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The Higman Theorem for Primitive-Recursive Groups - A Preliminary Report
THE HIGMAN THEOREM FOR PRIMITIVE-RECURSIVE GROUPS - A PRELIMINARY REPORT
R.W. GATTERDAM”
University of Wisconsin, Parkside
An extensive study of g n ( A )groups can be found in this volume,
[ 1 1 . A group is an A-p.r. group (primitive recursive relative t o
A C N ) if it is g n ( A ) for some (finite) n ; it is a p.r. group if it is g n ( N ) for some n. As mentioned in [ 1 I the following version of the Higman theorem, [ 51, holds for A-p.r. groups:
Theorem. L e t A C N be a recursively enumerable set. Then an A p.r. group can be embedded as an A-p.r. decidable subgroup of u finitely presented A-p.r. group. The embedding and its inverse, where defined, are A-p.r. computable. The complete proof of this theorem is contained in [ 41 . Also observe the similarity t o the Clapham result, [ 21 and [ 31 , which can be viewed as the same statement replacing “A-p.r.” by “ A recursive”. In [ 4 ] both the theorem and our version of the Clapham result are proved by a technique similar to that of Schoenfield [ 6 ] using the computability of the group constructions free product with amalgamation and strong Britton extension as discussed in [ 1 1 . The purpose of this report is t o briefly outline the proof given in [ 41 considering two crucial facets in some detail. First we see how countably many applications of the strong Britton extension (as in Lemma 5.10 of [ 1 ] ) are used in the proof. The proof is then completed by a direct induction based on the A-p.r. computable * AFOSR-I 321-67
and AFOSR-70-1870 (Grants).
421
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R. W. Gatterdam, The Higman theorem for primitive-recursivegroups
decision function for the word problem of a finitely generated, A-p.r. group. This report should be viewed as preliminary in that it does not utilize the full information o n the computability of the strong Britton construction as found in [ 11 b u t rather the weaker results of Chapters 2 and 3 of [ 41 . With these newer results it is hoped that the Higman construction can be shown to induce at most a jump of a few computability levels of the relativized Grzegorczyk hierarchy. An embedding of an A-p.r. group into another A-p.r. group is an A-p.r. embedding if it is an Gn(A) embedding for some n as in [ 1 1 . Similarly an A-p.r. isomorphism cp in G is an g n ( A ) isomorphism and we consider the A-p.r. group G, = ( G, t ; tht-l = ~ ( h ) , ii E H = domain cp ) as in [ 1 3 . In view of Theorem 5.5 o f [ 11 we restrict o u r attention t o finitely generated (f.g.), A-p.r. groups and following [ 61 say such a group is A-p.r. Higman if it can be A-p.r. embedded in a f.p., A-p.r. group. Again following [ 61 an A-p.r. isomorphism cp in an A-p.r. Higman group G is said t o be A-p.r. benign if G, is A-p.r. Higman and, in particular, an A-p.r. decidable subgroup H < G is A-p.r. benign if G, = G‘, l H is A-p.r. Higman. Then paralleling [ 61 the following are proved in [ 41 (replacing p.r. by A-p.r.1: 1 . H < K < G for G A-p.r. Higman, K f:g. and A-p.r. decidable, tiien I€ is A-p.r. benign in G ijf it is A-p.r. benign in K.
3. The intersection 0j.A-p.r. benign subgroups is A-p.r. benign.
3. Tile (group theoretic) union of A - p r . benign subgroups is A-p. r. benign if A-11.r. decidable. 4. The image of an A-p.r. benign subgroup under an A-p.r. computable Iiornoniorphism is A-p.r. benign if A-p.r. deciduble.
5 . The preimage of an A-p.r. benign subgroup under un A-p.r. computable homomorphism is A-p.r. benign. 6. The restriction of a n A-p.r. isomorphism in G with domain G to an A-p.r. benign subgroup is an A-p.r. isornorphism iii G.
R . W. Gatterdam, The Higman theorem for primitive-recursive groups
423
As might be expected the proofs of 1 . through 6. given in [ 4 ] are similar t o proofs in [ 6 ] with attention given t o the decidability of the various subgroups involved and the computability of the homomorphisms (both in the statements and internal t o the proofs). Using the above facts the link between A-p.r. Higman groups and A-p.r. benign subgroups is established by
7 . Let G be j:g., A-p.r. and let 1 + K + F + G + 1 be a presentation of G f b r F free and f g . Then G is A-p.r. Higrnan iff K is A-p.r. berz ign. In view of 7. we turn our attention t o A-p.r. decidable subsets of a f.g. groups F . F o r P C F , an A-p.r. decidable subset we define the subgroup E p < F * ( z ; ) t o be that subgroup generated by all words of the form XzX-'for X E P. We say P is an A-p.r. benign subset of F if E p < F * ( z ;) is an A-p.r. benign subgroup. We are now in a position to prove the crucial lemma (Lemma 5.1 of [ 4 ]). It is the proof of this lemma which requires countably many applications of the strong Britton extension and we consider it in some detail.
Lemma. I f P < F is an A-p.r. decidable subgroup o f the f : g . f r e e group F which is A-p.r. benign as u subset then it is A-p.r. benign as a subgroup.
To prove the lemma, set G = F * ( c, d ; ) and for every word X E F define px : { c } -+ { dX } an isomorphism of infinite cyclic subgroups of G. Let X I , X,, .. enumerate the words of F and consider GIFl = ( G, t X 1 ,t X 2 ,...; t Xc t X - l = d X b' X E F ) . The proof proceeds in the following steps:
1 . GIFl is an A-p.r. group (by Lemma 5.10 of [ 1 1 ; it is here that the strong form of Lemma 5.10 as compared to Lemma 3.2 of [ 41 may be used t o strengthen the result). 2. G I F ~is A-p.r. Higman (by a construction). 3. GIPi = { G, t X for all X E P } < GIFl is A-p.r. decidable (by a n indwtive argument during the construction of GIFl ).
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4. { c, d, t, for all X E P } = { c, d, t X for all X E P, P } < G p l is A-p.r. decidable (an application of Lemma 5.1 1 of [ I ] ).
5. { c, d, t , for X E P } is A-p.r. benign (as the image of { c, a', XzX-I for all X E P}< F * ( c ,d, z;) under an A-p.r. homo-
morp hism).
6 . P = F n { c, d, t, for all X E P } completing the proof of the lemma.
Next we consider certain subsets of F = ( a , b ; ) .We say an A-p.r. decidable subset B C N k (k-tuples of natural numbers) is A-p.r. benign if the set of all words in F of the form, ax' baX*b ... baXk for ( x l , _..,x k ) E B is an A-p.r. benign subset of F.A n A-p.r. computable function J' : N k + N is A-p.r. benign if the subset of k+ I-tuples of the form ( x l , ..., xk, f ( x l , ..., x k ) ) is A-p.r. benign. We then establish the following:
Lemma. If A C N is recursively enumerable and f:N k + N is A-p.r. computable then J'is A-p.r. benign. The proof of the above in [ 4 ] differs somewhat from the proof given in [ 61 in that the characterization of A-p.r. computable functions as the smallest class of functions containing the initial functions Z , U m n , f o ,E and cA and closed under substitution and (unlimited) iteration (see [ 13 ), is used directly. In particular we show that cA is in the class of A-p.r. benign functions since x E A ++ 3 yQ(x,y ) for Q a p.r. (actually g 3 )predicate related to the Kieene T predicate. Also the closure of the class of A-p.r. benign functions under (unlimited) iteration is shown by a direct construction (Proposition 5.10 of [ 41 ). The proof of the theorem is now completed by proving
Lemma. Let A C N be recursively enumerable and P C F be an A-p.r. decidable subset o f a j:g. free group F. Then P is A-p.r. benign.
R. W. Gatterdam, The Higman theorem for primitive-recursive groups
425
The proof uses the previous lemma applied t o the characteristic function of P. The construction given in [ 41 follows that of [ 61 closely verifying A-p.r. computability at each step. References [ 11 F.B. Cannonito and R.W. Gatterdam, The computability of group constructions, part 1, this volume. [ 2 ] C.R.J. Clapham, Finitely presented groups with word problem of arbitrary degrees of insolubility, Proceedings of the London Mathematical Society 14 (1964) 633-676. [ 3) C.R.J. Clapham, An embedding theorem for finitely generated groups, Proceedings of the London Mathematical Society 17 (1967) 419-430. [ 4 ] R.W. Gatterdam, Embeddings of primitive recursive computable groups, doctoral dissertation, University of California, Irvine, 1970. Submitted for publication. 1.51 G. Higman, Subgroups of finitely presented groups, Proceedings of the Royal Society, A 262 (1961) 455-475. [ 6 ] J.R. Schoenfield, Mathematical Logic (Addison Wesley, 1967).
R.W. GATTERDAM”
University of Wisconsin, Parkside
An extensive study of g n ( A )groups can be found in this volume,
[ 1 1 . A group is an A-p.r. group (primitive recursive relative t o
A C N ) if it is g n ( A ) for some (finite) n ; it is a p.r. group if it is g n ( N ) for some n. As mentioned in [ 1 I the following version of the Higman theorem, [ 51, holds for A-p.r. groups:
Theorem. L e t A C N be a recursively enumerable set. Then an A p.r. group can be embedded as an A-p.r. decidable subgroup of u finitely presented A-p.r. group. The embedding and its inverse, where defined, are A-p.r. computable. The complete proof of this theorem is contained in [ 41 . Also observe the similarity t o the Clapham result, [ 21 and [ 31 , which can be viewed as the same statement replacing “A-p.r.” by “ A recursive”. In [ 4 ] both the theorem and our version of the Clapham result are proved by a technique similar to that of Schoenfield [ 6 ] using the computability of the group constructions free product with amalgamation and strong Britton extension as discussed in [ 1 1 . The purpose of this report is t o briefly outline the proof given in [ 41 considering two crucial facets in some detail. First we see how countably many applications of the strong Britton extension (as in Lemma 5.10 of [ 1 ] ) are used in the proof. The proof is then completed by a direct induction based on the A-p.r. computable * AFOSR-I 321-67
and AFOSR-70-1870 (Grants).
421
422
R. W. Gatterdam, The Higman theorem for primitive-recursivegroups
decision function for the word problem of a finitely generated, A-p.r. group. This report should be viewed as preliminary in that it does not utilize the full information o n the computability of the strong Britton construction as found in [ 11 b u t rather the weaker results of Chapters 2 and 3 of [ 41 . With these newer results it is hoped that the Higman construction can be shown to induce at most a jump of a few computability levels of the relativized Grzegorczyk hierarchy. An embedding of an A-p.r. group into another A-p.r. group is an A-p.r. embedding if it is an Gn(A) embedding for some n as in [ 1 1 . Similarly an A-p.r. isomorphism cp in G is an g n ( A ) isomorphism and we consider the A-p.r. group G, = ( G, t ; tht-l = ~ ( h ) , ii E H = domain cp ) as in [ 1 3 . In view of Theorem 5.5 o f [ 11 we restrict o u r attention t o finitely generated (f.g.), A-p.r. groups and following [ 61 say such a group is A-p.r. Higman if it can be A-p.r. embedded in a f.p., A-p.r. group. Again following [ 61 an A-p.r. isomorphism cp in an A-p.r. Higman group G is said t o be A-p.r. benign if G, is A-p.r. Higman and, in particular, an A-p.r. decidable subgroup H < G is A-p.r. benign if G, = G‘, l H is A-p.r. Higman. Then paralleling [ 61 the following are proved in [ 41 (replacing p.r. by A-p.r.1: 1 . H < K < G for G A-p.r. Higman, K f:g. and A-p.r. decidable, tiien I€ is A-p.r. benign in G ijf it is A-p.r. benign in K.
3. The intersection 0j.A-p.r. benign subgroups is A-p.r. benign.
3. Tile (group theoretic) union of A - p r . benign subgroups is A-p. r. benign if A-11.r. decidable. 4. The image of an A-p.r. benign subgroup under an A-p.r. computable Iiornoniorphism is A-p.r. benign if A-p.r. deciduble.
5 . The preimage of an A-p.r. benign subgroup under un A-p.r. computable homomorphism is A-p.r. benign. 6. The restriction of a n A-p.r. isomorphism in G with domain G to an A-p.r. benign subgroup is an A-p.r. isornorphism iii G.
R . W. Gatterdam, The Higman theorem for primitive-recursive groups
423
As might be expected the proofs of 1 . through 6. given in [ 4 ] are similar t o proofs in [ 6 ] with attention given t o the decidability of the various subgroups involved and the computability of the homomorphisms (both in the statements and internal t o the proofs). Using the above facts the link between A-p.r. Higman groups and A-p.r. benign subgroups is established by
7 . Let G be j:g., A-p.r. and let 1 + K + F + G + 1 be a presentation of G f b r F free and f g . Then G is A-p.r. Higrnan iff K is A-p.r. berz ign. In view of 7. we turn our attention t o A-p.r. decidable subsets of a f.g. groups F . F o r P C F , an A-p.r. decidable subset we define the subgroup E p < F * ( z ; ) t o be that subgroup generated by all words of the form XzX-'for X E P. We say P is an A-p.r. benign subset of F if E p < F * ( z ;) is an A-p.r. benign subgroup. We are now in a position to prove the crucial lemma (Lemma 5.1 of [ 4 ]). It is the proof of this lemma which requires countably many applications of the strong Britton extension and we consider it in some detail.
Lemma. I f P < F is an A-p.r. decidable subgroup o f the f : g . f r e e group F which is A-p.r. benign as u subset then it is A-p.r. benign as a subgroup.
To prove the lemma, set G = F * ( c, d ; ) and for every word X E F define px : { c } -+ { dX } an isomorphism of infinite cyclic subgroups of G. Let X I , X,, .. enumerate the words of F and consider GIFl = ( G, t X 1 ,t X 2 ,...; t Xc t X - l = d X b' X E F ) . The proof proceeds in the following steps:
1 . GIFl is an A-p.r. group (by Lemma 5.10 of [ 1 1 ; it is here that the strong form of Lemma 5.10 as compared to Lemma 3.2 of [ 41 may be used t o strengthen the result). 2. G I F ~is A-p.r. Higman (by a construction). 3. GIPi = { G, t X for all X E P } < GIFl is A-p.r. decidable (by a n indwtive argument during the construction of GIFl ).
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R. W. Gatterdam, The Higrnan theorem for primitive-recursive groups
4. { c, d, t, for all X E P } = { c, d, t X for all X E P, P } < G p l is A-p.r. decidable (an application of Lemma 5.1 1 of [ I ] ).
5. { c, d, t , for X E P } is A-p.r. benign (as the image of { c, a', XzX-I for all X E P}< F * ( c ,d, z;) under an A-p.r. homo-
morp hism).
6 . P = F n { c, d, t, for all X E P } completing the proof of the lemma.
Next we consider certain subsets of F = ( a , b ; ) .We say an A-p.r. decidable subset B C N k (k-tuples of natural numbers) is A-p.r. benign if the set of all words in F of the form, ax' baX*b ... baXk for ( x l , _..,x k ) E B is an A-p.r. benign subset of F.A n A-p.r. computable function J' : N k + N is A-p.r. benign if the subset of k+ I-tuples of the form ( x l , ..., xk, f ( x l , ..., x k ) ) is A-p.r. benign. We then establish the following:
Lemma. If A C N is recursively enumerable and f:N k + N is A-p.r. computable then J'is A-p.r. benign. The proof of the above in [ 4 ] differs somewhat from the proof given in [ 61 in that the characterization of A-p.r. computable functions as the smallest class of functions containing the initial functions Z , U m n , f o ,E and cA and closed under substitution and (unlimited) iteration (see [ 13 ), is used directly. In particular we show that cA is in the class of A-p.r. benign functions since x E A ++ 3 yQ(x,y ) for Q a p.r. (actually g 3 )predicate related to the Kieene T predicate. Also the closure of the class of A-p.r. benign functions under (unlimited) iteration is shown by a direct construction (Proposition 5.10 of [ 41 ). The proof of the theorem is now completed by proving
Lemma. Let A C N be recursively enumerable and P C F be an A-p.r. decidable subset o f a j:g. free group F. Then P is A-p.r. benign.
R. W. Gatterdam, The Higman theorem for primitive-recursive groups
425
The proof uses the previous lemma applied t o the characteristic function of P. The construction given in [ 41 follows that of [ 61 closely verifying A-p.r. computability at each step. References [ 11 F.B. Cannonito and R.W. Gatterdam, The computability of group constructions, part 1, this volume. [ 2 ] C.R.J. Clapham, Finitely presented groups with word problem of arbitrary degrees of insolubility, Proceedings of the London Mathematical Society 14 (1964) 633-676. [ 3) C.R.J. Clapham, An embedding theorem for finitely generated groups, Proceedings of the London Mathematical Society 17 (1967) 419-430. [ 4 ] R.W. Gatterdam, Embeddings of primitive recursive computable groups, doctoral dissertation, University of California, Irvine, 1970. Submitted for publication. 1.51 G. Higman, Subgroups of finitely presented groups, Proceedings of the Royal Society, A 262 (1961) 455-475. [ 6 ] J.R. Schoenfield, Mathematical Logic (Addison Wesley, 1967).