Multiplicative equations over commuting matrices
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
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J. Tits (1972) has shown that a finitely generated linear group either contains a nonabelian free group or has a solvable subgroup of finite index. We give a polynomial time algorithm for deciding which of these two conditions holds for a given finitely generated matrix group over an algebraic number field. Noting that many computational problems are undecidable for groups with nonabelian free subgroups, we investigate the complexity of problems relating to linear groups with solvable subgroups of finite index. For such a group G, we are able in polynomial time to compute a homomorphism /spl phi/ such that /spl phi/(G) is a finite matrix group and the kernel of /spl phi/ is solvable. If in addition G has a nilpotent subgroup of finite index, we obtain much stronger results. These include an effective encoding of elements of G such that the encoding length of an element obtained as a product of length /spl les/l over the generators is O(logl) times a polynomial in the input length. This result is the best possible. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks including membership testing and computing a presentation. This generalizes recent work of R. Beals and L. Babai (1993), who give a Las Vegas algorithm for the case of finite groups.