The Isomorphism Problem for Finite Extensions of Free Groups Is In PSPACE

Authors Géraud Sénizergues, Armin Weiß

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Géraud Sénizergues
  • LABRI, Bordeaux, France
Armin Weiß
  • Universität Stuttgart, FMI, Germany

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Géraud Sénizergues and Armin Weiß. The Isomorphism Problem for Finite Extensions of Free Groups Is In PSPACE. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 139:1-139:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We present an algorithm for the following problem: given a context-free grammar for the word problem of a virtually free group G, compute a finite graph of groups G with finite vertex groups and fundamental group G. Our algorithm is non-deterministic and runs in doubly exponential time. It follows that the isomorphism problem of context-free groups can be solved in doubly exponential space. Moreover, if, instead of a grammar, a finite extension of a free group is given as input, the construction of the graph of groups is in NP and, consequently, the isomorphism problem in PSPACE.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Grammars and context-free languages
  • Theory of computation → Computational complexity and cryptography
  • virtually free groups
  • context-free groups
  • isomorphism problem
  • structure tree
  • graph of groups


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