Evaluation of Circuits Over Nilpotent and Polycyclic Groups

Abstract

We study the circuit evaluation problem (also known as the compressed word problem) for finitely generated linear groups. The best upper bound for this problem is coRP (the complements of problems in randomized polynomial time), which is shown by a reduction to polynomial identity testing for arithmetic circuits. Conversely, the compressed word problem for the linear group \({\mathsf {SL}}_3({\mathbb {Z}})\) is equivalent to polynomial identity testing. In the paper, we show that the compressed word problem for every finitely generated nilpotent group is in \({\mathsf {DET}} \subseteq {\mathsf {NC}}^2\). Within the larger class of polycyclic groups we find examples where the compressed word problem is at least as hard as polynomial identity testing for skew arithmetic circuits. It is a major open problem, whether polynomial identity testing for skew arithmetic circuits can be solved in polynomial time.