Let Fm be a free group with m generators and let R be a normal subgroup such that Fm /R projects onto ℤ. We give a lower bound for the growth rate of the group Fm / R′(where R′ is the derived subgroup of R) in terms of the length ρ= ρ(R) of the shortest non-trivial relation in R. It follows that the growth rate of Fm / R′ approaches 2m-1 as ρ approaches infinity. This implies that the growth rate of an m-generated amenable group can be arbitrarily close to the maximum value 2m- 1. This answers an open question of P. de la Harpe. We prove that such groups can be found in the class of abelian-by-nilpotent groups as well as in the class of virtually metabelian group

Arzhantseva, G. N.

Guba, V. S.

Guyot, L.

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J. Group Theory 8 (2005), 389–394 Journal of Group Theory( de Gruyter 2005Growth rates of amenable groupsG. N. Arzhantseva, V. S. Guba and L. Guyot*(Communicated by A. Yu. Olshanskii)Abstract. Let Fm be a free group with m generators and let R be a normal subgroup such thatFm=R projects onto Z. We give a lower bound for the growth rate of the group Fm=R 0 (whereR 0 is the derived subgroup of R) in terms of the length r ¼ rðRÞ of the shortest non-trivial re-lation in R. It follows that the growth rate of Fm=R0 approaches 2m 1 as r approaches in-ﬁnity. This implies that the growth rate of an m-generated amenable group can be arbitrarilyclose to the maximum value 2m 1. This answers an open question of P. de la Harpe. Weprove that such groups can be found in the class of abelian-by-nilpotent groups as well as in theclass of virtually metabelian groups.1 IntroductionLet G be a ﬁnitely generated group and A a ﬁxed ﬁnite set of generators for G. Wedenote by lðgÞ the word length of an element g A G in the generators A, i.e. the lengthof a shortest word in the alphabet AG1 representing g. Let BðnÞ denote the ballfg A G j lðgÞc ng of radius n in G with respect to A. The growth rate of the pairðG;AÞ is the limitoðG;AÞ ¼ limn!yﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃjBðnÞjnp:(Here jX j denotes the number of elements of a ﬁnite set X .) This limit exists due to thesubmultiplicativity property of the function jBðnÞj; see for example [5, VI.C, Propo-sition 56]. Clearly oðG;AÞd 1. A ﬁnitely generated group G is said to be of expo-nential growth if oðG;AÞ > 1 for some (and hence in fact for any) ﬁnite generating setA. Groups with oðG;AÞ ¼ 1 are groups of subexponential growth.Let jAj ¼ m. It is known that oðG;AÞ ¼ 2m 1 if and only if G is freely generatedby A; see [3, Section V]. In this case G is non-amenable whenever m > 1.A ﬁnitely generated group which is non-amenable is necessarily of exponentialgrowth [1]. The following interesting question is due to P. de la Harpe.*This work was supported by the Swiss National Science Foundation, No. PP002-68627.Question (see [5, VI.C, 62]). For an integer md 2, does there exist a constant cm with1 < cm < 2m 1, such that G is not amenable whenever oðG;AÞd cm?We show that the answer to this question is negative. Thus, given md 2, thereexists an amenable group on m generators with growth rate as close to 2m 1 as onelikes.It is worth noticing that for every md 2 there exists a sequence of non-amenablegroups (even containing non-abelian free subgroups) whose growth rates approach 1(see [4]).For a group H, we denote by H 0 its derived subgroup, that is, H 0 ¼ ½H;H .The authors thank A. Yu. Ol’shanskii for helpful comments.2 ResultsLet Fm be a free group of rank m with free basis A. Suppose that R is a normal sub-group of Fm. Assume that there is a homomorphism f from Fm onto an inﬁnite cyclicgroup whose kernel contains R (that is, Fm=R has the additive group Z as a homo-morphic image). By a we denote a letter from AG1 such thatfðaÞ ¼ maxffðxÞ j x A AG1g:Clearly fðaÞd 1.Throughout the paper, we ﬁx a homomorphism f from Fm onto Z, the letter adescribed above and the value C ¼ fðaÞ. By R we will usually denote a normal sub-group of Fm that is contained in the kernel of f.A word w over AG1 is called good whenever it satisﬁes the following conditions:(1) w is freely irreducible;(2) the ﬁrst letter of w is a;(3) the last letter of w is not a1;(4) fðwÞ > 0.Let Dk be the set of all good words of length k and let dk ¼ jDkj.Lemma 1. The number of good words of length kd 4 satisﬁes the following inequality:dkd 4mðm 1Þ2ð2m 1Þk4: ð1ÞIn particular, limk!yd1=kk ¼ 2m 1.Proof. Let W be the set of all freely irreducible words v of length k  1 satisfyingfðvÞd 0. The number of freely irreducible words of length k  1 equals2mð2m 1Þk2. At least half of them have non-negative image under f, and sojWjdmð2m 1Þk2.G. N. Arzhantseva, V. S. Guba and L. Guyot390Let W1 be the subset of W that consists of all words whose initial letter is di¤erentfrom a1. We show that jW1jd ðð2m 2Þ=ð2m 1ÞÞjWj. It is su‰cient to prove thatjW1 VAG1ujd ðð2m 2Þ=ð2m 1ÞÞjWVAG1uj for any word u of length k  2. Sup-pose that a1u belongs to W. For every letter b one has fðbÞd fða1Þ. Thereforebu A W1 for every letter b0 a1 if bu is irreducible. There are exactly 2m 2 ways tochoose a letter b with the above properties. Hence jW1 VAG1uj and jWVAG1uj have2m 2 and 2m 1 elements, respectively. If a1u B W, then the two sets coincide.Now let W2 denote the subset of W1 that consists of all words whose terminal letteris di¤erent from a1. A similar argument implies thatjW2jd ðð2m 2Þ=ð2m 1ÞÞjW1j:It is obvious that av is good if v A W2. Therefore the number of good words is at leastjW2jd 2m 22m 1 jW1jd2m 22m 1 2jWjd 4mðm 1Þ2ð2m 1Þk4:To every word w in AG1 one can uniquely assign a path pðwÞ in the Cayley graphC ¼ CðF=R;AÞ of the group F=R with A the generating set. This is the path that haslabel w and starts at the identity. We say that a path p is self-avoiding if it never visitsthe same vertex more than once.Let r ¼ rðRÞ be the length of the shortest non-trivial element in a normal subgroupR of Fm.Lemma 2. Let R be a normal subgroup of Fm that is contained in the kernel of a ho-momorphism f from Fm onto Z. Suppose that kd 2 is chosen in such a way that thefollowing inequality holds:rðRÞ > Ckð2k  3Þ þ 2k  2: ð2ÞThen any path in the Cayley graph C of Fm=R labelled by a word of the formg1g2 . . . gt, where td 1 and gs A Dk for all 1c sc t, is self-avoiding.Proof. Suppose that p is not self-avoiding, and consider a minimal subpath qbetween two equal vertices. Clearly jqjd rd k. Therefore q can be represented asq ¼ g 0gi . . . gjg 00, where gi; . . . ; gj are in Dk, the word g 0 is a proper su‰x of someword in Dk and g00 is a proper preﬁx of some word in Dk. We have jg 0j; jg 00jc k  1so that jgi . . . gj j > Ckð2k  3Þ. This implies that j  i þ 1 (the number of sectionsthat are completely contained in q) is at least Cð2k  3Þ þ 1. Obviouslyfðg 0ÞdCðk  1Þ and fðg 00ÞdCðk  2Þ (we recall that g 00 starts with a if it isnon-empty). On the other hand, fðgsÞd 1 for all s. Hencefðgi . . . gjÞd j  i þ 1dCð2k  3Þ þ 1and so fðg 0gi . . . gjg 00Þd 1, which is obviously impossible because for every r A R onehas fðrÞ ¼ 0.Growth rates of amenable groups 391Theorem 1. Suppose that R is a normal subgroup of the free group Fm that is containedin the kernel of a homomorphism f from Fm onto Z. Let C be the maximum value of fon the generators and their inverses. Let r ¼ rðRÞ be the length of the shortest cyclic-ally irreducible non-empty word in R. If the number kd 4 satisﬁes the inequalityrdCkð2k  3Þ þ 2k  1; ð3Þthen the growth rate of Fm=R0 with respect to the natural generators is at leastð2m 1Þ  4mðm 1Þ2ð2m 1Þ4 !1=k:Proof. We use the following known fact [2, Lemma 1]: a word w belongs to R 0 if andonly if, for any edge e, the path labelled by w in the Cayley graph of the group Fm=Rhas the same number of occurrences of e and e1. Hence distinct self-avoiding pathsof length n in the Cayley graph of Fm=R represent distinct elements of the groupFm=R0. Moreover, all of the corresponding paths in the Cayley graph of Fm=R 0 aregeodesic and so these elements have length n in the group Fm=R0.Suppose that the conditions of the theorem hold. For every n, one can consider theset of all words of the form g1g2 . . . gn, where each gi belongs to Dk. By Lemma 2these elements give us distinct self-avoiding paths in the Cayley graph of Fm=R.Hence for any n we have at least d nk distinct elements in Fm=R0 that have length kn.Therefore the growth rate of Fm=R0 is at least d 1=kk . It remains to apply Lemma 1.One can summarize the statement of Theorem 1 as follows: if all relations of Fm=Rare long enough, then the growth rate of the group Fm=R0 is big enough. Notice thatwe cannot avoid the assumption that Fm=R projects onto Z. Indeed, for any numberr, there exists a ﬁnite index normal subgroup in Fm all of whose non-trivial elementshave length greater than r. If R were such a subgroup, then F=R 0 would be a ﬁniteextension of an abelian group and its growth rate would be equal to 1.Theorem 2. Let Fm be a free group of rank m with free basis A and let f be a homo-morphism from Fm onto Z. Suppose thatker fdR1dR2d   dRnd   is a sequence of normal subgroups in Fm with trivial intersection. Then the growth ratesof the groups Fm=R0n approach 2m 1 as n approaches inﬁnity, that is,limn!yoðFm=R0n;AÞ ¼ 2m 1:Proof. Since the subgroups Rn have trivial intersection, the lengths of their shortestnon-trivial relations approach inﬁnity, that is, rðRnÞ !y as n !y. LetkðnÞ ¼ ½ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃrðRnÞ=2Cp;G. N. Arzhantseva, V. S. Guba and L. Guyot392where C is deﬁned in terms of f as above. Obviously the inequality (3) holds andkðnÞ !y. Theorem 1 implies that the growth rates of the groups Fm=R 0n approach2m 1.Now we show that for every m there exists an amenable group with m generatorswhose growth rate is arbitrarily close to 2m 1.Theorem 3. For every md 1 and for every e > 0, there exists an m-generated amenablegroup G, which is an extension of an abelian group by a nilpotent group such that thegrowth rate of G is at least 2m 1 e.Proof. It su‰ces to take the lower central series in the statement of Theorem 2 (thatis, R1 ¼ F 0m, Riþ1 ¼ ½Ri;Fm for all id 1). The subgroups Rn have trivial intersectionand they are contained in F 0m and hence certainly lie in kernels of epimorphisms to Z.The groups Gn ¼ Fm=R 0n are extensions of (free) abelian groups Rn=R 0n by (free) nil-potent groups Fm=Rn and so they are all amenable. Their growth rates approach2m 1.One can take instead the sequence Rn ¼ F ðnÞm of iterated derived subgroups (thatis, R1 ¼ F 0m, Riþ1 ¼ R 0i for all id 1). It is not hard to show that rðRnÞ grows ex-ponentially. The groups Fm=R0n ¼ Fm=Rnþ1 are free soluble. Their growth ratesapproach 2m 1 very quickly. For instance, the growth rate of the free soluble groupof degree 15 with 2 generators is greater than 2.999.One more application of Theorem 3 can be obtained as follows. The group Fmhas countably many ﬁnite index normal subgroups and so one can enumerate themas N1;N2; . . . ;Ni; . . . : Let Mi ¼ N1 VN2 V   VNi and let Ri ¼ M 0i for all id 1.Obviously the subgroups Mi (and thus the subgroups Ri) have trivial intersectionsince Fm is residually ﬁnite. As above, all subgroups Ri are contained in F0m and so inkernels of epimorphisms to Z. Hence the growth rates of the groups Fm=R 0i ¼ Fm=M 00iapproach 2m 1. These groups are extensions of Mi=M 00i by Fm=Mi, that is, they areﬁnite extensions of (free) metabelian groups.Therefore there exist m-generated groups with growth rates approaching 2m 1in each of the following: (1) the class of extensions of abelian groups by nilpotentgroups, and (2) the class of ﬁnite extensions of metabelian groups.Remark. A. Yu. Ol’shanskii suggested the following improvement. Let p be a prime.Since Fm is residually a ﬁnite p-group, there is a chain M1dM2d    of normalsubgroups with trivial intersection, where each Fm=Mi is a ﬁnite p-group. LetRi ¼ ker fVMi. The group Fm=Ri is a subdirect product of Z and a ﬁnite p-group. Inparticular, it is nilpotent. Moreover, it is also an extension of Z by a ﬁnite p-groupand an extension of a ﬁnite p-group by Z. So Fm=R 0i will be both abelian-by-nilpotentand metabelian-by-ﬁnite. (In fact, the metabelian part is an extension of an abeliangroup by Z.) Also Fm=R 0i is an extension of a virtually abelian group by Z.Growth rates of amenable groups 393References[1] G. M. Adel’son-Vel’skii and Yu. A. Sreider. The Banach mean on groups. Uspekhi Mat.Nauk (N.S.) 12, 131–136.[2] C. Droms, J. Lewin and H. Servatius. The length of elements in free solvable groups. Proc.Amer. Math. Soc. 119 (1993), 27–33.[3] R. Grigorchuk and P. de la Harpe. On problems related to growth, entropy, and spectrumin group theory. J. Dynam. Control Systems 3 (1997), 51–89.[4] R. Grigorchuk and P. de la Harpe. Limit behaviour of exponential growth rates for ﬁnitelygenerated groups. In Essays on geometry and related topics, Monogr. Enseign. Math. 38(Enseignement Math., 2001), pp. 351–370.[5] P. de la Harpe. Topics in geometric group theory (University of Chicago Press, 2000).Received 31 May, 2004G. N. Arzhantseva, Section de Mathe´matiques, Universite´ de Gene`ve, CP 240, 1211 Gene`ve24, SwitzerlandE-mail: Goulnara.Arjantseva@math.unige.chV. S. Guba, Department of Mathematics, Vologda State University, 6 S. Orlov St., Vologda160600, RussiaE-mail: guba@uni-vologda.ac.ruL. Guyot, Section de Mathe´matiques, Universite´ de Gene`ve, CP 240, 1211 Gene`ve 24, Swit-zerlandE-mail: Luc.Guyot@math.unige.chG. N. Arzhantseva, V. S. Guba and L. Guyot394

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