Infinitely Generated Subgroups of Finitely Presented Groups I

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Conversely , any presentation of form (3) has a naturally ordered collection of subpresentations. Such a polycyclic presentation will be termed honest if for each l^i^/c, the inclusion тг^.^^я, induces an embedding gp(Ä,_i)'^gp(^J. (Then the order of a^ mod gp(7rj_i) is e,.) Notice that even if %_i is honest тг^ may fail to be honest because (for instance) the type II relations specifying the action of a^ on gp(nj^_i) may not define an automorphism. Obviously, every polycyclic group has an honest polycyclic presentation. Below we shall give an algorithm to determine of an arbitrary polycyclic presentation where or not it is honest.

4 . 2 . Two facts concerning polycyclic groups beyond the definition will be needed: (i) polycyclic groups satisfy the maximum condition for subgroups (since cyclic groups satisfy the maximum condition this follows easily by induction on the length of a polycyclic series); and (ii) polycyclic groups are hopfian, i.e., every surjective endomorphism is an automorphism (see, for example, H.Neumann [16] Lemma32.1 and Corollary41.44).

We also observe the following fact which is easily established by induction on length.

Lemma 4.1. // л^ is an honest polycyclic presentation as in (3), and if gegp(n,^) has finite order, then \g\ divides Y\ ^v

ei < 00

We also observe the following:

Lemma 4.2. // щ is an honest polycyclic presentation as in (3), then there is a recursive procedure which, when applied to any word W in the generators of щ, gives the unique word in standard form equal to W. In particular, this algorithm solves the word problem for gp(^fc).

Proof . Use type II relations to push higher subscripted generators to right and type I relations to reduce mod в^<оо. Eventually, standard form will be arrived at which is unique since n^ is honest. This concludes the proof.

Theorem 4.3. There is an algorithm to determine of an arbitrary polycyclic presentation к^ as in (3) whether or not щ is honest. Moreover, if щ is honest, there are algorithms (uniform in л J for solving the following problems:

( i ) decide for arbitrary finite sets of words w^, ...,w„ in tt's and arbitrary gGTTfe whether or not gegp(w^, ...,w„) (called the generalized word problem, membership problem, occurrence problem, etc.),

( ii ) for an arbitrary finite set of words Wi,...,w„ in n^ find a finite presentation for gp(Wi,..., w„) (such a presentation exists since the subgroup is polycyclic), where the generators in the presentation naturally correspond to

Proof . By induction on the length к of the polycyclic presentation щ. For fc = 1, all assertions are easily checked. Suppose the theorem has been established for polycycUc presentations of length <k. Consider щ. Decide inductively if tu^.i is honest. If not, Щ is not honest and we are done. So suppose п^_1 is honest.