A CONTINUALLY DESCENDING ENDOMORPHISM OF A FINITELY GENERATED RESIDUALLY FINITE GROUP

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1 A CONTINUALLY DESCENDING ENDOMORPHISM OF A FINITELY GENERATED RESIDUALLY FINITE GROUP DANIEL T. WISE Abstract Let φ : G G be an endomorphism of a finitely generated residually finite group. R. Hirshon asked if there exists n such that the restriction of φ to φ n (G) is injective. We give an example to show that this is not always the case. 1. Introduction AgroupG is said to be Hopfian provided that every surjective endomorphism of G is an automorphism. A well-known theorem of Malćev [4] states that finitely generated residually finite groups are Hopfian. In [3], R. Hirshon gave the following interesting generalization of Malćev s theorem. Theorem 1 (Hirshon). Let G be a finitely generated residually finite group, and let φ be an endomorphism onto a finite index subgroup. Then there exists n such that φ restricted to φ n (G) is an injection. In particular, if G is torsion-free, then φ is an injection. A special case of Theorem 1 was proven later but independently in [6] foran application to knot groups. We note that the additional conclusion that φ is injective when G is torsion-free can be deduced as follows. First observe that φ n (G) is of finite index i in G for some i [G : φ(g)] n, and thus for each g G, we have g i φ n (G). Since G is torsion-free, we see that g i 1 unless g = 1. But since φ is injective on φ n (G), we see that φ(g i ) 1, and consequently φ(g) 1 unless g =1. Hirshon posed the following problem in [3] concerning a tempting generalization of Theorem 1. We found this problem in the collection of open problems in [7]. Problem 1. Let G be a finitely generated residually finite group, and let φ be an endomorphism of G. Does there exist n such that the restriction of φ to φ n (G) is injective? The purpose of this paper is to give an example illustrating that the answer to Hirshon s question is no. The group of the example is an amalgamated free product which is the double of a free group along a certain infinitely generated subgroup. This group and its endomorphism are described following two lemmas in Section 2. In [5], Z. Sela proved the impressive result that torsion-free, word-hyperbolic Received 4 November 1997; revised 19 May Mathematics Subject Classification 20E26. Author supported as an NSF postdoctoral fellow under grant no. DMS Bull. London Math. Soc. 31 (1999) 45 49

2 46 daniel t. wise groups are Hopfian. He points out that his proof actually yields the following more general result. Theorem 2 (Sela). Let G be torsion-free and word-hyperbolic, and let φ be an endomorphism of G. Then there exists N such that for all n>n,the restriction of φ to φ n (G) is injective. The example given in this paper shows that the analogy between the Hopficity of word-hyperbolic groups and finitely generated residually finite groups breaks down when we attempt to allow any endomorphism. In particular, word-hyperbolic groups have stronger endomorphism properties than do finitely generated residually finite groups. 2. The example We shall adopt the following notation. Given a group F, weletf denote an isomorphic copy of F. Given a feature x of F, weusex to denote the corresponding feature of x. So, for an element g F, weletg F denote the corresponding element, and for a subgroup H F, weleth F denote the corresponding subgroup. The example will be chosen so that it satisfies the hypotheses of the following two lemmas. We shall use Lemma 1 to show that the endomorphism of our example has the desired property. Lemma 1. Let F H=H F denote the double of F along the subgroup H. Let φ : F F be an endomorphism of F such that φ(h) H. Because φ(h) H, there is an obvious induced endomorphism Φ:F H=H F F H=H F. Suppose that φ n (H) is a strictly increasing sequence of subgroups of F. Then Kernel(Φ n ) is a strictly increasing sequence of subgroups. Proof. By hypothesis, for each n there exists an element g n such that φ n 1 (g n ) H but φ n (g n ) H. We conclude that Kernel(Φ n ) is a strictly increasing sequence of subgroups, because gn 1 g n Kernel(Φ n ), but by the Normal Form Theorem for amalgamated free products [4, Theorem IV.2.6], gn 1 g n Kernel(Φ n 1 ). We shall use Lemma 2 to show that the group of our example is residually finite. Recall that a subgroup H F is said to be closed (in the profinite topology) provided that H is the intersection of a set of finite index subgroups of F. Lemma 2. Let H be a closed subgroup of the residually finite group F. Then the double F H=H F is residually finite. Proof. By the Normal Form Theorem, an element w of F H=H F is non-trivial if and only if either w H 1 F or w can be represented as a product w = w 1 w 2 w n of elements alternately in F H and F H. In the first case, we consider the obvious homomorphism F H=H F F, and note that w is mapped to a non-trivial element in F, and consequently w can be mapped to a non-trivial element in some finite quotient of F. In the second case, we use the hypothesis that H is a closed subgroup to choose

3 continually descending endomorphism 47 a finite index subgroup K F such that H K, and such that for each i, in case w i F then w i K, and in case w i F then w i K.WeletF F be the finite quotient obtained by letting F act on the left cosets of K. Let H denote the image of H, and consider the induced map F H=H F F H= H F. For each i, let w i denote the image of w i in F H= H F, and note that w i is an element of either F or F, depending on whether w i is an element of F or F. However, in either case, our choice of K implies that w i H = H. It follows that the image of w, which is the product w 1 w 2 w n, is a non-trivial normal form in F H= H F. Since F H= H F is an amalgamated free product of finite groups, it is well known to be residually finite [1]. It follows that w survives in some finite quotient, as claimed. Theorem 3. There exists an endomorphism Φ:G G of a finitely generated residually finite group such that for each n, the restriction of Φ to Φ n (G) is not injective. Proof. Combining Lemmas 1 and 2, we see that it is sufficient to give an example of: (i) a finitely generated residually finite group F; (ii) an endomorphism φ : F F; (iii) a closed φ-invariant subgroup H such that φ n (H) is strictly increasing. We then let G = F H=H F be the double of F along H, and let Φ : G G be the endomorphism induced by φ : F F and φ : F F. One such example goes as follows. Let F = a, t be a rank 2 free group. Let φ : F F be induced by a a 2 and t t. Let H = (a 2m ) tm : m 0. (We use the notation x y = yxy 1.) It is clear that F is finitely generated and residually finite [2]. To see that φ(h) H, observe that φ maps each generator of H to its square. To see that φ n (H) is strictly increasing, we argue as follows. Let g n = a tn, and observe that φ n (g n )=φ n (a tn )=(a 2n ) tn H, but φ n 1 (g n )=φ n 1 (a tn )=(a 2n 1 ) tn H. An easy way to see that (a 2n 1 ) tn H is to consider the based covering space ˆB B (described below) which corresponds to H, and to observe that the path in B representing (a 2n 1 ) tn does not lift to a closed path at the basepoint of ˆB. To see that H is a closed subgroup, we first note that the finitely generated subgroups of a free group are closed [2], and of course the intersection of any set of closed subgroups is closed. It is therefore sufficient to show that H is the intersection of finitely generated subgroups of F. Let H r = (a 2m ) tm, t r+1 :0 m r. We shall show that H = r H r. For each r, we have H H r. To see this, we show that each generator of H is an element of H r. Consider the generator t m a 2m t m, and express m = a(r +1)+b, where 0 b<(r + 1) and a 0. Then t m a 2m t m = t a(r+1)+b a 2a(r+1)+b t (a(r+1)+b) =(t (r+1) ) a (t b a 2b t b ) 2a(r+1) (t (r+1) ) a, where the elements in parentheses are generators of H r. Finally, we must show that for an element g H, there exists r such that g H r. The proof is expressed in terms of covering spaces. Let B denote a bouquet of two oriented circles labelled a and t. Let ˆB denote the based covering space of B corresponding to H. First consider a based copy T of the universal cover of the t circle. Next, for each j 0, we attach to T a copy of the degree 2 j cover of the a circle. The jth such circle is attached at the end of the path t j beginning at the basepoint of T. Finally, we complete this labelled graph to a covering space of B

4 48 daniel t. wise by adding trees to the incomplete vertices in the obvious way. (By an incomplete vertex, we mean a vertex which does not have a neighbourhood isomorphic to a neighbourhood of the unique vertex of B, or, equivalently, its valence is < 4.) The core of this based covering space is partially illustrated in Figure 1. Fig. 1. The core of ˆB The basepoint is indicated by a large vertex, and the edges labelled by black and white arrows correspond to the letters a and t, respectively. Similarly, we form the based cover ˆB r corresponding to H r by beginning with a based copy T r+1 of the degree r + 1 cover of the t circle. Then for 0 j r we add a degree 2 j cover of the a circle at the end of the path t j beginning at the basepoint of T r+1. Finally, we complete this labelled graph to a covering space of B by adding trees to the incomplete vertices in the obvious way. The core of the based covering space ˆB 3 is illustrated in Figure 2, where we follow the same labelling conventions as indicated for Figure 1. Fig. 2. The core of ˆB 3 Observe that the balls of radius n about the basepoints of ˆB and ˆB n+1 are isomorphic as labelled graphs. Now suppose that g H, and let w denote a reduced word representing g. Then w does not lift to a closed path in ˆB. Let w denote the length of w. Since ˆB and ˆB w +1 have isomorphic balls of radius w, we see that w does not lift to a closed path in ˆB w +1 and hence g H w +1, and we are done. Acknowledgements. their helpful comments. I am grateful to Jon McCammond and to the referee for

5 continually descending endomorphism 49 References 1. G. Baumslag, On the residual finiteness of generalized free products of nilpotent groups, Trans. Amer. Math. Soc. 106 (1963) M. Hall Jr, Coset representations in free groups, Trans. Amer. Math. Soc. 67 (1949) R. Hirshon, Some properties of endomorphisms in residually finite groups, J. Austral. Math. Soc. Ser. A 24 (1977) R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Ergeb. Math. Grenzgeb. 89 (Springer, Berlin, 1977). 5. Z. Sela, Endomorphisms of hyperbolic groups I: The Hopf property, Topology, to appear. 6. D. S. Silver, Nontorus knot groups are hyperhopfian, Bull. London Math. Soc. 28 (1996) The Magnus list of topical open problems, Problem FP18, Department of Mathematics Cornell University Ithaca, NY USA