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. issues coated contain choice difficulties, finitely offered basic teams, combinatorial geometry and homology, and automated teams and similar subject matters.

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Thus we can effectively enumerate all homomorphisms Ii from G into finite groups K i . Now to decide whether w =a 1 we start both listing processes. As each Ii is enumerated, evaluate li(W) and check to see whether Ji(w) = 1 in K i . Since G is residually finite, if w i-a 1 then for some i one eventually finds li(W) i- 1 in K i · On the other hand if w =a 1 then w will appear in the first list. So by waiting until one of these two events occurs we can decide whether w =a 1. Despite the solvability of the word problem for finitely presented, residually finite groups the results explained in the previous section show the other fundamental decision problems are all unsolvable.

Thus every group G has a hereditary set of normal forms T with respect to a generating set X. Assuming X is finite, the set T constructed in this way is recursive if and only if G has solvable word problem. So normal forms exist, but what of "transforming" words into normal form? One such notion has been investigated by computer scientists ( see Decision Problems for Groups: Survey and Reflections 49 [64] or [68]). Define a rewrite rule to be an ordered pair (u, v) of words of * such that u =a v. *

7. There exist asynchronously automatic groups with unsolvable conjugacy problem. The isomorphism problem for asynchronously automatic groups is recursively unsolvable. For more information on asynchronously automatic groups see [17]. These various classes of (bi)automatic groups have number of other interesting properties. Thurston has shown that automatic groups are of type FP 00 (see [4]). Gersten and Short [39] have obtained useful information about subgroups of biautomatic and hyperbolic groups.