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 lined contain choice difficulties, finitely provided basic teams, combinatorial geometry and homology, and automated teams and similar subject matters.
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Extra info for Algorithms and Classification in Combinatorial Group Theory
This completes our commentary concerning the summary table. Decision problems for certain other algebraic classes will be considered in a subsequent section. 6. Algorithms for further classes of groups In this section we review the status of the fundamental decision problems for some further classes of groups which did not fit conveniently into the previous two sections. Nevertheless algorithmic questions concerning these groups have played an important role in ongoing developments. One relator groups: Consider a group G defined by a single defining relation, say G =< Xl, ...
The construction of the previous theorem can be combined with the construction used in proving the Adian-Rabin Theorem to show that the isomorphism problem for groups with such a very elementary structure is unsolvable. The details are somewhat more difficult. 9 (Miller ). Let U be a group with unsolvable word problem. F. Miller (2) each gp(7fw ) is the split extension of one finitely generated free group by another; (3) the word problem for each of the groups gp(7fw ) is solvable by a uniform method; (4) gp(7fw ) ~ gp( 7fd if and only if w =u 1.
To decide whether an arbitrary word w is equal to 1 in G begin recursively enumerating two lists of words. F. Miller equal to 1 in G. The second list consists of all words equal to 1 in G w . If W =e 1 then W will appear in the first list. But w i-e 1 if and only if U appears in the second list. By examining the lists until one of these events occurs we can determine whether or not w is equal to 1 in G. This completes the proof. In particular, finitely presented simple groups have solvable word problem.
Algorithms and Classification in Combinatorial Group Theory by Charles F. Miller III (auth.), Gilbert Baumslag, Charles F. Miller III (eds.)