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

TL;DR

This paper analyzes the dimensions of matrix embeddings of torsion-free nilpotent groups, establishing exponential lower bounds, upper bounds, and improvements for specific cases like free nilpotent and Heisenberg groups.

Contribution

It provides new bounds on the size of embeddings produced by Nickel's algorithm and compares them with Jennings' embeddings, highlighting cases with quadratic size.

Findings

Embedding dimension grows exponentially with group rank under certain conditions

Nickel's algorithm has exponential worst-case running time

Special Mal'cev bases can produce quadratic-sized embeddings

Abstract

Since the work of Jennings (1955), it is well-known that any finitely generated torsion-free nilpotent group can be embedded into unitriangular integer matrices $UT_N(Z)$ for some $N$. In 2006, Nickel proposed an algorithm to calculate such embeddings. In this work, we show that if $UT_n(Z)$ is embedded into $UT_N(Z)$ using Nickel's algorithm, then $N \geq 2^{n/2 -2}$ if the standard ordering of the Mal'cev basis (as in Nickel's original paper) is used. In particular, we establish an exponential worst-case running time of Nickel's algorithm. On the other hand, we also prove a general exponential upper bound on the dimension of the embedding by showing that for any torsion free, finitely generated nilpotent group the matrix representation produced by Nickel's algorithm has never larger dimension than Jennings' embedding. Moreover, when starting with a special Mal'cev basis, Nickel's embedding for $UT_n(Z)$ has only quadratic size. Finally, we consider some special cases like free nilpotent groups and Heisenberg groups and compare the sizes of the embeddings.