DETECTING LARGE GROUPS

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1 DETECTING LARGE GROUPS MARC LACKENBY Mathematcal Insttute, Unversty of Oxford St. Gles, Oxford, OX1 3LB, Unted Kngdom Abstract Let G be a fntely presented group, and let p be a prme. Then G s large (respectvely, p-large ) f some normal subgroup wth fnte ndex (respectvely, ndex a power of p) admts a non-abelan free quotent. Ths paper provdes a varety of new methods for detectng whether G s large or p-large. These relate to the group s profnte and pro-p completons, to ts frst L 2 -Bett number and to the exstence of certan fnte ndex subgroups wth rapd descent. The paper draws on new topologcal and geometrc technques, together wth a result on error-correctng codes. 1. Introducton A group s known as large f one of ts fnte ndex subgroups has a free nonabelan quotent. Large groups have many nterestng propertes, for example, superexponental subgroup growth and nfnte vrtual frst Bett number. (See also [4], [15]). It s therefore useful to be able to detect them n practce. In ths paper, we wll show how one may deduce that a fntely presented group s large usng an array of dfferent structures: ts profnte and pro-p completons, ts frst L 2 -Bett number and the homology growth of ts fnte ndex subgroups. The detecton of large groups was the am of [5], where the author gave a charactersaton of large fntely presented groups n terms of the exstence of a sequence of fnte ndex subgroups satsfyng certan condtons. In ths paper, we start by deducng the followng consequence. Theorem 1.1. Let G and K be fntely presented (dscrete) groups that have somorphc profnte completons Ĝ and ˆK. Then G s large f and only f K s large. In the above result, the term somorphc can be taken to mean somorphc as groups, snce any group somorphsm between profnte completons Ĝ and ˆK s automatcally contnuous. We do not requre that the somorphsm Ĝ ˆK be 1

2 nduced by a homomorphsm G K. One can also defne a group to be p-large, for some prme p, f t contans a normal subgroup wth ndex a power of p that has a free non-abelan quotent. In a smlar sprt to Theorem 1.1, we wll prove the followng. Theorem 1.2. Let G and K be fntely presented (dscrete) groups that have somorphc pro-p completons for some prme p. Then G s p-large f and only f K s p-large. A sample applcaton of Theorem 1.2 s to weakly parafree groups, whch are defned n terms of the lower central seres, as follows. Denote the th term of the lower central seres of a group by γ ( ). A group s weakly parafree f there s some non-trval free group F wth the same lower central seres as G. Ths means that there s an somorphsm F/γ (F) G/γ (G), for each postve nteger, and that these somorphsms are compatble wth each other n the obvous way. A group s known as parafree f t s weakly parafree and resdually nlpotent. Many nterestng examples of parafree groups are gven n [1] and ther propertes are nvestgated n [2]. A consequence of Theorem 1.2 s the followng result. Theorem 1.3. Let G be a fntely presented, weakly parafree group wth b 1 (G) > 1. Then G s large. Here, b 1 (G) denotes the frst Bett number of G. Ths has the followng corollary. Corollary 1.4. Any fntely presented, parafree group s ether large or nfnte cyclc. There are also applcatons of Theorem 1.2 to low-dmensonal topology, ncludng the followng. Theorem 1.5. If two closed 3-manfolds M 1 and M 2 are topologcally Z p -cobordant, for some prme p, then π 1 (M 1 ) s p-large f and only f π 1 (M 2 ) s p-large. Recall that two closed 3-manfolds M 1 and M 2 are topologcally Z p -cobordant f there s a compact topologcal 4-manfold X such that X = M 1 M 2, and such that the ncluson of each M nto X nduces somorphsms of homology groups wth mod p coeffcents. Thus, Theorem 1.5 represents an unexpected lnk between two very dfferent areas of low-dmensonal topology: the theory of fnte covers of 3-manfolds, and 4-dmensonal topology. 2

3 One of the goals of ths paper s also to relate largeness to L 2 -Bett numbers. The frst L 2 -Bett number s defned n [13] for any fntely presented group G and s denoted here by b (2) 1 (G). Theorem 1.6. Let G be a fntely presented (dscrete) group that s vrtually resdually p-fnte for some prme p, and such that b (2) 1 (G) > 0. Then G s large. Ths gves new examples of large groups. By applyng results of Shalom from [16], we obtan the followng. Corollary 1.7. Let G be a fntely presented, non-amenable, dscrete subgroup of SO(n, 1) or SU(n, 1), wth n 2 and wth crtcal exponent strctly less than 2. Then G s large. Ths s a consequence of Theorem 1.6 because Shalom showed that such a group G has b (2) 1 (G) > 0 (Theorem 1.5 n [16]). And snce t s fntely generated and lnear over a feld of characterstc zero, t s vrtually resdually p-fnte, for all but fntely many prmes p (Proposton 9 n Wndow 7 of [11]). Corollary 1.7 can be vewed as a generalsaton of the followng mportant result of Cooper, Long and Red (Theorem 1.3 of [3]) to Le groups other than SO(3, 1). Theorem 1.8. Let G be a fntely generated, dscrete subgroup of SO(3, 1) that s nether vrtually abelan nor cocompact. Then G s large. Theorem 1.6 s proved usng two results. The frst s due to Lück (Theorem 0.1 of [12]). It relates b (2) 1 (G) to the ordnary frst Bett number b 1(G ) of fnte ndex normal subgroups G. Theorem 1.9. (Lück [12]) Let G be a fntely presented group, and let {G } be a nested sequence of fnte ndex normal subgroups such that G = 1. Then lm b 1 (G ) [G : G ] exsts and equals b (2) 1 (G). Ths theorem s concerned wth the growth rate of b 1 (G ) for fnte ndex subgroups G. Recent work of the author has nstead focused on the growth rate of homology wth coeffcents modulo some prme. Let us fx some termnology. Let F p be the feld of order a prme p. For a group G, let d p (G) be the dmenson of the homology group H 1 (G; F p ). 3

4 The second result formng the bass for Theorem 1.6 s the followng, whch s a consequence of the results of the author n [5]. Theorem Let G be a fntely presented group wth a surjectve homomorphsm φ: G Z. Let G = φ 1 (Z), and let p be a prme. Then 1. lm d p (G )/[G : G ] exsts; 2. ths lmt s postve f and only f d p (G ) s unbounded; 3. f the lmt s postve, then G s large. Thus, fast growth of d p (G ) as a functon of the ndex [G : G ] appears to be a strong and useful property. We say that a nested sequence of fnte ndex subgroups {G } of a group G has lnear growth of mod p homology f nf d p (G )/[G : G ] s strctly postve. A notable stuaton where ths arses s the followng, whch was the man result of [6]. Theorem Let G be a lattce n PSL(2, C) satsfyng one of the followng: 1. G contans a non-trval torson element, or 2. G s arthmetc. Then G contans a strctly nested sequence of fnte ndex subgroups {G } wth lnear growth of mod p homology, for some prme p. It seems very lkely that these lattces are large. But t remans unclear whether the concluson of the theorem s strong enough to mply ths. However, the followng theorem provdes an affrmatve answer when {G } s the derved p-seres for G. Recall that ths s a sequence of fnte ndex subgroups {D (p) (G)} defned recursvely by settng D (p) 0 (G) = G and D(p) +1 (G) = [D(p) (G), D (p) (G)](D (p) (G)) p. Thus, D (p) (G)/D (p) +1 (G) s smply H 1(D (p) (G); F p ). Theorem Let G be a fntely presented group, and let p be a prme. Suppose that the derved p-seres for G has lnear growth of mod p homology. Then G s p-large. Ths has mplcatons for other seres of fnte ndex subgroups of G, for example the lower central p-seres, whch s defned as follows. The frst term γ (p) 1 (G) s G. The remanng terms are defned recursvely, settng γ (p) +1 (G) = [γ(p) (G), G](γ (p) (G)) p. 4

5 Corollary Let G be a fntely presented group, and let p be a prme. Suppose that the lower central p-seres for G has lnear growth of mod p homology. Then G s p-large. The reason that 1.12 mples 1.13 s as follows. Each term D (p) (G) of the derved p-seres contans γ (p) j (G) for all suffcently large j. Ths s because the lower central p-seres of the fnte p-group G/D (p) (G) termnates n the dentty element, snce ths s true for any fnte p-group. Moreover, we have the nequalty d p (D (p) (G)) 1 d p(γ (p) j (G)) 1. [G : D (p) (G)] [G : γ (p) j (G)] Ths s an applcaton of Lemma 3.3 n ths paper, usng the fact that γ (p) j (G) s normal n D (p) (G) and has ndex a power of p. Thus, the assumpton that the lower central p-seres of G has lnear growth of mod p homology mples that the same s true of the derved p-seres. Theorem 1.12 then mples that G s p-large, as requred. It s clear from ths proof that versons of Corollary 1.13 apply to other seres of subgroups, for example, the dmenson subgroups modulo Z p. Theorem 1.12 s a consequence of a more general result, whch we now descrbe. An abelan p-seres for a group G s a sequence of fnte ndex subgroups G = G 1 G 2 G 3... such that G /G +1 s an elementary abelan p-group for each natural number. We nvestgate fntely presented groups G havng an abelan p-seres {G } whch descends as fast as possble, n the sense that the ndex [G : G +1 ] s (approxmately) as bg as t can be. Clearly, the fastest possble descent occurs for the derved p-seres of a non-abelan free group F, of rank n, say. In ths case, [D (p) (F) : D (p) +1 (F)] = pd p(d (p) (F)) = p [F:D(p) (F)](n 1)+1, and so d p (D (p) (F)/D (p) +1 (F)) = [F : D(p) (F)](n 1) + 1. Thus, we say that an abelan p-seres {G } has rapd descent f nf d p (G /G +1 ) [G : G ] > 0. In Sectons 4-7, we wll prove the followng theorems. Theorem Let G be a fntely presented group, and let p be a prme. Then the followng are equvalent: 5

6 1. G s large; 2. some fnte ndex subgroup of G has an abelan p-seres wth rapd descent. Theorem Let G be a fntely presented group, and let p be a prme. Then the followng are equvalent: 1. G s p-large; 2. G has an abelan p-seres wth rapd descent. These theorems represent a sgnfcant mprovement upon the results n [5], and can be vewed as the strongest theorems n ths paper. They are nterestng for two reasons. Frstly, (2) n each theorem does not obvously mply that G has a fnte ndex subgroup wth postve b 1, although ths s of course a consequence of the theorems. Secondly, the proof of these results nvolves some genunely new technques. As n [5] and [7], topologcal and geometrc methods play a central role. But n ths paper, some basc deas from the theory of error-correctng codes are also used. In partcular, we apply a generalsaton of the so-called Plotkn bound [14]. Acknowledgement. I would lke to thank Yehuda Shalom for pontng out Corollary 1.7 to me. 2. Profnte completons and weakly parafree groups Our goal n ths secton s to prove Theorems Our startng pont s the followng result, whch s one of the man theorems n [5]. Theorem 2.1. Let G be a fntely presented group. Then the followng are equvalent: 1. G s large; 2. there exsts a sequence G 1 G 2... of fnte ndex subgroups of G, each normal n G 1, such that () G /G +1 s abelan for all 1; () lm ((log[g : G +1 ])/[G : G ]) = ; () lm sup (d(g /G +1 )/[G : G ]) > 0. Here, d( ) denotes the rank of a group, whch s the mnmal sze of a generatng 6

7 set. Remark 2.2. In the proof of (2) (1), one actually deduces that G admts a surjectve homomorphsm onto a non-abelan free group, for all suffcently large. (See the comments after Theorem 1.2 n [5].) Theorem 1.1 s a rapd consequence of the above result and the followng elementary facts, whch follow mmedately from the defnton of the profnte completon of a group. Proposton 2.3. Let G and K be fntely generated (dscrete) groups, and let φ: Ĝ ˆK be an somorphsm between ther profnte completons. Then the followng hold. 1. There s an nduced bjecton (also denoted φ) between the set of fnte ndex subgroups of G and the set of fnte ndex subgroups of K. 2. If G s any fnte ndex subgroup of G, then G s normal n G f and only f φ(g ) s normal n K. In ths case, there s an nduced somorphsm G/G K/φ(G ), agan denoted φ. 3. If G and G j are fnte ndex subgroups of G, then G G j f and only f φ(g ) φ(g j ). 4. If G G j are fnte ndex subgroups of G, and G s normal n G, then the somorphsm φ: G/G K/φ(G ) sends G j /G to φ(g j )/φ(g ). We can now prove the followng. Theorem 1.1. Let G and K be fntely presented (dscrete) groups that have somorphc profnte completons Ĝ and ˆK. Then G s large f and only f K s large. Proof. Let φ: Ĝ ˆK be the gven somorphsm. Suppose that G s large. It therefore contans a nested sequence of fnte ndex subgroups G satsfyng each of the condtons n Theorem 2.1. These condtons are all detectable by the profnte completon, as follows. Let G be the ntersecton of the conjugates of G. Proposton 2.3 (1) gves fnte ndex subgroups φ(g ) and φ( G ) of K, whch we denote by K and K respectvely. By Proposton 2.3 (2), K s normal n K. By Proposton 2.3 (3), K s contaned n K. By Proposton 2.3 (2) and (4), there s an somorphsm G/ G K/ K whch 7

8 takes G j / G to K j / K for any j. The normalty of G n G 1 s equvalent to the normalty of G / G n G 1 / G. Thus, K s normal n K 1. The somorphsm G/ G +1 K/ K +1 takes G / G +1 and G +1 / G +1 onto K / K +1 and K +1 / K +1 respectvely. Hence, K /K +1 s somorphc to G /G +1. Thus, the sequence K satsfes the condtons of Theorem 2.1. So, K s large. Remark 2.4. A modfed verson of Proposton 2.3 holds, where the phrase profnte completons s replaced by pro-p completons for some prme p, and fnte ndex subgroup(s) s replaced throughout by subnormal subgroup(s) wth ndex a power of p. We now embark upon the proof of Theorem 1.2. For ths, we need the followng varant of Theorem 2.1. Theorem 2.5. Let G be a fntely presented group and let p be a prme. Then the followng are equvalent: 1. G s p-large; 2. there exsts a sequence G 1 G 2... of subgroups of G, each wth ndex a power of p n G, such that G 1 s normal n G, and each G s normal n G 1, and where the followng hold: () G /G +1 s abelan for all 1; () lm ((log[g : G +1 ])/[G : G ]) = ; () lm sup (d(g /G +1 )/[G : G ]) > 0. Proof. (1) (2): Snce G s p-large, some fnte ndex normal subgroup G 1, wth ndex a power of p, admts a surjectve homomorphsm φ onto a non-abelan free group F. Defne the followng subgroups of F recursvely. Set F 1 = F, and let F +1 = [F, F ](F ) p. Set G = φ 1 (F ). Then t s trval to check that the condtons of (2) hold. (2) (1): By Theorem 2.1 and Remark 2.2, some G admts a surjectve homomorphsm φ onto a non-abelan free group F. By assumpton, G s subnormal n G and has ndex a power of p. Set G to be the ntersecton of the conjugates of G. Then G s normal n G and also has ndex a power of p. The restrcton of φ to G s a surjectve homomorphsm onto a fnte ndex subgroup of F, whch s therefore free non-abelan. Thus, G s p-large. 8

9 Theorem 1.2. Let G and K be fntely presented (dscrete) groups that have somorphc pro-p completons for some prme p. Then G s p-large f and only f K s p-large. Proof. Ths s very smlar to the proof of Theorem 1.1. Suppose that G s p-large. It therefore has a sequence of subgroups G where the conclusons of Theorem 2.5 hold. Usng Remark 2.4, K also has such a sequence of subgroups. Thus, by Theorem 2.5, K s p-large. Our am now s to prove Theorem 1.3. Theorem 1.3. Let G be a fntely presented, weakly parafree group wth b 1 (G) > 1. Then G s large. As stated n the Introducton, ths has the followng corollary. Corollary 1.4. Any fntely presented, parafree group s ether large or nfnte cyclc. Proof. Any parafree group G wth b 1 (G) 1 s nfnte cyclc. Theorem 1.3 s a consequence of a stronger result concernng weakly p-parafree groups, whch we now defne. A group G s weakly p-parafree, for some prme p, f there s some non-trval free group F such that G/γ (p) (G) s somorphc to F/γ (p) (F) for each 1. Recall that γ (p) ( ) denotes the lower central p-seres of a group. Proposton 2.6. A weakly parafree group s weakly p-parafree for each prme p. Proof. By assumpton, there s an somorphsm between G/γ (G) and F/γ (F) for some non-trval free group F. Ths nduces an somorphsm between the lower central p-seres of G/γ (G) and F/γ (F). Hence, But γ (p) G/γ (G) γ (p) (G/γ (G)) = F/γ (F) γ (p) (F/γ (F)). (G/γ (G)) s somorphc to γ (p) (G)/γ (G), snce γ (G) s contaned n γ (p) (G). So, the left-hand sde s somorphc to G/γ (p) (G). Smlarly, the rght-hand sde s somorphc to F/γ (p) (F). Thus, we obtan an somorphsm between G/γ (p) (G) and F/γ (p) (F). Note that, when defnng the weakly p-parafree group G, we do not make the assumpton that the somorphsms G/γ (p) 9 (G) F/γ (p) (F) are compatble. Ths

10 means that the followng dagram commutes, for each 2: G/γ (p) (G) F/γ (p) (F) G/γ (p) 1 (G) F/γ(p) 1 (F). Here, the horzontal arrows are the gven somorphsms and the vertcal maps are the obvous quotent homomorphsms. However, we can assume ths, wth no loss, as the followng lemma mples. Lemma 2.7. Let G and K be fntely generated groups. Suppose that, for each nteger 1, there s an somorphsm θ : G/γ (p) (G) K/γ (p) (K). Then there s a collecton of such somorphsms that are compatble. Proof. For each j, θ restrcts to an somorphsm between γ (p) j (K)/γ (p) (K), and hence, quotentng G/γ (p) (G) by γ (p) (G)/γ (p) γ (p) j j (G)/γ (p) (G) and (G), we obtan an somorphsm θ,j : G/γ (p) j (G) K/γ (p) j (K). As G/γ (p) 2 (G) s fnte, some θ,2 occurs nfntely often. Take ths to be the gven somorphsm φ 2 : G/γ (p) 2 (G) K/γ(p) 2 (K), and only consder those θ for whch θ,2 = φ 2. Among these, some θ,3 occurs nfntely often. Defne ths to be φ 3, and so on. Then the φ form the requred compatble collecton of somorphsms. The above lemma s elementary and well-known, as s the followng result. They are ncluded for the sake of completeness. Lemma 2.8. Let G and K be fntely generated groups, and let p be a prme. Then the followng are equvalent: 1. the pro-p completons of G and K are somorphc; 2. for each 1, there s an somorphsm G/γ (p) (G) K/γ (p) (K). Proof. (1) (2): Let Ĝ(p) denote the pro-p completon of G. Then, for each 1, Ĝ (p) /γ (p) (Ĝ(p)) s somorphc to G/γ (p) (G). The clam follows mmedately. (2) (1): The pro-p completon Ĝ(p) can be expressed as the nverse lmt of... G/γ (p) 2 (G) G/γ(p) (G). Ths s because the kernel of any homomorphsm 1 of G onto a fnte p-group contans γ (p) (G) for all suffcently large. Suppose now that, for each 1, there s an somorphsm G/γ (p) (G) K/γ (p) (K). Accordng to Lemma 2.7, these somorphsms can be chosen compatbly. Ths mples there s an 10

11 somorphsm between the nverse lmts: lm G/γ (p) (G) = lmk/γ (p) (K). Thus, Ĝ (p) and ˆK (p) are somorphc. Settng K to be a free group n Lemma 2.8 gves the followng charactersaton of weakly p-parafree groups n terms of pro-p completons. Corollary 2.9. Let G be a fntely generated (dscrete) group and let p be a prme. Then G s weakly p-parafree f and only f ts pro-p completon s somorphc to the pro-p completon of a non-trval free group. Thus, Theorem 1.3 s a consequence of the followng. Theorem Let G be a fntely presented group that s weakly p-parafree for some prme p. Suppose that d p (G) > 1. Then G s p-large. Proof. The assumpton that G s weakly p-parafree mples that the pro-p completon of G s somorphc to the pro-p completon of a free group F, by Corollary 2.9. Snce d p (G) > 1, F s a non-abelan free group. In partcular, t s p-large. Thus, by Theorem 1.2, G s p-large. We close ths secton wth a topologcal applcaton of Theorem 1.2. Theorem 1.5. If two closed 3-manfolds M 1 and M 2 are topologcally Z p -cobordant, for some prme p, then π 1 (M 1 ) s p-large f and only f π 1 (M 2 ) s p-large. Ths s an mmedate consequence of Theorem 1.2 and the followng result. Theorem If two closed 3-manfolds M 1 and M 2 are topologcally Z p - cobordant, for some prme p, then the pro-p completons of ther fundamental groups are somorphc. Ths s, n turn, a consequence of the followng theorem, appled to the homomorphsms π 1 (M 1 ) π 1 (X) and π 1 (M 2 ) π 1 (X) nduced by the ncluson maps of M 1 and M 2 nto X, where X s the 4-manfold n the defnton of Z p -cobordant. Theorem Let φ: G K be a homomorphsm between fntely presented groups that nduces an somorphsm H 1 (G; F p ) H 1 (K; F p ) and a surjecton H 2 (G; F p ) H 2 (K; F p ), for some prme p. Then, the pro-p completons of G and K are somorphc. 11

12 Proof. For brevty, let G = γ (p) (G) and K = γ (p) (K). We wll show that, for each 1, φ nduces an somorphsm G/G K/K. The theorem then follows from Lemma 2.8. We wll prove ths by nducton on. For = 1, t s trval, and for = 2, t follows from the natural somorphsm between G/G 2 and H 1 (G; F p ). So, let us suppose that t s known for a gven nteger 2. In the followng commutatve dagram, the rows are the Stallngs exact sequence [17] and the vertcal homomorphms are nduced by φ. All homology groups are wth respect to mod p coeffcents. H 2 (G) H 2 (G/G ) G /G +1 H 1 (G) H 1 (G/G ) H 2 (K) H 2 (K/K ) K /K +1 H 1 (K) H 1 (K/K ) The frst vertcal homomorphsm s surjectve by hypothess. The second, fourth and ffth are somorphsms by nducton and by hypothess. So, by the 5-lemma, the thrd homomorphsm s an somorphsm. Ths s the frst vertcal map n the followng commutatve dagram: 1 G /G +1 G/G +1 G/G 1 1 K /K +1 K/K +1 K/K 1 Here, the rows are exact and the vertcal homomorphsms are nduced by φ. The thrd vertcal map s also an somorphsm, by nducton, and hence so s the mddle vertcal map. Ths proves the theorem by nducton. 3. Homology growth n cyclc covers The goal of ths secton s to prove the followng result and then Theorem 1.6. Theorem Let G be a fntely presented group wth a surjectve homomorphsm φ: G Z. Let G = φ 1 (Z), and let p be a prme. Then 1. lm d p (G )/[G : G ] exsts; 2. ths lmt s postve f and only f d p (G ) s unbounded; 3. f the lmt s postve, then G s large. We wll need the followng lemma. 12

13 Lemma 3.1. Let k be a non-negatve nteger and let f: N >0 R be a functon satsfyng for any, j N. Then 1. lm f / exsts; f( + j) f() f(j) k, 2. ths lmt s non-zero f and only f f s unbounded. Proof. Note frst that a trval nducton establshes that f() (f(1) + k) for each postve N. Hence lm sup f()/ s fnte, M say. Clam. Suppose that f(m) > 2k, for some postve m N. Then lm nf f()/ > 0. We wll show that f(nm) > (n + 1)k, for each postve n N, by nducton on n. The nducton starts trvally. For the nductve step, note that f((n + 1)m) f(m) + f(nm) k > 2k + (n + 1)k k = (n + 2)k. Ths establshes the nequalty. The clam now follows by notng that f = nm + r, for 0 r < m, then f() f(nm) k + max 1 r<m f(r). ( ) Now let g() = M f(). By the defnton of M, lm nf g()/ = 0. Now, g satsfes ( ) and so by the clam, g s bounded above by 2k. Hence, f() M 2k. Thus, lm nf f()/ = M. Ths proves (1). To prove (2), note that f f satsfes ( ), then so does f. Thus, applyng the clam, we deduce that ether f() 2k for all postve or lm f()/ s non-zero. Proof of Theorem We clam that there s a non-negatve nteger k such that, for all, j 1, d p (G +j ) d p (G ) d p (G j ) k. Ths and Lemma 3.1 wll then mply (1) and (2). Let K be a fnte connected 2-complex wth fundamental group G. We may fnd a map F: K S 1 so that F : π 1 (K) π 1 (S 1 ) s φ: G Z. Let b be a pont n S 1. Then, after a small homotopy, we may assume that F 1 (b) s a fnte graph Γ, that a regular neghbourhood of Γ s a copy of Γ I and that the restrcton of F to ths neghbourhood s projecton onto the I factor, followed by ncluson of I nto S 1. Let K be the result of cuttng K along Γ. Let K be the -fold cover of K correspondng to G. Then K can be obtaned from copes of K glued together n a crcular 13

14 fashon. Cut K along one of the copes of Γ n K, and let K be the result. The Mayer-Vetors sequence (appled to the decomposton of K nto K and Γ I) gves the followng nequaltes: d p (Γ) d p (K ) d p (K ) Γ. Smlarly, snce the dsjont unon of K and K j s obtaned by cuttng K +j along a copy of Γ, we have d p (Γ) d p (K +j ) d p (K ) d p (K j ) Γ. Snce d p (K ) = d p (G ), the clam now follows, lettng k = 2d p (Γ) + 2 Γ. Concluson (3) s a consequence of the followng result (Theorem 1.2 of [5]), settng H = G, J = G and K = [G, G ](G ) p. Theorem 3.2. Let G be a fntely presented group and suppose that, for each natural number, there s a trple H J K of fnte ndex normal subgroups such that () H /J s abelan for all ; () lm ((log[h : J ])/[G : H ]) = ; () nf (d(j /K )/[G : J ]) > 0. Then, K admts a surjectve homomorphsm onto a non-abelan free group, for all suffcently large. To prove Theorem 1.6, we need one more fact, whch s well known. It appears as Proposton 3.7 n [6], for example. Lemma 3.3. Let G be a fntely generated group and let K be a normal subgroup wth ndex a power of a prme p. Then d p (K) (d p (G) 1)[G : K] + 1. We can now prove Theorem 1.6. Theorem 1.6. Let G be a fntely presented (dscrete) group, that s vrtually resdually p-fnte, for some prme p, and such that b (2) 1 (G) > 0. Then G s large. 14

15 Proof. Snce G s vrtually resdually p-fnte, t has a fnte ndex normal subgroup G 1 that s resdually p-fnte. Thus, G 1 contans a nested sequence of normal subgroups G, each wth ndex a power of p, such that G = 1. By the multplcatvty of b (2) 1 (Theorem 1.7 (1) of [13]), b (2) 1 (G 1) = b (2) 1 (G)[G : G 1] > 0. By Lück s theorem (Theorem 1.9), lm b 1 (G )/[G 1 : G ] exsts and equals b (2) 1 (G 1), whch s postve. Hence, by relabellng the G, we may assume that b 1 (G 1 ) > 0. Let φ: G 1 Z be a surjectve homomorphsm, and let K = φ 1 (p Z). We clam that lm nf d p (K )/[G : K ] s postve. Consder the subgroups G K. Each s a fnte ndex normal subgroup of G 1 and ther ntersecton s the dentty. Hence, by Theorem 1.9, lm b 1 (G K )/[G 1 : G K ] exsts and s postve. In partcular, lm nf d p (G K )/[G : G K ] s postve. Now, the quotent K /(G K ) s somorphc to K G /G, whch s a subgroup of G 1 /G, and so has order a power of p. Hence, by Lemma 3.3, d p (G K ) [G : G K ] (d p(k ) 1)[K : G K ] + 1. [G : G K ] Therefore, lm nf d p (K )/[G : K ] s postve, as clamed. In partcular, d p (K ) s unbounded. Thus, by Theorem 1.10, G s large. 4. Cocycle sze and Property (τ) Most of the remander of the paper s devoted to the proof of Theorems 1.14 and Theorem Let G be a fntely presented group and let p be a prme. Then the followng are equvalent: 1. G s large; 2. some fnte ndex subgroup of G has an abelan p-seres wth rapd descent. Theorem Let G be a fntely presented group, and let p be a prme. Then the followng are equvalent: 1. G s p-large; 2. G has an abelan p-seres wth rapd descent. The dffcult drecton n each of these theorems s (2) (1). For ths, one needs a method for provng that a fntely presented group s large or p-large. Varous 15

16 technques have been developed wth ths am. The one we wll use deals wth the relatve sze of cocycles on 2-complexes. Let K be a connected fnte 2-complex wth fundamental group G. Let K be the coverng space correspondng to a fnte ndex subgroup G. The key to our approach s to consder cellular 1-cocycles on K representng non-trval elements of H 1 (K ; F p ). For a cellular 1-dmensonal cocycle c on K, let ts support supp(c) be those 1-cells wth non-zero evaluaton under c. For an element α H 1 (K ; F p ), consder the followng quantty, whch was defned n [7]. The relatve sze of α s mn{ supp(c) : c s a cellular cocycle representng α} relsze(α) =. Number of 1-cells of K The followng result was proved n [7] and s central to our approach. Theorem 4.1. Let K be a fnte connected 2-complex, and let {K K} be a collecton of fnte-sheeted coverng spaces. Suppose that {π 1 (K )} has lnear growth of mod p homology for some prme p. Then one of the followng must hold: () π 1 (K ) s p-large for nfntely many, or () there s some ǫ > 0 such that the relatve sze of any non-trval class n H 1 (K ; F p ) s at least ǫ, for all. Ths s a slghtly modfed verson of Theorem 6.1 of [7]. In that result, t s not explctly stated that π 1 (K ) s p-large for nfntely many, merely that π 1 (K) s large. But the proof does ndeed gve a normal subgroup of π 1 (K ) wth ndex a power of p that has a free non-abelan quotent. In ths secton, we wll relate the relatve sze of cocycles to Property (τ). Whle not drectly needed n the remander of the paper, ths s a potentally mportant lnk. We now recall the defnton of Property (τ). Let G be a group wth a fnte generatng set S. Let {G } be a collecton of fnte ndex subgroups of G. Let X = X(G/G ; S) be the Schreer coset graph of G/G wth respect to the generatng set S. The Cheeger constant h(x ) s defned to be { } A h(x ) = mn : A V (X ), 0 < A V (X ) /2. A 16

17 Here, V (X ) denotes the vertex set of X, and for a subset A of V (X ), A denotes the set of edges wth one endpont n A and one not n A. Then G has Property (τ) wth respect to {G } f nf h(x(g/g ; S)) > 0. Ths turns out not to depend on the choce of fnte generatng set S. When a group G has Property (τ) wth respect to an nfnte collecton of fnte ndex subgroups {G }, there are lots of nce consequences. For example, the resultng Schreer coset graphs have applcatons n theoretcal computer scence and codng theory. But when a group does not have Property (τ), there are other useful conclusons one can often draw, whch we now brefly descrbe. Ths s partcularly the case n low-dmensonal topology, where the followng mportant conjecture remans unresolved. Conjecture. [10] (Lubotzky-Sarnak) Let G be the fundamental group of a fntevolume hyperbolc 3-manfold. Then G does not have Property (τ) wth respect to some collecton {G } of fnte ndex subgroups. To apprecate the sgnfcance of ths conjecture, note the followng theorem, whch appears as Corollary 7.4 n [9]. Theorem. [9] The Lubotzky-Sarnak conjecture and the Geometrsaton Conjecture together mply that any arthmetc lattce n PSL(2, C) s large. Thus, t s mportant to develop new methods for showng that a group does not have Property (τ). The followng result, whch s the man theorem n ths secton, may be a useful tool. Theorem 4.2. Let K be a fnte connected 2-complex wth fundamental group G. Let {G } be a collecton of fnte ndex subgroups of G, and let {K } be the correspondng coverng spaces of K. Suppose that there s a prme p and, for each, there s a non-trval class α n H 1 (K ; F p ), such that relsze(α ) 0 as. Let G be the kernel of the homomorphsm G Z/pZ nduced by α. Then G does not have Property (τ) wth respect to { G }. In the proof of ths theorem, we wll need the followng constructon, whch wll also be mportant later n the paper. Let K be a fnte connected 2-complex wth some 0-cell as a basepont b. Let c be a cocycle on K representng a non-trval element of H 1 (K; F p ) and let ( K, b) be a (based) coverng space of (K, b). Suppose 17

18 that π 1 ( K, b) les n the kernel of the homomorphsm π 1 (K, b) Z/pZ determned by c. Then one can defne, for any 0-cell v of K, ts c-value c(v), whch s an nteger mod p. It s defned as follows. Pck a path β from b to v n the 1-skeleton of K, project t to a path n K and defne c(v) to be the evaluaton of c on ths path. To see that ths s ndependent of the choce of β, let β be any other path n K from b to v n the 1-skeleton of K. Then β.β 1 s a closed loop n K. Ths projects to a closed loop n K. By our hypothess on the coverng space K, the evaluaton of c on any such closed loop s trval. Ths mmedately mples that c(v) s ndeed well-defned. An example s useful. Let K be the wedge of 3 crcles, labelled x, y and z. Let c be the mod 2 cocycle supported on the x labelled edge. Let K be the coverng space correspondng to the second term of the derved 2-seres of K. Ths s shown n Fgure 1. (Note that the dotted edges jon up wth each other.) The c-value of ts vertces s shown b ~ 1 1 y z x Fgure 1. Lemma 4.3. Let K be a fnte connected 2-complex, and let α be a non-trval element of H 1 (K; F p ). Let K be a fnte-sheeted coverng space of K, such that π 1 ( K) les n the kernel of the homomorphsm π 1 (K) Z/pZ determned by α. Let X be the 1-skeleton of K. Let V (K) and E(K) be the 0-cells and 1-cells of K respectvely. Then h( X) E(K) V (K) /p relsze(α). 18

19 Proof. Let c be a cocycle on K representng α and for whch supp(c) s mnmal. Let A be the vertces n K wth zero c-value. We clam that A = V ( K) /p. Let l be any loop n K based at the basepont b such that c(l) = 1. Then [l] represents an element of the coverng group π 1 (K)/π 1 ( K), whch ncreases the c-value of every vertex by 1 modulo p. Hence, the number of vertces wth gven c-value s V ( K) /p, whch proves the clam. As a consequence, A = d V (K) /p, where d s the degree of the cover K K. Any edge n A must le n the nverse mage of the support of c. Thus, A d supp(c) = d E(K) relsze(α). So, h( X) A A and the requred formula now follows. d E(K) d V (K) /p relsze(α), Proof of Theorem 4.2. Let X be the 1-skeleton of the coverng space of K correspondng to G. By Lemma 4.3, h( X ) E(K ) V (K ) /p relsze(α ) = E(K) V (K) /p relsze(α ). Snce we are assumng that the relatve sze of α tends to zero, and snce the other terms on the rght-hand sde of the above formula depend only on K, we deduce that h( X ) 0. Hence, G does not have Property (τ) wth respect to { G }. 5. Cocycles n coverng spaces Our am over the next few chapters s to prove (2) (1) of Theorems 1.14 and 1.15, and thereby establsh that the group G n these theorems s large or p-large as approprate. The proof wll be topologcal, and so we consder a fnte connected 2-complex K wth fundamental group G. We are assumng that G has a fnte ndex subgroup G 1 wth a rapdly descendng p-seres G. (In the proof of Theorem 1.15, take G 1 to be G.) Let K be the correspondng coverng spaces of K. Theorem 4.1 gves a crteron for establshng that G 1 s p-large, n terms of the exstence of 1-cocycles on K wth relatve sze tendng to zero. We therefore, n ths secton, nvestgate how 1-cocycles on a 2-complex can be used to construct 1-cocycles n coverng spaces (wth potentally smaller relatve sze). If U s a set of 1-cocycles on a cell complex K, we defne ts support supp(u) to be the unon of the supports of the cocycles n U. Our man result s the followng. Theorem 5.1. Let K be a fnte connected 2-complex wth r 2-cells. Let U be a collecton of cocycles on K that represent lnearly ndependent elements of H 1 (K; F p ). 19

20 Let u = U. Let q: K K be a fnte regular cover such that π 1 (K)/π 1 ( K) s an elementary abelan p-group wth rank n. Then there s a collecton Ũ of cocycles on K representng lnearly ndependent elements of H 1 ( K; F p ) such that 1. supp(ũ) q 1 (supp(u)); 2. Ũ (n u)u r. The pont behnd Theorem 5.1 s that t provdes not just a lower bound on the dmenson of H 1 ( K; F p ) but also gves nformaton about certan cocycles on K representng ths cohomology. We now embark on the proof of Theorem 5.1. Consder the followng subspaces of H 1 (K; F p ): 1. the space spanned by the elements of U; 2. the classes that have trval evaluaton on all elements of π 1 ( K). Let V 1 and V 2 be these two subspaces. Then the dmensons of V 1 and V 2 are u and n respectvely. Pck a complementary subspace for V 1 V 2 n V 2, and let C be a set of cocycles on K that represents a bass for ths subspace. Note that C n u. Note also that, by constructon, C U forms a lnearly ndependent set of elements n H 1 (K; F p ). For each c 1 U and c 2 C, we wll show how to construct a cochan on K, whch we denote c 1 c 2. These cochans wll play a vtal role n the proof of Theorem 5.1. Pck an orentaton on each of the 1-cells of K. Ths pulls back to gve an orentaton on each 1-cell e of K. Let (e) denote the ntal vertex of e. Let c 1 be the nverse mage of c 1 n K. Ths s a cocycle on K. Snce each 1-cell e s orented, c 1 (e) s a well-defned element of F p. Fx a basepont b n the 0-skeleton of K, and let b be a basepont for K n the nverse mage of b. Recall from Secton 4 that each vertex v of K then has a well-defned c 2 -value, denoted by c 2 (v). We now defne c 1 c 2. Snce the edges of K are orented, t suffces to assgn an 20

21 nteger (c 1 c 2 )(e) modulo p to each edge e. We defne ths to be (c 1 c 2 )(e) = c 1 (e) c 2 ((e)), where the product s multplcaton n F p. Note that supp(c 1 c 2 ) supp( c 1 ) q 1 (supp(u)). It may be helpful to consder the case where K s the wedge of 3 crcles labelled x, y and z, where p = 2 and where π 1 ( K) s the second term n the derved 2-seres for π 1 (K). Let c 1 and c 2 be the cochans supported on the x-labelled and y-labelled edges of K, respectvely. Then the followng s a dagram of K, and the edges n the support of c 1 c 2 are shown n bold. y z x b ~ Fgure 2. We denote by U C the space of cochans on K spanned by elements c 1 c 2, where c 1 U and c 2 C. Let Z 1 ( K) denote the space of 1-cocycles on K wth mod p coeffcents. We wll establsh the followng. Clam 5.2. The dmenson of U C s U C, whch s at least u(n u). Clam 5.3. The subspace Z 1 ( K) U C of cocycles n U C has codmenson at most r (the number of 2-cells of K). Clam 5.4. The map from Z 1 ( K) U C nto H 1 ( K; F p ) s an njecton. Thus, settng Ũ to be a bass for Z1 ( K) U C wll establsh Theorem

22 Lemma 5.5. Let g and h be loops n K based at the same pont. Let [g, h] denote any lft of ghg 1 h 1 to K. Then (c 1 c 2 )([g, h]) = c 1 (h)c 2 (g) c 1 (g)c 2 (h), where equalty s n F p. Proof. Each letter n the word g corresponds to an edge e, say, n the frst part of the loop [g, h]. The nverse of ths letter appears n g 1, where the loop runs over the edge e. Between the vertces (e) and (e ) s a word w conjugate to h. We clam that c 2 ((e )) c 2 ((e)) = c 2 (h). Pck a path from the basepont b of K to (e). Then c 2 ((e)) s the evaluaton under c 2 of the projecton of ths path to K. If we extend ths path usng the word w, we obtan a path from b to (e ). Thus, the dfference between c 2 ((e )) and c 2 ((e)) s the evaluaton of c 2 on the projecton of w. The parts of g and g 1 n w project to the same edges n K, but wth reverse orentatons. Hence, c 2 ((e )) c 2 ((e)) equals c 2 (h), as requred. Therefore, the evaluaton of the loop [g, h] along the edges n g and g 1 s c 1 (g)c 2 (h) n total. Smlarly, along the edges n h and h 1, t s c 1 (h)c 2 (g). So, the total evaluaton s c 1 (h)c 2 (g) c 1 (g)c 2 (h), as requred. Let C 1 ( K) and B 1 ( K) be the space of 1-cochans on K, wth mod p coeffcents, and the subspace of coboundares. Lemma 5.6. The cochans {c 1 c 2 : c 1 U, c 2 C} map to lnearly ndependent elements n C 1 ( K)/B 1 ( K). Proof. Snce U C forms a lnearly ndependent set of classes n H 1 (K; F p ), there are loops l n K, based at the basepont of K, where U C, such that, for all c U C, c(l ) = { 1 f = c 0 otherwse. Let U and j C. Then, by Lemma 5.5, for any c 1 U and c 2 C, (c 1 c 2 )([l j, l ]) = c 1 (l )c 2 (l j ) c 1 (l j )c 2 (l ) = { 1 f = c1 and j = c 2 ; 0 otherwse. Snce every element of B 1 ( K) has trval evaluaton on any loop n K, we deduce the lemma. Lemma 5.6 mples Clam 5.2. It also mples that the restrcton of the quotent homomorphsm C 1 ( K) C 1 ( K)/B 1 ( K) to U C s an njecton. Thus, t s an 22

23 njecton on any subspace of U C. Ths gves Clam 5.4. We now verfy Clam 5.3. Lemma 5.7. Let l and l be the boundary loops of two 2-cells of K that dffer by a coverng transformaton of K. Then, (c 1 c 2 )(l ) = (c 1 c 2 )(l). Proof. Let g be a path n the 1-skeleton of K from the basepont of l to the basepont of l. Thus, the loop glg 1 runs from the basepont of l to the basepont of l, then goes around l and then returns to the basepont of l. Snce g and l project to loops n K based at the same pont, Lemma 5.5 gves that (c 1 c 2 )([g, l]) = c 1 (l)c 2 (g) c 1 (g)c 2 (l). Ths s zero because l s the boundary of a 2-cell and so has zero evaluaton under the cocycles c 1 and c 2. So, (c 1 c 2 )(l ) = (c 1 c 2 )([g, l]) + (c 1 c 2 )(l) = (c 1 c 2 )(l). The cocycles n U C are precsely those cochans n U C that have zero evaluaton on the boundary of any 2-cell n K. But Lemma 5.7 states that f two 2-cells dffer by a coverng transformaton, then they have the same evaluaton. Thus, one need only check the evaluaton of the boundary of just one 2-cell n each orbt of the coverng acton. There are precsely r such orbts, where r s the number of 2-cells n K. Thus, the codmenson of Z 1 ( K) U C n U C s at most r. Ths proves Clam 5.3 and hence Theorem 5.1. Remark 5.8. Although Theorem 5.1 suffces for the purposes of ths paper, t s possble to strengthen t a lttle. One can n fact fnd a set Ũ satsfyng the requrements of Theorem 5.1, but wth the stronger nequalty d(d + 1) Ũ un r, 2 where d = dm(v 1 V 2 ). Ths s proved as follows. Pck a bass for V 1 + V 2 so that t contans a bass for V 1 V 2, a bass for V 1 and a bass for V 2. Pck a total order 23

24 on the bass for V 1 V 2. Then consder all cochans c 1 c 2, where c 1 les n the bass for V 1, c 2 les n the bass for V 2, and c 1 < c 2 f c 1 and c 2 both le n V 1 V 2. The number of such cochans s un d(d + 1)/2. It s possble to prove the correspondng versons of 5.2, 5.3 and 5.4 for these cochans. Hence, the requred nequalty follows. Remark 5.9. The cochans c 1 c 2 we have consdered n ths secton are, n fact, specal cases of a much more general constructon. In [8], a more general class of cochan was used to provde new lower bounds on the homology growth and subgroup growth of certan groups. These more general cochans had a certan nteger l, known as ther level, assgned to them. The cochans c 1 c 2 are those wth level one. 6. The subspace reducton theorem Let E be a fnte set, and let F E p be the vector space over F p consstng of functons E F p. The support of an element φ of F E p s supp(φ) = {e E : φ(e) 0}. The support of a subspace W of F E p s supp(w) = supp(φ). φ W The man example we wll consder s where E s the set of 1-cells n a fnte 2-complex K (wth some gven orentatons). Then F E p s just C 1 (K), the space of 1-cochans on K. Recall that our goal s to fnd cocycles representng non-trval elements of H 1 (K; F p ) and wth small relatve sze. The followng result, whch s the man theorem of ths chapter, wll be the tool we use. Theorem 6.1. Let V be a subspace of F E p wth dmenson v, and let w be a postve nteger strctly less v. Then, V contans a subspace W wth dmenson w such that supp(w) pv p v w p v 1 supp(v ) pw+1 p p w+1 supp(v ). 1 In our case, V wll be a subspace of C 1 (K) spanned by v cocycles, representng lnearly ndependent elements of H 1 (K; F p ). We wll use Theorem 6.1 to pass to a set of w cocycles (where w s a fxed nteger less than v) spannng a subspace W wth 24

25 support whch s smaller than the support of V by a defnte factor, ndependent of v. We now embark on the proof of Theorem 6.1. The followng lemma gves a formula relatng the support of a subspace to the support of each of ts elements. Lemma 6.2. For a non-zero subspace W of F E p, supp(w) = 1 (p 1)p dm(w) 1 φ W supp(φ). Proof. Focus on an element e E n the support of W. Then, there s a ψ n W such that ψ(e) 0. Decompose W as a drect sum ψ W. Then we may express W as a unon of translates of W, as follows: p 1 W = (ψ + W ). =0 Now, for any element φ W, ψ(e) + φ (e) = 0 for exactly one value of between 0 and p 1. Denote the ndcator functon of an element φ n F E p by I φ: E {0, 1}. Ths s defned as follows: I φ (e) = { 0 f φ(e) = 0; 1 otherwse. Then, φ W I φ (e) = φ W =0 p 1 I φ +ψ (e) = (p 1)p dm(w) 1. Summng ths over all e n the support of W gves as requred. supp(φ) = (p 1)p dm(w) 1 supp(w), φ W Theorem 6.3. Let V be a non-zero subspace of F E p wth dmenson v. Then there s a codmenson one subspace W of V such that supp(w) pv p p v supp(v ). 1 25

26 Proof. Note that the theorem holds trvally f v = 1. We therefore assume v 2. There are (p v 1)/(p 1) codmenson one subspaces W of V. Summng the formula of Lemma 6.2 over each of these gves: 1 supp(w) = (p 1)p v 2 supp(φ). W The number of tmes a non-zero element φ of V appears n the sum W s ndependent of the element φ. Snce there are (p v 1)/(p 1) codmenson one W φ W φ W subspaces W, each contanng p v 1 1 non-zero elements, and there are p v 1 nonzero elements of V, the number of tmes a non-zero element φ of V appears n the sum W Hence, φ W By Lemma 6.2, Hence, s therefore (p v 1) (p 1) supp(w) = W (p v 1 1) (p v 1) = pv 1 1 p 1. 1 (p v 1 1) (p 1)p v 2 supp(φ). (p 1) φ V supp(φ) = (p 1)p v 1 supp(v ). φ V W supp(w) = (pv p) supp(v ). (p 1) The average, over all W, of supp(w) s therefore p v p p v supp(v ). 1 Hence, there s a codmenson one subspace W wth support at most ths sze. Proof of Theorem 6.1. We prove the frst nequalty by nducton on the codmenson v w. When ths quantty s 1, ths s Theorem 6.3. For the nductve step, suppose that we have a subspace W of V wth dmenson w + 1 such that supp(w ) pv p v w 1 p v supp(v ). 1 By Theorem 6.3, W has a subspace W wth dmenson w such that supp(w) pw+1 p p w+1 1 supp(w ) ( p w+1 ) ( p p v p v w 1 ) p w+1 1 p v supp(v ) 1 = pv p v w p v supp(v ), 1 26

27 as requred. The second nequalty s trval, because p v p v w / p w+1 p p v 1 p w+1 1 = pv w 1 (p w+1 1) p v = pv p v w 1 1 p v 1 1. It s nstructve to consder the case w = 1 n Theorem 6.1. Ths states that n any subspace V of F E p wth dmenson v > 1, there s an element wth at most p v p v 1 p v 1 E p2 p p 2 1 E non-zero co-ordnates. Ths s a theorem n the theory of error-correctng codes, known as the Plotkn bound [14]. For, a lnear code s just a subspace of F E p, and the Hammng dstance of such a code s the mnmal number of non-zero co-ordnates n any non-zero element of the subspace. Thus, Theorem 6.1 can be vewed as a generalsaton of the Plotkn bound, gvng nformaton not just about elements of V but whole subspaces. It s probably well-known to experts on error-correctng codes. 7. Proof of Theorems 1.14 and 1.15 One drecton of Theorems 1.14 and 1.15 s easy: the mplcaton (1) (2). The proof s as follows. Suppose that φ: G 1 F s a surjectve homomorphsm from a fnte ndex subgroup of G onto a non-abelan free group F. For the proof of Theorem 1.15, assume n addton that G 1 s normal n G and has ndex a power of p. Let {F } be the derved p-seres of F, and let G = φ 1 (F ). Snce {F } s an abelan p-seres for F wth rapd descent, {G } s therefore an abelan p-seres wth rapd descent, as requred. The dffcult part of Theorems 1.14 and 1.15 s the mplcaton (2) (1). So, suppose that some fnte ndex subgroup G 1 of G has an abelan p-seres {G } wth rapd descent. In the proof of 1.15, take G 1 to be G. We wll show that G 1 s p-large, whch wll establsh the theorems. Snce d p (G /G +1 ) d p (G ), the rapd descent of {G } mples that t has lnear growth of mod p homology. Let K be a connected fnte 2-complex wth fundamental group G. Let K be the fnte-sheeted coverng space correspondng to the subgroup G. Recall from Secton 4 the defnton of the relatve sze of an element of H 1 (K ; F p ), and the followng result. 27

28 Theorem 4.1. Let K be a fnte connected 2-complex, and let {K K} be a collecton of fnte-sheeted coverng spaces. Suppose that {π 1 (K )} has lnear growth of mod p homology for some prme p. Then one of the followng must hold: () π 1 (K ) s p-large for nfntely many, or () there s some ǫ > 0 such that the relatve sze of any non-trval class n H 1 (K ; F p ) s at least ǫ, for all. Thus, our plan s to prove that () of Theorem 5.1 does not hold, and therefore deduce that G 1 s p-large. We wll keep track of a set U of cellular 1-dmensonal cocycles on K that represent lnearly ndependent elements of H 1 (K ; F p ). The cardnalty U wll be some fxed postve nteger u ndependent of. (The precse sze of u wll depend on data from the group G and the seres {G }.) Our am s to ensure that supp(u ) 0. ( ) Number of 1-cells of K In partcular, the relatve sze of any element of U tends to zero, whch means that () does not hold. We establsh ( ) usng the followng method. Let q : K +1 K be the coverng map. We wll fnd a set of cocycles U + +1 on K +1 representng lnearly ndependent elements of H 1 (K +1 ; F p ), wth the followng propertes: I. supp(u + +1 ) q 1 (supp(u )); II. U + +1 > u. Note that the nequalty n (II) s strct. Let E denote the set of 1-cells of K +1 wth gven orentaton. Then C 1 (K +1 ) s somorphc to F E p, the vector space of functons E F p. Let V be the subspace of C 1 (K +1 ) spanned by U + +1, and let w = u. We apply Theorem 6.1 to V. Theorem 6.1. Let V be a subspace of F E p wth dmenson v, and let w be a postve nteger strctly less v. Then, V contans a subspace W wth dmenson w such that supp(w) pv p v w p v 1 supp(v ) pw+1 p p w+1 supp(v ). 1 Let U +1 be a bass for the subspace W gven by Theorem 6.1. Note that the factor (p w+1 p)/(p w+1 1) s strctly less than 1, and s dependent only on the 28

29 fxed ntegers u and p. Now, q 1 (supp(u )) s obtaned from supp(u ) by scalng by the degree of the cover q. The same relaton holds between the number of 1-cells n K +1 and the number of 1-cells n K. Thus, ( ) follows. The key, then, s to construct the cocycles U + +1 wth propertes (I) and (II). For ths, we use Theorem 5.1 (settng K = K, U = U and K = K +1 ) Theorem 5.1. Let K be a fnte connected 2-complex wth r 2-cells. Let U be a collecton of cocycles on K that represent lnearly ndependent elements of H 1 (K; F p ). Let u = U. Let q: K K be a fnte regular cover such that π 1 (K)/π 1 ( K) s an elementary abelan p-group wth rank n. Then there s a collecton Ũ of cocycles on K representng lnearly ndependent elements of H 1 ( K; F p ) such that 1. supp(ũ) q 1 (supp(u)); 2. Ũ (n u)u r. We defne U + +1 to be the set Ũ provded by ths theorem. Condton (1) of the theorem s just (I) above. We need to ensure that Condton (II) holds. Thus, we requre (n u)u r > u. We wll ensure that ths holds by usng the hypothess that G has rapd descent and by a sutable choce of u. Let X R be a fnte presentaton for G. Let λ be lm nf d p (G /G +1 ) 2. [G : G ] Snce {G } s rapdly descendng, λ s postve. Let u be 4 R λ whch s a postve nteger. Now pck a suffcently large nteger j, such that, for all j, and, d p (G /G +1 ) 2 [G : G ] λ[g : G ] > 4u. 29 > λ 2

30 Hence, Now, λ[g : G ]/2 u > λ[g : G ]/4. n = d p (G /G +1 ) λ[g : G ]/ So, (n u)u r (λ[g : G ]/2 + 2 u)u r λ[g : G ]u/4 + 2u r 2u, snce λu 4 R = 4r/[G : G ]. Ths proves (2) (1) of Theorems 1.14 and References 1. G. Baumslag, Groups wth the same lower central sequence as a relatvely free group. I. The groups. Trans. Amer. Math. Soc. 129 (1967) G. Baumslag, Groups wth the same lower central sequence as a relatvely free group. II. Propertes. Trans. Amer. Math. Soc. 142 (1969) D. Cooper, D. Long, A. Red, Essental closed surfaces n bounded 3-manfolds, J. Amer. Math. Soc. 10 (1997) M. Edjvet, S. Prde, The concept of largeness n group theory II, n Groups - Korea 1983, Lecture Notes n Math. 1098, Sprnger, Berln, 1984, pp M. Lackenby, A charactersaton of large fntely presented groups, J. Algebra. 287 (2005) M. Lackenby, Coverng spaces of 3-orbfolds, Duke Math. J. 136 (2007) M. Lackenby, Large groups, Property (τ) and the homology growth of subgroups, Preprnt. 8. M. Lackenby, New lower bounds on subgroup growth and homology growth, Preprnt. 9. M. Lackenby, D. Long, A. Red, Coverng spaces of arthmetc 3-orbfolds, Int. Math. Res. Not. (to appear) 30