We prove that given a finitely generated group G with a homomorphism of G onto G × H, H non-trivial, or a finitely generated group G with a homomorphism of G onto G × G, we can always find normal subgroups N ≠ G such that G/N G/N × H or G/N G/N × G/N respectively. We also show that given a finitely presented non-Hopfian group U and a homomorphism φ of U onto U, which is not an isomorphism, we can always find a finitely presented group H ⊇ U and a finitely generated free group F such that φ induces a homomorphism of U * F onto (U * F) × H. Together with the results above this allows the construction of many examples of finitely generated groups G with G G × H where H is finitely presented. A finitely presented group G with a homomorphism of G onto G × G was first constructed by Baumslag and Miller. We use a slight generalisation of their method to obtain more examples of such group

Hirshon, Ron

Meier, David

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BULL. AUSTRAL. MATH. SOC. 20F05 , 20E06, 20E34VOL. 45 (1992) [513-520]GROUPS WITH A QUOTIENT THAT CONTAINS THE ORIGINALGROUP AS A DIRECT FACTORR O N HIRSHON AND DAVID MEIERWe prove that given a finitely generated group G with a homomorphism of G ontoG X H, H non-trivial, or a finitely generated group G with a homomorphism ofG onto G X G, we can always find normal subgroups N ^ G such that G/N ==G/N X H or G/N ^ G/N X G/N respectively. We also show that given a finitelypresented non-Hopfian group U and a homomorphism tp of U onto U, which isnot an isomorphism, we can always find a finitely presented group H D U and afinitely generated free group F such that tp induces a homomorphism of U * Fonto (U * F) X H. Together with the results above this allows the construction ofmany examples of finitely generated groups G with G = GxH where H is finitelypresented. A finitely presented group G with a homomorphism of G onto G X Gwas first constructed by Baumslag and Miller. We use a slight generalisation oftheir method to obtain more examples of such groups.1. INTRODUCTIONFinitely generated groups G with G = G x H, H^1,OTG = GXG werefirst constructed by Tyrer Jones [4], but it is still an open question whether there existnon-trivial finitely presented groups G with these properties. More insight into thestructure and a new construction of finitely generated groups with the above propertiesare therefore still useful. The starting point of our investigations was the constructionof a finitely presented group G with a homomorphism of G onto G x G by Baumslagand Miller [1].Our paper consists of four sections. The main results are the following theoremswhich are proved in Sections 3 and 5 respectively. Theorem 2: A group G with ahomomorphism of G onto G x H always contains a normal subgroup N such thatG/N = G/N x H. Theorem 4: A non-trivial group G with a homomorphism from Gonto GxG always contains a normal subgroup N ^ G such that G/N == G/N x G/N.Since in Sections 3 and 5 we need finitely presented groups G with homomorphismsof G onto G x H or of G onto G x G, we study the construction of such groups inReceived 28 June 1991We would like to thank the referee for his constructive comments to an earlier version of our paperwhich have improved its presentation.Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/92 $A2.00+0.00.513available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700030422Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 10 Jul 2017 at 15:11:06, subject to the Cambridge Core terms of use,514 R. Hirshon and D. Meier [2]Sections 2 and 4. In Section 2 we use ideas contained in [3] to give a fairly generalconstruction for groups G with a homomorphism of G onto G x H. More precisely, weshow that given a finitely presented non-Hopfian group U and a homomorphism ip ofU onto U, which is not an isomorphism, we can always find a finitely presented groupH ~DU and a finitely generated free group F such that ip induces a homomorphism •&of U * F onto (U * F) x H. We use this to give an example of a one-relator group Gwith a homomorphism of G onto G x H, H ^ 1. Together with Theorem 2 this allowsthe construction of many examples of finitely generated groups G with G = G x Hwhere H is finitely presented. The construction of groups G with a homomorphism ofG onto G x G given in Section 4 is just a slight generalisation, again based on ideas in[3], of the Baumslag-Miller group [1]. We start with a finitely generated abelian groupA and a one-one homomorphism fi of A to A satisfying some additional conditionsand construct a finitely presented group G ^ A and a homomorphism •& from G ontoG x G induced by fi.2. FINITELY PRESENTED GROUPS G WITH A HOMOMORPHISMFROM G ONTO G x HThe following lemma is needed in the proof of Theorem 1.LEMMA 1. Let U be finitely presented and 1 ^ x e U. Then there exists afinitely presented group H ^ U, such that x generates H as a normal subgroup.A proof can, for example, be modelled after the proof of Theorem 3.5 of [2], p.190.THEOREM 1 • Let U be finitely presented and non-Hopfian and let ip be a homo-morphism of U onto U with non-trivial kernel. Then there exists a finitely presentedgroup H ~D U and a finitely presented group V such that <p induces a homomorphismi? of V*U onto (V*U) xH.PROOF: Let 1 ^ x be an element in the kernel of (p. Form the group H as inLemma 1 and let fi: U —* H be the embedding. Let V be any finitely presented groupfor which there exists a homomorphism p of V onto H. The map •& of V * U onto(V * U) x H is given by•&: v -> (v, vp) for v G V of V * U•9: u —y (uip, ufi) for u € U of V * U.In (wp, ufi) the left-hand side, utp, is in U of V * U. Because V * U is a free productand tp and p are homomorphisms, i? extends to a homomorphism. Note, xft = (1, xfi).The map p is onto. (V * J7)i? contains therefore an element (g, h) for all h £ H. Sincexfj, generates H as a normal subgroup, (V * U)ti contains (1, h) for all h G H andavailable at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700030422Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 10 Jul 2017 at 15:11:06, subject to the Cambridge Core terms of use,[3] Groups with a quotient 515therefore (1, H). Therefore (V * U)d contains (v, 1) for all v £ V and (wp, 1) forall u £ U. Because ip is onto this implies that also (V * U, 1) ot (V * U) x H is in( F * Z7)t?. Together we get that i? is onto and the proof is complete. UREMARK. There are two natural choices for V:(i) V = H and p the identity map.(ii) V free. More exactly if hi, ..., hn is a generating set of H then we maytake V = ( / i , . . . , / n ) , the free group freely generated by / i , . . . , / « andp the homomorphism given by fop — hi for i = 1, . . . , n . This gives thefollowing corollary:COROLLARY 1 . Let U be finitely presented and non-Hopfian. Then there existsa finitely generated free group F and a finitely presented group H containing U suchthat there is a homomorphism d of F *U onto (F *U) X H.EXAMPLE 1: Let U = (t, a-^'^a^t — a3) be the well known non-Hopfian group ofBaumslag and Soli tar. See [2], p.197, where it is also shown that 1 ^  x = [ t " 1 ^ , a] isin the kernel of ip given by tip = t and mp = a2. The groupH = (t, a, s, b;t-la2t = a3, s~xl?a = b3,s = [t^at, a], t = [s'Hs, b}),which contains U and has the property that x generates IT as a normal subgroup, wasconstructed in [3]. Let G\ be the free product of H and U:d = (t, a, a, b, r, c^ati = a3, s^^s = b3, s = [t^at, a],t = [s-1bs,b),r-1c2r = c3)By Theorem 1, the homomorphism i> of G\ to G\ x H given bytf: t -» (i, t), tf: a -> (a, a), tf: s -» (a, « ) , * : & - » (6, 6),tf:r-»(r,i), tf: c-> (c2, a)is onto.EXAMPLE 2: Let Z7 and ff be the groups of Example 1. The relation a —[t-1a<, a] of H can be used to eliminate the generator s, so that H has the followingpresentation on three generators and three relations:H = (t, a, bit-ifi = a3, [a, r 1 at^t'1 at, a) - b3 ,t = [[a, r 1 at}b[rl at, a], 6]) .Let G^ be the one-relator groupG2 = {r,c, ft, / . , fh-v-1c2r^c3).Then by Corollary 1, the map 1? of G2 to G2 x H given by*: ft - (/*, *), * : / . - (/., «). * = / » -tf:r^(r,*), t > : c ^ ( C 2 , a )is onto.available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700030422Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 10 Jul 2017 at 15:11:06, subject to the Cambridge Core terms of use,516 R. Hirshon and D. Meier [4]3 . FINITELY GENERATED GROUPS G WITH G = G XE ,H FINITELY PRESENTEDTHEOREM 2. Let i? be a homomoTphism of A onto Ax B. Then A contains anormal subgroup K such that A/K = A/K x B.PROOF: Let K be a maximal normal subgroup such that K& is contained in(K, 1). Then i? induces a map t\ of A/K onto A/K x B. Let L = ker?? and let Nbe its preimage under the quotient map A —> A/K. Then N contains K and has theproperty that TVi? is contained in (N, 1). By the maxdmality of K, we get that K — Nand therefore that 77 is an isomorphism and the proof is complete. DWe now show that a maximal normal subgroup K of A with the property thatK"& is contained in (K, 1) is unique.T H E U P P E R KERNEL OF A HOMOMORPHISM 1? OF A ONTO AXB. Let Ax = (A, l ) ^ " 1and for i > 0, let Ai+i — (Ai, l ) i ? - 1 . The Ai form a descending chain of normalsubgroups. The intersection of the Ai contains the kernel of 1?. We call it the upperkernel of 1? and write uker (1?) for it and claim that it has the property A/ uker (1?) =A/ uker (1?) X B. Since 1? maps all elements of uker(i?) to (uker(i?), 1), 1? inducesa map 1?1 from .4/uker (1?) to .A/uker (1?) X B, and since ai? G (uker(i?), 1) impliesa G uker (1?), 1?' is an isomorphism.REMARK. If ai? G (A, 1), we can consider ai? as an element of A and apply •& againto get ai?2 . If o&2 is again in (A, 1) we can continue in the same way to get ai?3 andso on. The upper kernel uker(i?) consists now precisely of those elements a £ A withoi?n in (A, 1) for all n .PROPOSITION 1 . Let 1? be a homomorphism of A onto AxB. If K is a max-imal normal subgroup of A such that Kti is contained in (K, 1), then K — uker (1?).PROOF: Since K-9 is contained in (K, 1), A\ = (A, l ) i ? - 1 2 K. Assume nowthat Ai 2 K. Then K D K# implies Ai D #1? and 4.-+1 = (Aif lji?"1 2 K. Byinduction we get Ai 5 K for all i and therefore uker(i?) 3 K. By maximality of Kthis implies K = uker (1?). DT H E LOWER KERNEL OF A HOMOMORPHISM 1? OF A ONTO AxB. The upper kernelis a method of finding the largest normal subgroup K with .K"i? in (K, 1) and A/K =A/K X B as an intersection of a chain of subgroups. The smallest normal subgroup withKti in (K, 1) and A/K = A/K x B can be found as follows. Let Kx - ker(i?). Then1? induces a map i?j of A/K\ onto A/K\ x B. Let K2 be the preimage of ker(i?i)under the quotient map A to A/K\. Inductively if i?j maps A/Ki onto A/Ki x B wedefine Ki+i to be the preimage of ker (i?j) under the quotient map A to A/Ki. TheKi form an ascending chain of subgroups. Let K — UKi. We call K the lower kernelavailable at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700030422Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 10 Jul 2017 at 15:11:06, subject to the Cambridge Core terms of use,[5] Groups with a quotient 517of i? and write lker (i?) for it. •& induces a map i?' from A/K onto A/K x B which isan isomorphism.PROPOSITION 2 . Let •& be a homomorphism of A onto AxB. If N is a normalsubgroup of A such that Nd is contained in (N, 1) and i? induces an isomorphism ofA/N to A/N x B, then N D lker (tf).PROOF: Obviously, N I) Ki = ker(tf). Assume now that N D K*. ThenKi+iN/N is in the kernel of the homomorphism A/N to A/N x B induced by tf.Since this map is an isomorphism, N Q -Ki+i. By induction we get N 13 Ki for all t,therefore ND lker (tf). DREMARK. In general lker (i?) ^ uker(i?). If G = G x B and ip is an isomorphism ofG to G x B and H is any group then the map •& of H x G to (H x G) x B given by(h, g)d = (h, gip) is an isomorphism. In this case we get lker (i?) = 1 and uker (i?) D H.COROLLARY 2 . Let U be a finitely presented non-Hopfian group. Form thefinitely presented group H as in Lemma 1. Then there exists a finitely generated groupG such that G = G x H.PROOF: By Theorem 1, there exists a homomorphism -9 of H*U onto H*U X H.By Theorem 2, the group G = H*U/uker(d) has the property G = GxH. DEXAMPLE 3: Let G2 and H be as in Example 2. From the above, there exists aquotient G3 of G2 such that G3 = G3 x H.4. FINITELY PRESENTED GROUPS G WITH A HOMOMORPHISMFROM G ONTO G X GLet A = (ai, . . . , an) be a finitely generated abelian group and let / j , v be two one-one homorphisms from A to A which are not onto, such that fiv = i/fi. It was shownin [3] that the HNN extension U = {t, >l;<~1Oj/it = a,{V, i = 1, . . . , n) is non-Hopfianif A = {Afi, Av). Theorem 3 generalises the construction of the group of Theorem Cof Baumslag and Miller [1].THEOREM 3 . Let A, n, v be as above. If tie eiements ai/^aji/)"1, i — 1, . . . , ngenerate A, then there exists a finitely presented group G containing A with a homo-morphism •& of G onto G x G induced by p.PROOF: Let B = (&i, . . . , bn) and C — (ci, . . . , cn) be isomorphic to A such thataj —> bi and a; —* Ci, i = 1, . . . , n induce isomorphisms and form the HNN extensionsU - (t, A; t^amt = aw, i = 1, . . . , n),V = (s, B x C; s^bifis = biu, a^au, i = l n).available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700030422Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 10 Jul 2017 at 15:11:06, subject to the Cambridge Core terms of use,518 R. Hirshon and D. Meier [6]Since fi and v are not onto, we can find ai 6 A — Afj. and a,j E A — Av. Thenby Britton's Lemma (see [2], p.181), the element [t^atf, a,] is not trivial and t and[t~1ait, a,j] generate a free subgroup of U. Similarly [a~1bi8, bj] is not trivial and aand [a~1biS, bj] generate a free subgroup of V.We can therefore construct the amalgamationG={U,V;t = [s^bta, bj], a = [i^tai, a,-]).We claim tha t t he m a p •& from G to G x G given on the generators by1?: t—>(t,t), tf: a<-> (a*, a*), i = 1, . . . , n ,i?: a -> (a, a), •d: 6j-> (&,-, &;), i = 1, . . . , n ,is a homomorphism and onto.To show that •& is a homomorphism, we have to prove that i? maps relations inG to relations of G x G. Since all elements except Ci, i — 1, . . . , n , are mapped tothe corresponding diagonal elements of G x G, it suffices to show that a~1Cifia = c^v,i — 1, . . . , n, and [&i, CJ] = 1, for i, j =1, ... ,n, are mapped to relations.Note,This can be simphned by the relations of V:{a~1bifj,aa~1 (ct-/x2)a, a~1Ci/j.2a) = (We now use the fact that fiu — vfi to derive (biVCifiv, C^LV) = {biVCiVfi, CiVfi). But(bit/CiUfi, CiUfi) — (CJI/)I?. [b{, Cj\9 = (1, 1) is true because B x C is abelian.It remains now to show that •& is onto. We observe, [a~1Ci8, bj\9 = ([s~16jCi/xs, bj],[a~1Ciiia, bj\) — ([s~16jSCji/) bj], [c{V, bjfj = (^8~1bi8, bj, 1]). From the relations ofG we derive (t, 1) 6 Gi?. Because Gi? contains (o,-, Oj), t = 1, . . . , n , it con-tains (oj/i, aifi) for all i . Also (0,71, am)(t, I)'1 (am, Oi/x)"1^, 1) = ^ ^ ( a j i / ) " 1 , l j ,i = 1, . . . , n . Since the elements Oj^(aji/)~ , i = 1, ... ,n generate A, (A, 1) is inG$. From a — [t~1ait, aj], it follows that also (a, 1) £ Gti. The consideration abovewith a, B and C shows now that (B, 1) and (C, 1) are in Gti. Together we get that(G, 1) G Gti. If we look at the values of 1? on the generators we see that therefore(1, t), (1, a), (1, aj) , (1 , bi), (1, Ci/z), i = 1, . . . , n, are in Gt?. As before we concludein a similar way that (1 , G) € Gi?. This completes the proof of Theorem 3. 0available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700030422Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 10 Jul 2017 at 15:11:06, subject to the Cambridge Core terms of use,[7] Groups with a quotient 519The next example is the group constructed by Baumslag and Miller [1].EXAMPLE 4: Let A = (a) be infinite cyclic, and fi, v given by a\i = a2, av — a3.Then the conditions of the theorem are satisfied. Also a £ A — A\i and a £ A — Av.By Theorem 3 or rather the proof of Theorem 3, we construct the groupsU = (t,a;t-1a2t = a3),V = {a, b, c; [6, c] = 1, s^tfs = bs, »~xi?a = c3)and conclude that for the groupG4 = (t, a, s, b, c j i - V t = a3, [6, c] = 1, a"1*2* = 63, J ' V J = c3,s = [t-'at, a], t - [s-'bs, b])the map •& from G* to G\ x G± given by•&:t-*{i,t), •&: a -» (a, a), •&: a -> {a, a), d:b^> (b, b),t ? : c ^ {bc\c2)is a homomorphism and onto.EXAMPLE 5: Let A = (ai , o2) be free abelian on two generators, and fi, v givenby ai/x = af, 02/1 = 02, o,\V = a i , 02^ = a j . Then the conditions of the theorem aresatisfied. Also a\ £ A — A/x and an £ A — Av.Again by the proof of Theorem 3, we construct the groupsU - (t, a i , a 2 ; [ a i , a2] = 1, t 1 o j t = o i , t 1a2t = a\),V = (a, bi, b2, ci, c2; [61, 62] = 1, a^bla = blt 8~*b2a - b[ci, c2] = 1, a'^cla - ci, a~1c2a = c\,[bi, Cj) - 1, i, j = 1, 2)and conclude that for the groupGs — (i, 01, a2) s, 61, 62, ci , c2;[a!, o2] = 1, t 1a\t = alt t 1a2t — a2,[61, ft,] = 1, a-Hla = bu a^ha - b2,i, c2] = 1, a - 1 c j a = a, a~xc2a - c\,[bi, cs] = 1, i, j = 1, 2,a = [t-1a1t,a2],t = [athere exists a homomorphism •& from G$ onto Gs x Gs.available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700030422Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 10 Jul 2017 at 15:11:06, subject to the Cambridge Core terms of use,520 R. Hirshon and D. Meier [8]5. FINITELY GENERATED GROUPS G = G x GTHEOREM 4 . If a non-trivial finitely generated group G has a homomorphism •&from G onto G X G, then there exists a non-trivial quotient E of G isomorphic to itsown direct square.PROOF: Let if be a maximal normal subgroup, K ^ G, such that, Jfi? is con-tained in K x K. Such a normal subgroup exists for a finitely generated group G byZorn's lemma. Set E = G/K. Then i? induces a map 77 of E onto E X E. LetL = kerT/; then L ^ E and its preimage N under the quotient map G —> E containsK and has the property that N-& is contained in N x N. By maximality of K, weget that K = N and therefore that 77 is an isomorphism. This completes the proof ofTheorem 4.As in Section 3 we can find a normal subgroup K* with G/K* = G/K* X G/K*as follows. Let K\ — ker(i?); then •& induces a map •&! of G/K\ onto G/K\ x G/K\.Let K2 be the preimage of ker (i?i) under the quotient map G to G/Ki. Inductivelyif 1?; maps G/Ki onto G/Ki x G/Ki we define .Ki+i to be the preimage of ker(i?j)under the quotient map G to G/Ki. The Ki form an ascending chain of subgroups.Let K* = UKi. Then 1? induces a map iT from G/K* onto G/K* X G/K* which isan isomorphism. DPROPOSITION 3 . Let •& be a homomorphism of G onto GxG. If N isanormaisubgroup of G such that JVi? is contained in N X N and •& induces an isomorphism ofG/N to G/N x G/N, then N D K* for K* = UKi as above.PROOF: Obviously, N D Kt = ker(i?). Assume now that N D Ki. ThenKi+\N/N is in the kernel of the homomorphism G/N to G/N x G/N induced by1?. Since this map is an isomorphism, N D -Ki+i • Therefore by induction, N D Ki forall i and hence N D K*. UREFERENCES[1] G. Baumslag and C.F. Miller III, 'Some odd finitely presented groups', Bull. LondonMath. Soc. 20 (1988), 239-244.[2] R.C. Lyndon and P.E. Schupp, Combinatorial Group Theory (Springer-Verlag, Berlin,Heidelberg, New York, 1977).[3] D. Meier, 'Non-hopfian groups', / . Lond Math. Soc. 26 (1982), 265-270.[4] J.M. Tyrer Jones, 'Direct products and the Hopf property', J. Austral. Math. Soc. 17(1974), 174-196.Polytechnic University Pilgerweg 1Brooklyn, NY 11201 8044 ZurichUnited States of America Switzerlandavailable at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700030422Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 10 Jul 2017 at 15:11:06, subject to the Cambridge Core terms of use,

Groups with a quotient that contains the original group as a direct factor

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