Algorithms and Classification in Combinatorial Group Theory by Charles F. Miller III (auth.), Gilbert Baumslag, Charles F.

By Charles F. Miller III (auth.), Gilbert Baumslag, Charles F. Miller III (eds.)

The papers during this quantity are the results of a workshop held in January 1989 on the Mathematical Sciences study Institute. subject matters lined comprise choice difficulties, finitely provided basic teams, combinatorial geometry and homology, and automated teams and comparable themes.

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A Dehn's algorithm for C is a finite set of words ~ c N such that any non-empty word wEN can be shortened by applying a relator in~. F. Miller non-empty wEN has the form w == ubv where there is an element of the form bc E ~ with Icl < Ibl. Thus applying the relator bc to w, we can deduce that w =a uc-1v where luc-1vl < Iwl. If G has Dehn's algorithm ~ then one can solve the word problem for G in a particularly straight forward way: repeatedly try to shorten the word in question by replacing a subword using a relator in~.

8 n }, then each 'Trw can be written as a presentation on the same generating symbols. For by the Main Technical Lemma the presentations could be given on two generators, say 81 and 82. In case n > 2 the same group can then also be presented by adding additional generators {83, ... , 8 n } and additional defining relations 83 = 1, ... ,8 n = 1. Now in the above lemma take M = F and let Lw be the subgroup generated by the indicated elements using the presentation 'Trw as H. Let Hw = gp('Trw). -.

As an example of a conjugation in G, consider the following: d d t€d- 1d- 1 =c (d 1d 2 t €d-1d-1)-1 i21 q12i21 =c =c =c =c =c d d t€d- 1d-1 1 d 1d2ti-€d-1d-1 2 1q12i2 -1 d t€d- 1d-1 1 d 1 d 2 t i-€d-1 2 8 1 q8 1 2 i 2 1d- 1 d 1 d 2 t i-€ 8 1-1 8 2-1 q8 2 8 1 t€di 2 1 1d- 1 d 1 d 281-1 8 2-1 qR€i 82 8 1 d2 1 d 181-1 q82-lR€i 8 2 8 1 d-1 1 q8 -1 8 -lR€i 82 8 1 1 2 Generalizing this calculation, one can find a word Y on {d 1 , ... , dn , it, ... , t n } determined by the representation of w as a product of conjugates of the defining relations of U such that y-1qY =c qw which is the desired result (see [77] for more details of the calculation).

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