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

ISBN-10: 1461397308

ISBN-13: 9781461397304

ISBN-10: 1461397324

ISBN-13: 9781461397328

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

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**Extra resources for Algorithms and Classification in Combinatorial Group Theory**

**Sample text**

There exists a finitely presented group G of cohomological dimension 2 having unsolvable word problem. Indeed, G can be obtained from a free group by applying three successive HNN-extensions where the associated subgroups are finitely generated free groups. THEOREM Of course the associated subgroups in the second and third HNN-extension are only free subgroups of the previous stage in the construction, not subgroups of the original free group. The proof is obtained by making minor variations to the one given in Rotman's textbook [92].

Be new generating symbols and form the presentation u(G) =< G,x,tl,t2,··. I t;l[Ui,X]ti = ViX-1UiX,i = 1,2, ... >. Then 0-( G) is an HNN-extension of the free product of G with the infinite cyclic group generated by x. The associated subgroups are just the infinite cyclic groups genrated by the [Ui,X] and the ViX-1UiX. A routine argument shows that the word problem for u( G) in the indicated presentation can be solved using the given solution to the word problem for G. Also observe that in 0-( G) each Vi lies in the normal closure of the corresponding Ui.

In this case one obtains a recursive family of presentations {'Trw I w E U} indexed by words in a group U with unsolvable word problem such that 'Trw ~ 1 if and only if w =u 1. Moreover, if we suppose F has free basis {81, ... , 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, ...

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

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