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 examine Institute. themes coated contain determination difficulties, finitely provided uncomplicated teams, combinatorial geometry and homology, and automated teams and comparable themes.

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**Sample text**

An example of the first type was given by Gorjaga and Kirkinskii [41], while examples of both of these phenomena were given by Collins and Miller [30]. The proofs are somewhat technical. A further aspect of the above results is the following: denote by lITo the set of finitely generated free groups. Then denote by lIT 1 the collection of groups formed from groups in lITo by either free product with finitely generated amalgamation or HNN-extension with finitely many stable letters and finitely generated associated subgroups.

Observe that C'(A) implies C(p) for A :::; 1/(P - 1). Thus C'(~) implies C(7). F. Miller condition T(q) for q a natural number. R satisfies T(q) if for every sequence rI,"" rm (3 ::; m ::; q) with no successive inverse pairs, at least one of the products rIr2, ... ,rm-Irm , rmrI is reduced without cancellation. (This condition turns out to be dual to to the condition C(p) when 1/p+1/q = 1/2 in a suitable geometric sense. ) Small cancellation theory was initiated by Tartakovskii [101] [102] [103] who solved the word problem for groups whose defining relators R satisfy C(7).

In particular, if c is a constant larger than the lengths of all the words in ~ then ~c = {w E N Ilwl :::; c} is also a Dehn's algorithm. 3 below, one can check that having a Dehn's algorithm is independent of the choice of generating set and hence is an abstract property of the group. The following are examples of groups having presentations with a Dehn's algorithm: free groups, finite groups (multiplication table presentation), and groups satisfying the cancellation condition G'(i) (Greendlinger's Lemma).