Conjugacy in Baumslag’s Group, Generic Case Complexity, and Division in Power Circuits

Abstract

The conjugacy problem is the following question: given two words x, y over generators of a fixed group G, decide whether x and y are conjugated, i.e., whether there exists some z such that zx z − 1 = y in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the conjugacy problem for two prominent groups: the Baumslag-Solitar group BS 1,2 and the Baumslag(-Gersten) group G 1,2. The conjugacy problem in BS 1,2 is TC 0-complete. To the best of our knowledge BS 1,2 is the first natural infinite non-commutative group where such a precise and low complexity is shown. The Baumslag group G 1,2 is an HNN-extension of BS 1,2 and its conjugacy problem is decidable G 1,2 by a result of Beese (2012). Here we show that conjugacy in G 1,2 can be solved in polynomial time in a strongly generic setting. This means that essentially for all inputs conjugacy in G 1,2 can be decided efficiently. In contrast, we show that under a plausible assumption the average case complexity of the same problem is non-elementary. Moreover, we provide a lower bound for the conjugacy problem in G 1,2 by reducing the division problem in power circuits to the conjugacy problem in G 1,2. The complexity of the division problem in power circuits is an open and interesting problem in integer arithmetic.