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

The Isomorphism Problem for Finite Extensions of Free Groups Is In PSPACE Géraud Sénizergues LABRI, Bordeaux, France Armin Weiß Universität Stuttgart, FMI, Germany Abstract 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. 2012 ACM Subject Classification Mathematics of computing → Graph theory, Theory of computation → Grammars and context-free languages, Theory of computation → Computational complexity and cryptography Keywords and phrases virtually free groups, context-free groups, isomorphism problem, structure tree, graph of groups Digital Object Identifier 10.4230/LIPIcs.ICALP.2018.139 Related Version A full version of the paper is available at [23], https://arxiv.org/abs/1802. 07085. Acknowledgements G.S. thanks the FMI for hosting him from October to the end of the year 2017. Both authors acknowledge the financial support by the DFG project DI 435/7-1 “Algorithmic problems in group theory” for this work. 1 Introduction The study of algorithmic problems in group theory was initiated by Dehn [6] when he introduced the word and the isomorphism problem. The word problem asks whether a given word over a (finite) set of generators represents the identity of the group. It also can be viewed as a formal language, namely ϕ−1 (1) ⊆ Σ∗ for some surjective monoid homomorphism ϕ : Σ∗ → G. The isomorphism problem receives two finite presentations as input, the question is whether the groups they define are isomorphic. Although both these problems are undecidable in general [18, 3], there are many classes of groups where at least the word problem can be decided efficiently. One of these classes are the finitely generated virtually free groups (groups with a free subgroup of finite index). It is easy to see that the word problem of a finitely generated virtually free group can be solved in linear time. Indeed, it forms a deterministic context-free language. A seminal paper by Muller and Schupp [16] shows the converse: every group EA TC S © Géraud Sénizergues and Armin Weiß; licensed under Creative Commons License CC-BY 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Editors: Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella; Article No. 139; pp. 139:1–139:14 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 139:2 The Isomorphism Problem for Finite Extensions of Free Groups Is In PSPACE with a context-free word problem is virtually free. Since then, also a wide range of other characterizations of virtually free groups have emerged – for a survey we refer to [1, 9]. The isomorphism problem of virtually free groups is also decidable as Krstić showed in [15] (indeed, later Dahmani and Guirardel showed that even the isomorphism problem for the larger class of hyperbolic groups is decidable [5]). Here the input consists of two arbitrary finite presentations with the promise that both define virtually free groups. Unfortunately, the approach in [15] does not give any bound on the complexity. For the special case where the input is given as finite extensions of free groups or as context-free grammars for the word problems, Sénizergues [21, 22] showed that the isomorphism problem is primitive recursive. Krstić’s and Sénizergues’ approaches both compute so-called graphs of groups, which encode groups acting on trees, and then check these graph of groups for “isomorphism”. By the work of Karrass, Pietrowski and Solitar [14], a finitely generated group is virtually free if and only if it is the fundamental group of a finite graph of groups with finite vertex groups. Contribution. We improve the complexity for the isomorphism problem by showing: (A) Given a context-free grammar for the word problem of a context-free group G, a graph O(n2 ) of groups for G with finite vertex groups can be computed in NTIME(22 ) (Theorem 33). (B) Given a virtually free presentation for G, a graph of groups for G with finite vertex groups can be computed in NP (Theorem 34). (C) The isomorphism problem for context-free groups given as grammars is in 2 2O(n ) DSPACE(2 ) (Theorem 37). (D) The isomorphism problem for virtually free groups given as virtually free presentations is in PSPACE (Theorem 38). Here, a virtually free presentation for G consists of a free group F plus a set of representatives S for the quotient F \G together with relations describing pairwise multiplications of elements from F and S. Typical examples of virtually free presentations are finite extensions of free groups (i. e., where the free sugroup F is normal in G). For non-deterministic function problems we use the convention, that every accepting computation must yield a correct result; but the results of different accepting computations might differ1 . The results C and D can be seen be to follow from A and B rather easily. Indeed, we conclude from Forester’s work on deformation spaces [10] that two graphs of groups with finite vertex groups and isomorphic fundamental groups can be transformed one into each other by a sequence of slide moves (Proposition 35). Our approach for proving A and B is as follows: in both cases the algorithm simply guesses a graph of groups together with a map and afterwards it verifies deterministically whether the map is indeed an isomorphism. The latter can be done using standard results from formal language theory. The difficult part is to show the existence of a “small” graph of groups and isomorphism (within the bounds of A and B). For this, we introduce the structure tree theory by Dicks and Dunwoody [7] following a slightly different approach by Diekert and Weiß [8] based on the optimal cuts of the Cayley graph (Section 2.3). The optimal cuts can be seen as the edge set of some tree on which the group G acts. By Bass-Serre theory, this yields the graph of groups we are aiming 1 Thus, B means that the graph of groups can be computed in NPMV in the sense of [20]. More precisely, it can be rephrased as follows: the multi-valued function mapping a virtually free presentation for G into a pair (G, ϕ), where G is a graph of groups and ϕ : π1 (G) → G is an isomorphism of polynomial size, is everywhere defined and belongs to the class FNP as defined in [19]. G. Sénizergues and A. Weiß 139:3 for. Vertices in the graph of groups are defined in terms of equivalence classes of optimal cuts. The key in the proof is to bound the (...truncated)