Conjugacy in Baumslag's group, generic case complexity, and division in power circuits

TL;DR

This paper analyzes the complexity of the conjugacy problem in Baumslag's groups, showing low complexity in some cases and high complexity in others, and connects it to division in power circuits.

Contribution

It provides the first precise complexity classification for conjugacy in BS(1,2), demonstrates polynomial-time solutions in a generic setting for G(1,2), and links the problem to division in power circuits.

Findings

Conjugacy in BS(1,2) is TC^0-complete.

In G(1,2), conjugacy is generically decidable in polynomial time.

Average case complexity of conjugacy in G(1,2) is non-elementary under plausible assumptions.

Abstract

The conjugacy problem belongs to algorithmic group theory. It 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 zxz^{-1} = y in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the complexity of 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). We show that the conjugacy problem is decidable (which has been known before); but our results go far beyond decidability. In particular, we are able to 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.