** x Sx (take all the components to be I’), but in general not all internal sets are of this form. The definition of internal function can be obtained from (3.11) (as (3.10) was from (3.9)) and turns out to be equivalent to: (3.13) F: ST x ... x S,-, -+ S, is internal iff there exists for each i E I a function Fi : S, x ... x S,- i --t S, such that for all f, E S{ ,..., f, _, E SL I F(f,/D,.-,f,/D) = (i + Fi(fi(i),***,fn- I(i>>>/D. We shall need the following lemmas later; they are examples of remark (b) of (3.2).
(3.14) LEMMA. (i) Suppose W is an internal subset of S, with components Wi, and n E n\i and # Wi ,< n p.p.i. Then #W,< n. (ii) No infinite subset of S is an internal subset of S. ProoJ We leave the proof of (i) to the reader. For (ii) suppose A c S, A infinite and internal. Let (Ai)ier be a family of components for A. Suppose a,, a2 ,..., a, ,... are distinct elements of A, and say Z = {i, , i, ,..., i, ,... } (recall that Z is countable). Define f E S’ by f (i,) = uj, wherej is maximal such that aj E Ai,, if j exists, a,,j otherwise, where j is minimal such that a, +j E A in ; since f (i) E Ai for all i E Z, we have f/D EA. Therefore f/D = a, for some m E R\i, i.e., f/D =6,/D, i.e., f(i) = a,,, p.p.i. However, this clearly implies a ,,,+, E Ai p.p.i., so 6,+,/D 6?A, i.e., a,,, &A-contradiction. [
(3.15) LEMMA. Suppose g, E S* for n E N. Then there is an internal function F: R\l* -+ S* such that F(n) = g, for all % E R\i. (We make no claim here for the values of F(n) when n is nonstandard, except of course that they lie in S’.)
Proof Say g, =f,/D, where f,, E S’, for n E N. For each i E Z define the function Fi : N -+ S by Fi(n) = f,(i) (n E N). Let F be the function N * + S* (necessarily internal-see (3.2)) (^) with components {Fi: i E I}. Then for n E N, F(i,D) = i ++ Fi(fi(i>) D (by 3.13),
i t-+ Fi(n) D
(by definition of n^),
GROUPS OF POLYNOMIAL GROWTH 361
of w, )...) W,,^ f, /D ,...,^ fi/D^ which^ results^ from^0 by^ changing^ “3x^ E Sk”, “Vx E Sk” to “3x E Sk,” “V/x E Sk,” respectively, and “3X c Z7,” “VX c Ill” to “there is an internal subset of lZ...” and “for all internal subsets of n...,” respectively; here, if IZ is, say S, x S,, then ZI* is S: x S:. (For example, let us see what @,, @f, and @T (of (3.16)) say. Q;” asserts that the relation < totally orders R ; @f asserts that every nonempty internal subset of R * which is (< -) bounded above has a (<* -) supremum; @F asserts that every nonempty internal subset of N * has a (<* -) least element.) Then Los’s theorem states:
(3.18) w,..., W,,^ filD^ ,...,^ f,/D^ have^ property^ @*^ lyf^ WIi ,..., Wmi, f,(i),...,fi(i) have property @ p.p.i. In particular, Qi holds of (the standard
sets and elements) V, ,..., V,, s, ,..., sI zg @* holds of Vf ,..., V,* S,,...,s,.
A full discussion of Los’s theorem (in the elementary case) may be found in [2]; see also [9, Chap. 11. However, the proof of (3.18) for some particular Q’s conveys the flavour of the general result. Since @, (of (3.16)) holds of <, we must show @r holds of <, i.e., we must show < totally orders R . So suppose x, y E R , x < y and y < x.
Say x =f/D, y = g/D. Then f(i) <g(i) p.p.i. and g(i) <f(i) p.p.i. (by
definition of <*). Hence by (3.3)(ii), (iii), f(i) = g(i) p.p.i., so x = y. We
leave the proof of the other two conjuncts in @$ (the third requires (3.3)(iv)) to the reader. Let us now show that @F (of 3.16)) holds of <. Let Xc R * be internal and assume f/D E R * is an (< -) upper bound for X. Let (Xi)iE, be a family of components for X. We claim that Xi is bounded above (in R) p.p.i. For otherwise Xi would be unbounded above p.p.i. (by (3.3)(iv)) and so we could choose, for each i E (i E I: Xi unbounded above} an element g(i) E Xi such that f(i) < g(i). Setting g(i) = 0 (say) if Xi is bounded above, gives f(i) < g(i) p.p.i., and g(i) E Xi p.p.i., and hence f/D < * g/D and g/D E X, which contradicts the assumption that f/D is an <* - upper bound for X. Now define
qER’ (^) by I?(i) =
SUP Xi if Xi is (4 -) bounded above, 0 (say) otherwise.
We leave the reader to check that q/D is the <* - supremum of X. As a further example, we recommend the exercise of proving (3.18) for the property Q3 of (3.16). We hope these examples go some way towards convicing the reader why the definition of internal set, and the restriction of set quantifiers to these sets, guarantees the truth of (3.18). Of course, in using (3.18) we shall not always write out the property @J under consideration in strict logical notation, since we hope it will be fairly clear what @* is saying.
362 VAN DEN DRIES AND WILKIE
Indeed, if @ is in fact elementary, then @* expresses the same property of the nonstandard extension as @ does of the original structure, although more care must be taken if @ is not known to be elementary. For example, it is not the case that every subset of R, which is bounded above, has a supremum as we shall see below. (3.19) An immediate corollary of (3.18) is that any subset of S,” x .f X Sz which can be defined from internal setsusing (our restricted) quantifiers and boolean operations is also internal and hence, if it is infinite, must contain a nonstandard element (by (3.14)(ii)). This latter phenomenon is called overspill. It is crucial in many applications of nonstandard analysis becausethe nonstandard elementsthat arise in this way often turn out to do the coding mentioned in remark (3.2)(b).
(3.20) We now look more closely at the structure of R^ and f ^ (the nonstandard extension of the group r) in the light of (3.16). For convenience of giving examples, we take our index set I to be the set of natural numbers. (3.18) implies that the nonstandard extension to R: +,-, ., <, etc., of the usual operations and relations on R, makes R into an ordered field (this is an elementary property). Suppose v E R . If -r < n <* r for some r E R, n is called finite; otherwise, it^ is^ called^ injhite.^ For^ example, i t--+3 + (l/i + 1)/D is a finite element of R * (which is not in R); i b i/D is infinite (by (3.3)(i)). If --r <* q <r for all positive r E R, q is called infinitesimal. Thus i t-+ l/i + l/D is an example of a nonzero infinitesimal. Define Rfi” = {n E R: r] finite}, R” = {q E R: v infinitesimal}. Warning: these sets are not internal subsets of R. (Proof: They are both bounded above (in R) but neither has a supremum.) Clearly R G R”“. We leave the reader to check that IR’” is a subring of R, that R” is a maximal ideal in R”” and that the map p: R + Rfi”/Ro: r t-+ r + R” is a (field) isomorphism. Let h: R”” --) R”“/R’ be the natural homomorphism. The homomorphism
P-’^ 0 h: R”” -+ R is called the standard-part map and is usually denoted by st. Thus for each n E Rfi”, St(r) is the unique real number “infinitesimally close to q”, i.e., it satisfies St(q) - n E R”.
Let us also note here that N * n R”” = N, so that all nonstandard elements
of N* are infinite. (3.20) We now investigate r. Let us suppose I- is infinite so that r #I- (by (3.5)). By (3.18) r* is certainly a group under the nonstandard extension, 0 , of the group operation, 0, on r (the reader may like to verify this directly from (3.10)) and r is a subgroup of r by (3.10). Now suppose X is a finite generating set for r. By (3.5) X=X; but now we appear to have a conflict with (3.18). While the statement “X generatesI”’ is true, the statement “X (i.e., X) generatesr” cannot be literally true (since r f r). The clue is that “X generatesr*’ is not an elementary statement about X and
364 VAN DEN DRIES AND WILKIE
- NONSTANDARD CONSTRUCTION OF THE SPACE Y
(4.1) We now return to the situation of Section 1, so that r is a group with finite generating subset X, and length function 1 1:r-, N. Our aim in this section is to construct a space Y having the properties listed in (2.4). To this end we consider (uniform) non-standard extensions, r, R, n\l* etc., of r, R, N, as described in Section 3, although we shall now use the same symbols to denote the nonstandard extensions of familiar functions and relations defined on these sets. (For example, we shall use just 1.1 for I. /, < for <.) Fix a positive infinite hyperreal number R, i.e., R E R* and R > n for all n E R\l, and define rtR) as the subgroup { g E r* I / g(/R < c for some c > 0, c E R} of r, and let ,U=,u(~) be the subgroup {g E r ) I gI/R < c for all cm, 00) 0f r (R). The quotient I. I/R defines a map rCR) -+ R”” = {xER*:-c<x<c, some c E R }, and clearly (^) Igl/R-IhlIR is infinitesimal, whenever gp = hp in the set of left cosets rCR’/,u. So we can factor out ,D and apply the standard map st: R”” -+ R to obtain a commuting diagram: r(R) - 1. I/R p,fin
I P II w II= st(l d/R).
(Note that the construction here is rather similar to the discussion of R in (3.20).) From the definitions it follows that II gp /I = 0 o g E ,u, so by putting &x hp) = II g-‘WI, we obtain^ a metric^ space (rCR)/p, d) which^ we denote by YCR’, or simply by Y if no confusion results. The reader can easily verify that example (1.3)(a) leads to the space Y = (R*, N.Y.-distance).
(4.2) PROPOSITION. The metric space Y has the following properties. (a) Y is homogeneous; (b) for each two points p, q E Y with d(p, q) = r there is an isometry of [0, r] into Y sending 0 to p and r to q; so Y is connected and locally connected; (c) Y is complete. Proof. (a) i-CR)acts on^ Y on the left by isometries:
d(ag~,ah~)=Ih~‘a-‘ag~l=Ih-‘g~l=d(g~,h~),
and clearly this action is transitive.
GROUPS OF POLYNOMIAL GROWTH 365
(b) For simplicity we assume that p= ep, (^) q=m and d(p, q) = st(/ g]/R) = 1 (the general case is very similar). Replacing if necessary g by another member of gp, we may assume that / g] = [RI, [ ] denoting the integral part operator defined on IR * (with values in Z *). So g has a shortest representation as a (nonstandard) word g, ... gtR,, where all gi E XUX-‘. (See the discussion in (3.20).) We define f: [0, 1] + Y by
f@-> = g, “’ g[rR]. A^ straightforward^ computation^ shows^ that^ f^ is^ the required isometry. (4 Let k4nEN be^ a^ Cauchy^ sequence^ in^ Y.^ For^ simplicity^ we assume 11g,,pl/ < 1 for all n. As in (b), there is no loss of generality in further assuming that 1g,l <R. Extend (g,),, N to an internal sequence (g,),, N, this being possible by Lemma (3.15). For each k E N >O, take M(k) E N such that j g;‘g,l < R/k for all (standard) integers m, n > M(k). By “overspill” this remains true for all m, n E N * greater than M(k) but less than some infinite N(k) E N . (See (3.19). We are applying the remark there to the internal set {tEN:VmENVnEN*(M(k)<m,k<t-+~g~’g,~~ R/k)}.) By a similar argument using (3.15) and (3.19) there is o E N * greater than all M(k) and less than all N(k), k E N >O. Clearly we have limg,p =g,p in Y. I (4.3) Remarks. (1) We have to keep in mind that the metric space Y depends not only on the hyperreal number R but also on the generating set X, so let us (temporarily) write Yx and its metric as d,. Take another finite generating set X’ (but keep the same R). Then, with
we have (for Z-f {e}):
c-‘Iglx4&~cl& for all g E r,^ hence for all g E I’*.
So rcR’ does not change, nor does piR), and the metric spaces Yx and Y,, have the same underlying set rcR)/p, and their metrics are related by: c-‘d,<(5,,<c& (2) The functoriality mentioned in the introduction amounts to the following: Let o: (r,, X,) + (r,, X,) be a morphism of groups with distinguished finite generating sets X, , X,, i.e., o(X,) c X,. Clearly ( gl > ]rp( g)i for all g E r, (the norms are taken w.r.t. the generating sets X, and X,, respectively). So a, induces naturally a group morphism r, + ry) sending ,uy’ into ,u,(R) Hence. p induces a map @: Y, -+ Y,, where Y, , Y, are the spaces attached to (I-,, X,, R) and (r,, X,, R), respectively. Clearly we have II 41 > IlP(w>ll = II(w>,4, g E rl”‘. So the as$wmnts (r,X)+ K cp-+ U, define a functor, still depending on R, from the category of groups with distinguished finite generating sets to the category of metric spaces with
GROUPS OF POLYNOMIALGROWTH 361
(5.5) PROPOSITION. Supposer has no abelian subgroups offinite index. Thenfor each neighborhood U of 1y there are /I E (r’)* and s E S such that ,&‘r’p c rcR) and lBmlsBE U\ { ly}.
Proof. ForyE~andO<rERweputB(y,r)=max(d(ya,a):lal<r} ( maximum “displacement”^ effected by^ y among the points of B,(r)).^ We claim:
Q-‘x, 4 <W, r>+ 2 Igl (gEr) (1)
Let Ia I < r. Then 4 g- ‘yga,a>= d(yga,ga) < d(y,l g l + r) < 6(y, r) + 2 1g/.
The last inequality follows by writing an element of B,(l gl + r) as bc with
lb/Gr, Icl<lgl;then
d(ybc, bc) < d(ybc, yb) + d(yb, b) + d(b, bc)
= d(yb, 6) + 2d(bc, b) < 6(y, r) + 2 1gl.
inequality (1) is established. FIX a neighborhood U = U,,,, k E N >O, E > 0. A nonstandard translation of Lemma (5.3) gives us s E S and g E (P)* such that:
1g-‘sgl > ER. (^) (2)
Write g=s,. ..s.,~~ES,t~N*.ForO<i<tweput:
gi=s, “‘Si and Mi = max{b(gIT1sgi, kR): s E S}.
Further we let C = max{lsI: s E S}, so C E N. Then we have:
M, < ER (becauser’^ acts trivially^ on Y),^ (3) M, > ER (^) (by (211, (4)
IM,+,-MiI<2C for^ 0 < i < t -^ 1 (by 1)).^ (5)
From (3), (4) and (5) we derive the existence of an i E {O,..., t} with:
IM,-ERI<~C. (^) (‘5)
ForthisiwedefineP=gi,sopE(r’)*.NotethatifyEr’,thenP~‘YPisa finite product of elements of the form p-‘s/3, s E S, each of which is in rcR) by (6), so:
p-‘z-‘p c rcR). (7)
From (6) we also obtain the existence of s E S such that S@-‘sjI, kR) differs from ER by at most 2C. For u = I,- ,S4this meansthat u # 1,. Moreover, for
368 VAN DEN DRIES AND WILKIE
(a ) 4 kR: d(o(a,u), up) = st(d(au, a)/R) < E, by (6), so u E U. The propo- sition is proved. 1
- FURTHER PROPERTIES OF YRESULTING FROM GROWTH RESTRICTIONS ON r
(6.1) The properties of the space Y = Y(R’ discussed in Sections 4 and 5 hold for any finitely generated group r, and any positive infinite R E IR*. We now show that if r has near polynomial growth, then R can be chosen so that YcR) is locally compact and of tinite dimension. The proof of the following lemma contains the crucial argument.
(6.2) LEMMA. Let R, be positive infinite and suppose G(R,) < c. Rf where 0 < c E IR, d E N. Then there is a positive infinite S < R,, such that for every i E N, i > 4, the following property Pi(S) holds:
P,(S): if g, ,..., g, E B&S/4), t E N *, and B,,(S/i) ,..., B,,(S/i) are pairwise disjoint, then t < id+ ‘.
ProoJ Suppose the lemma is false. Thus, for all S E R* with log R, < S <R, there is some i E N, i > 4, such that Pi(S) fails. In fact, clearly the function mapping S to the least i such that P,(S) fails (log R, < S <R,), is internal, so its range must be internal; since this range is a subset of N, it is bounded by some K E N (by (3.14)(ii)). Hence we may define internally, by induction, natural numbers i, ,..., i,, u E N * to be chosen below, and elements g(l, j) E I’*, for 1 < I < U, 1 <j < t,, where t, = [if”] + 1, such that for I= l,..., u:
4<i,<K, (^) (1)
g(Lj) E Be (^) (4i, “Pi,-I) for^ 1 <j,<^ t,,^ (2)
for 1 <j<j’<t,. (^) (3)
(As 1 goes from 1 to u, the radii R,/i, .a- i,-, represent^ decreasing^ values of S.) Clearly the obvious inductive definition of the i,‘s and g(l,j)‘s may proceed as long as the condition PI(R,/i,... i[-,) fails, and this will be guaranteed if we choose u to satisfy
