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. subject matters lined comprise determination difficulties, finitely awarded uncomplicated teams, combinatorial geometry and homology, and automated teams and comparable subject matters.

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**Example text**

G. p. resid. P. Miller As we have mentioned before, the structure of a finitely generated abelian group can be completely and effectively determined from a finite presentation for such a group. In particular this enables one to solve the word problem in each such group and to solve the ismorphism problem for the class of such groups. Now for abelian groups conjugacy is the same as equality so the conjugacy problem is also solvable. The generalized word problem for a subgroup H of an abelian group G is equivalent to the word problem for G / H so GW P( G) is also solvable.

In small cancellation theory one consider various cancellation hypotheses on a symmetrized set of words R and uses them to deduce properties of the group G = F / N. In what follows we assume R is a symmetrized set of words. If R contains two distinct words of the form rl == bCI and r2 == bC2 then the word b is called a piece relative to R or simply a piece when R is understood. Observe that, in forming the product r 1l r2 and freely reducing, such a piece b is cancelled. Thus a piece is simply a subword of an element of R which can be cancelled by the multiplication of two noninverse elements of R.

It should be emphasized again that rewrite rules are not required to be length reducing. If a group G has a finite, length reducing, complete rewriting system, then that system gives a Dehn's algorithm and, moreover, it is known (see [68]) that G must be virtually free. One interesting feature of groups with a finite complete rewriting system is that they are of type FP oo (see [6J, [25J, [46J and [100]). Moreover, one can in principle effectively calculate free resolutions and carry out certain homological calculations for such groups.

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

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