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The algorithmic theory of polycyclic-by-finite groups. (English) Zbl 0774.20019
A polycyclic group is a group having a normal series with cyclic factors; a polycyclic-by-finite group is a group containing a polycyclic subgroup of finite index. Polycyclic groups were introduced by K. A. Hirsch in 1938 [Proc. Lond. Math. Soc., II. Ser. 44, 53–60 (1938; Zbl 0018.14505); 44, 336–344 (1938; Zbl 0019.15602; JFM 64.0066.01)]; an exposition of the basic theory of these groups can be found in D. Segal’s book [Polycyclic groups. Cambridge Tracts Math. 82. Cambridge etc.: Cambridge University Press (1983; Zbl 0516.20001)].
The paper under review is concerned with producing algorithms for answering various group-theoretic questions about polycyclic-by-finite groups. Some results (such as solutions to the generalized word problem and the conjugacy problem) have already been obtained by other authors; these results are discussed in detail in the introduction to the paper. The new results obtained in the paper are quite extensive. Taken together with the paper by D. Segal [Proc. Lond. Math. Soc., III. Ser. 61, 497–528 (1990; Zbl 0674.20020)] they provide a comprehensive treatment of algorithmic questions concerning polycyclic-by-finite groups.
There are far too many results to discuss in a short review, but statements of two key results should give a flavour of the material: (1) There is an algorithm which on being given a finite presentation of a polycyclic-by-finite group and a finite set \(X\) of words in the generators, produces a finite presentation of the subgroup generated by \(X\) (Theorem 3.4); (2) There is an algorithm which, on being given a finite subset \(Y\) of \(\mathrm{GL}(n,\mathbb{Z})\), decides if the subgroup generated by \(Y\) is polycyclic-by-finite, and, if so produces a finite presentation of it (Theorem 4.1). [In relation to (2), recall that every polycyclic-by- finite group can be embedded in \(\mathrm{GL}(n,\mathbb{Z})\) for some \(n\) (L. Auslander, Ann. Math. (2) 86, 112–116 (1967; Zbl 0149.26904); R. G. Swan, Proc. Am. Math. Soc. 18, 573–574 (1967; Zbl 0153.03801)].

MSC:
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F05 Generators, relations, and presentations of groups
20F16 Solvable groups, supersolvable groups
20F19 Generalizations of solvable and nilpotent groups
20E07 Subgroup theorems; subgroup growth
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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