Many fundamental problems are undecidable for innnite matrix groups. Polycyclic matrix groups represent a large class of groups for which these same problems are known to be decidable. In this paper we describe a suite of new algorithms for studying polycyclic matrix groups | algorithms for testing membership and for uncovering the polycyclic structure of… (More)
Counter automata are more powerful versions of finite-state automata where addition and subtraction operations are permitted on a set of n integer registers, called counters. We show that the word problem of Z n is accepted by a nondeterministic m-counter automaton if and only if m ≥ n.
We present an algorithm to solve the orbit-stabilizer problem for a polycyclic group G acting as a subgroup of GL(d, Z) on the elements of Q d. We report on an implementation of our method and use this to observe that the algorithm is practical.
We study finitely generated groups whose word problems are accepted by counter automata. We show that a group has word problem accepted by a blind n-counter automaton in the sense of Greibach if and only if it is virtually free abelian of rank n; this result, which answers a question of Gilman, is in a very precise sense an abelian analogue of the… (More)
Let K be a number eld. We present several algorithms for working with polycyclic-by-nite subgroups of GL(n; K). Let G be a subgroup of GL(n; K) given by a nite generatingset of matrices. We describe an algorithm for deciding whether or not G is polycyclic-by-nite. For polycyclic-by-nite G, we describe an algorithm for deciding whether or not a given matrix… (More)
a r t i c l e i n f o a b s t r a c t MSC: 20F65 68Q15 Keywords: Logspace algorithm Logspace normal form Logspace embeddable Wreath product Baumslag–Solitar group Logspace word problem We consider the class of finitely generated groups which have a normal form computable in logspace. We prove that the class of such groups is closed under passing to finite… (More)
We describe a new algorithm for nding matrix representations for polycyclic groups given by nite presentations. In contrast to previous algorithms, our algorithm is eecient enough to construct representations for some interesting examples. The examples which we studied included a collection of free nilpotent groups, and our results here led us to a… (More)
We consider the class of finitely generated groups which have a normal form computable in logspace. We prove that the class of such groups is closed under passing to finite index subgroups, direct products, wreath products , and certain free products and infinite extensions, and includes the solv-able Baumslag-Solitar groups, as well as non-residually… (More)