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

Abstract

The conjugacy problem asks whether two words over generators of a fixed group G are conjugated, i.e., it is the problem to decide on input words x, y whether there exists z such that $$zx z^{-1} =y$$zxz-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 $$\mathbf{{BS}}_{1,2}$$BS1,2 and the Baumslag group $${\mathbf{{G}}}_{1,2}$$G1,2 (also known as Baumslag---Gersten group). The conjugacy problem in $${\mathbf{{BS}}}_{1,2}$$BS1,2 is complete for the circuit class $$\mathsf {TC}^0$$TC0. To the best of our knowledge $${\mathbf{{BS}}}_{1,2}$$BS1,2 is the first natural infinite non-commutative group where such a precise and low complexity is shown. The Baumslag group $${\mathbf{{G}}}_{1,2}$$G1,2 is an HNN-extension of $${\mathbf{{BS}}}_{1,2}$$BS1,2. Hence, decidability of the conjugacy problem in $$\mathbf{{G}}_{1,2}$$G1,2 outside the so-called black hole follows from Borovik et al. (Int J Algebra Comput 17(5/6):963---997, 2007). Decidability everywhere is due to Beese. Moreover, he showed exponential time for the set of elements which cannot be conjugated into $$\mathbf{{BS}}_{1,2}$$BS1,2 (Beese 2012). Here we improve Beese's result in two directions by showing that the conjugacy problem in $${\mathbf{{G}}}_{1,2}$$G1,2 can be solved in polynomial time in a strongly generic setting. This means that essentially for all inputs, conjugacy in $${\mathbf{{G}}}_{1,2}$$G1,2 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 $${\mathbf{{G}}}_{1,2}$$G1,2 by reducing the divisibility problem in power circuits to the conjugacy problem in $${\mathbf{{G}}}_{1,2}$$G1,2. The complexity of the divisibility problem in power circuits is an open and interesting problem in integer arithmetic. We conjecture that it cannot be solved in elementary time because we can show that it cannot be solved in elementary time by calculating modulo values in power circuits.