New PDF release: Word Problems II: The Oxford Books. Proceedings Oxford, 1976

By S.I. Adian, W.W. Boone and G. Higman (Eds.)

ISBN-10: 044485343X

ISBN-13: 9780444853431

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Extra info for Word Problems II: The Oxford Books. Proceedings Oxford, 1976

Sample text

The word problem for Gs is reduced to the process for transforming words of Gs into normal form. Since the word problem for G4 is solvable the process is reduced to the problem of whether or not a word 0 of the group G3 is equal to a word of the form V ( Qy)qW(ri,x). By Lemmas 6 and 7, this is reducible to the problem of equality with q for special words in T. To prove the converse reduction, it suffices to note that 2 = q in T if and only if Z - ' t Z k = k Z - ' t 2 in Gs. One part of this assertion follows from Lemma 6 and the other part from applications of Britton's lemma.

A. Bokut' 44 D = Uip :UzE,U;'p yEUY1. (25) Take E = - 1; as D contains only positive occurrences of p l and p2, Britton's lemma applied to (25) yields (26) Ek KE{p,E:, U2E = 21p8. This contradicts our choice of Uo. So we have (27) I/oEku~'=piYPzY+ Since Uo has no occurrences of p ; , p ; we can, without loss of generality assume that Ek KplXp2X'. Then Britton's lemma applied to (27) yields Bp,X21,= Y - in 21plh. It follows from Novikov's first theorem that X Y in K(2l). I). Inductively it suffices to Conversely suppose X consider the two cases: - XKA,X,, Y K X , B , and X F X I B , , Y K A , X , .

Two cases arise: 1) E = 1. Then, by Lemma 2, we have T3T,I;'T2, T2€ GI, V K Vi(qJ)B;'r,l,V2 where T2V,(qJ)B;'r, = V3(aJ) in G I . B,V4(q,)), where V3 and V 4are reduced words. The last equality shows that Tp2 is not normal. ) 2) E = - 1. Here we have Tz€ GI, T K TIl,T2, V K Vi(q,)l;'Vz where T 2 V l ( q , ) =V3(aJ). Thus T , = V3(aJ)V4(qJ), with V,, V 4 reduced and again Tp2 is not in normal form. We conclude that the leftmost letter 1: of V(B;'r,l,,q,)is not cancelled in reducing TV to normal form.

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Word Problems II: The Oxford Books. Proceedings Oxford, 1976 by S.I. Adian, W.W. Boone and G. Higman (Eds.)

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