Conjugacy in Baumslag’s Group, Generic Case Complexity, and Division in Power Circuits
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.
KeywordsAlgorithmic group theory power circuit generic case complexity
Unable to display preview. Download preview PDF.
- 2.Beese, J.: Das Konjugationsproblem in der Baumslag-Gersten-Gruppe. Diploma thesis, Fakultät Mathematik, Universität Stuttgart (2012) (in German)Google Scholar
- 4.Ceccherini-Silberstein, T., Grigorchuk, R.I., de la Harpe, P.: Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces. Tr. Mat. Inst. Steklova 224, 68–111 (1999)Google Scholar
- 7.Diekert, V., Miasnikov, A., Weiß, A.: Conjugacy in Baumslag’s group, generic case complexity, and division in power circuits. CoRR, abs/1309.5314 (2013)Google Scholar
- 8.Gersten, S.M.: Isodiametric and isoperimetric inequalities in group extensions (1991)Google Scholar
- 14.Lyndon, R., Schupp, P.: Combinatorial Group Theory, 1st edn. (1977)Google Scholar
- 16.Miller III, C.F.: On group-theoretic decision problems and their classification. Annals of Mathematics Studies, vol. 68. Princeton University Press (1971)Google Scholar
- 19.Myasnikov, A.G., Ushakov, A., Won, D.W.: Power circuits, exponential algebra, and time complexity. IJAC 22, 51 pages (2012)Google Scholar
- 23.Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Univ. Press (2000)Google Scholar