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. subject matters coated contain selection difficulties, finitely offered easy teams, combinatorial geometry and homology, and automated teams and similar subject matters.
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Additional info for Algorithms and Classification in Combinatorial Group Theory
Since the word problem for H is unsolvable, the problem of deciding membership in LH is recursively unsolvable. This proves the theorem. 3 (Mihailova ). Let F be a finitely generated free group of rank at least 2. Then the group G = F x F has a finitely generated subgroup L such that the generalized word problem for L in G is recursively unsolvable. The above lemma can also be used to obtain a number of other unsolvability results concerning the direct product of two free groups. 4 (Miller ).
If G is a finite extension of the finitely generated group H having solvable generalized word problem, then G has solvable generalized word problem. 11 (). (1) There is a finitely presented group G 1 with unsolvable conjugacy problem that has a subgroup M of index 2 which has solvable conjugacy problem. (2) There is a finitely presented group G 2 with solvable conjugacy problem that has a subgroup L of index 2 which has unsolvable conjugacy problem. An example of the first type was given by Gorjaga and Kirkinskii , while examples of both of these phenomena were given by Collins and Miller .
As above, either w appears on the list of words equal to 1 in H or else since S is simple U appears on the list of words equal to 1 in Hw (in which case w i-H 1 ). So by enumerating these two lists we can decide whether w is equal to 1 in G. For the converse, suppose G has solvable word problem. Then the set of all pairs of words (u, v) such that u i-e 1 and v i-e 1 is recursive and can be arranged in a recursive list as say (Ui,Vi),i = 1,2, .... Let X,tl,t2, ... be new generating symbols and form the presentation u(G) =< G,x,tl,t2,··.