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. issues lined contain choice difficulties, finitely provided easy teams, combinatorial geometry and homology, and automated teams and similar themes.
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Extra info for Algorithms and Classification in Combinatorial Group Theory
That the generalized word problem is solvable is a theorem of Nielsen . The isomorphism type of a free group is determined by its rank which can be easily computed from any presentation by considering its abelianization. Hence all of the basic decision problems are solvable for finitely generated free groups. In this section we survey what is known about the fundamental decision problems for classes of groups enjoying some of the more familiar algebraic properties, for example abelian, solvable, linear and so on.
Alternatively, the generalized word problem for polycyclic groups can be solved by a direct inductive method (see ). Quite remarkably, Grunewald and Segal  have shown that the isomorphism problem for finitely generated nilpotent groups is recursively solvable. And more recently Segal  has succeeded in solving the isomorphism problem for polycyclic groups. In fact all of the algorithms mentioned carry over to the larger class of polycyclic-by-finite groups. 7. The word problem, conjugacy problem, generalized word problem and isomoprphism problem for polycyclic-by-finite groups are recursively solvable.
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 ). Gersten and Short  have obtained useful information about subgroups of biautomatic and hyperbolic groups. Normal forms and rewriting systems: Continuing with the above notations, one difficulty with Dehn's algorithm for solving the word problem is the following: if R is a Dehn's algorithm and we apply the process of R-reduction to a word w i-c 1 the resulting word, say p( w) will be Rreduced but it is not unique.