On the dimension of matrix representations of finitely generated torsion free nilpotent groups

Abstract

It is well known that any polycyclic group, and hence any finitely generated nilpotent group, can be embedded into GLn(Z) for an appropriate n ∈ N; that is, each element in the group has a unique matrix representation. An algorithm to determine this embedding was presented in [6]. In this paper, we determine the complexity of the crux of the algorithm and the dimension of the matrices produced as well as provide a modification of the algorithm presented in [6].

On the dimension of matrix embeddings of torsion-free nilpotent groups

“…Up to now there are no bounds known neither on the running time nor on the dimension of the embedding obtained by Nickel's algorithm. In [3], a polynomial bound for both is claimed; however, there is a gap in the proof. Indeed, here we prove these results to be wrong: our main result (Theorem 3.9) establishes the lower bound of N ≥ 2 n/2−2 for the embedding of U T n (Z) (with the standard Mal'cev basis as in Nickel's paper) into U T N (Z) computed by Nickel's algorithm.…”

A linear decomposition attack

“…Graaf and Nickel [14] and Nickel in [33] studied the classical linear representations of finitely generated torsion-free nilpotent groups G into U T n (Z) from the algorithmic view-point, showing in particular that there are polynomial time (in the Hirsh length of G) algorithms to construct the representations. Recently, in [17] Habeeb and Kahrobaei, following [33], showed that the dimension of the classical representations above is O(n 2 ) where n is the Hirsh length of the group. It seems the dimension is not prohibitively high in this case.…”