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 coated contain choice difficulties, finitely provided uncomplicated teams, combinatorial geometry and homology, and automated teams and comparable themes.
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Extra resources for Algorithms and Classification in Combinatorial Group Theory
However, the foregoing results show that the generalized word problem, the conjugacy problem and the isomorphism problem can be recursively unsolvable for groups in WI. Since the generalized word problem is unsolvable for a group in WI, it follows that the word problem for a suitable group in W2 is unsolvable. For example, if F is free of rank at least two and L is a finitely generated subgroup such that the generalized word problem for L in F x F is unsolvable, one can form the HNN-extension G =< F x F, t I rIxt = x, x E L >.
It can be shown that if the finitely presented group G is defined by two finite presentations P and P1 , then there are constants Cl, C2 and C3 such that OPl (n) :::; C10p(C2n) + C3n. In a different direction, suppose that the finite presentation P2 =< X I R2 > of G is obtained from P by adding some elements of N to R so that R ~ R2. Then A p2 (w) :::; Ap(w) for all wEN and Op2(n) :::; Op(n). The above corollary can be restated now as follows: the word problem for G is solvable if and only if there is a recursive function f such that Decision Problems for Groups: Survey and Reflections 43 Dp(n) ::::; f(n) for all n > O.
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 .
Algorithms and Classification in Combinatorial Group Theory by Charles F. Miller III (auth.), Gilbert Baumslag, Charles F. Miller III (eds.)