Abstract
The circuit evaluation problem (also known as the compressed word problem) for finitely generated linear groups is studied. The best upper bound for this problem is coRP, which is shown by a reduction to polynomial identity testing (PIT). Conversely, the compressed word problem for the linear group \(\mathsf {SL}_3(\mathbb {Z})\) is equivalent to PIT. In the paper, it is shown 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 PIT for skew arithmetical circuits.
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