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The Compressed Word Problem for Groups Markus Lohrey The Compressed Word Problem for Groups – Monograph – April 3, 2014 Springer Dedicated to Lynda, Andrew, and Ewan. Preface The study of computational problems in combinatorial group theory goes back more than 100 years. In a seminal paper from 1911, Max Dehn posed three decision prob- lems [46]: The word problem (called Identitatsproblem¨ by Dehn), the conjugacy problem (called Transformationsproblem by Dehn), and the isomorphism problem. The first two problems assume a finitely generated group G (although Dehn in his paper requires a finitely presented group). For the word problem, the input con- sists of a finite sequence w of generators of G (also known as a finite word), and the goal is to check whether w represents the identity element of G. For the con- jugacy problem, the input consists of two finite words u and v over the generators and the question is whether the group elements represented by u and v are conju- gated. Finally, the isomorphism problem asks, whether two given finitely presented groups are isomorphic.1 Dehns motivation for studying these abstract group theo- retical problems came from topology. In a previous paper from 1910, Dehn studied the problem of deciding whether two knots are equivalent [45], and he realized that this problem is a special case of the isomorphism problem (whereas the question of whether a given knot can be unknotted is a special case of the word problem). In his paper from 1912 [47], Dehn gave an algorithm that solves the word problem for fundamental groups of orientable closed 2-dimensional manifolds. His algorithm is nowadays known as Dehn’s algorithm and can be applied to a larger class of groups (so called hyperbolic groups). But Dehn also realized that his three problems seem to be very hardin general.In [46], he wrote “Die drei Fundamentalprobleme fur¨ alle Gruppen mit zwei Erzeugenden ... zu losen,¨ scheint einstweilen noch sehr schwierig zu sein.”(Solving the three fundamentalproblems for all groups with two generators seems to be very difficult at the moment.) When Dehn wrote this sentence, a formal definition of computability was still missing. Only in the first half of the 1930’s, the foundations of the modern theory of computability where laid by Alonzo Church, Kurt G¨odel, Emil Post, and Alan Turing, see e.g. Chapter 1 in the textbook [40] for a brief historial outline. Another 20 years later, Pyotr Sergeyevich Novikov [136] and independently William Werner Boone [22] proved that Dehn’s intuition concerning 1 Implicitly, the isomorphism problem can be also found in Tietze’s paper [161] from 1908. vii viii Preface the difficulty of his problems was true: There exist finitely presented groups with an undecidable word problem (and hence an undecidable conjugacy problem). The isomorphism problem turned out to be undecidable around the same time [1, 145]. More details on the history around Dehn’s three decision problems can be found in [158]. Despite these negative results, for many groups the word problem is decidable. Dehn’s result for fundamental groups of orientable closed 2-dimensional manifolds was extended to one-relator groups by his student Wilhelm Magnus [121]. Other important classes of groups with a decidable word problem are automatic groups [57] (including important classes like braid groups [8], Coxeter groups, right-angled Artin groups, hyperbolic groups [68]) and finitely generated linear groups, i.e., groups of matrices over a field [144] (including polycyclic groups). This makes it interesting to study the complexity of the word problem. Computational complexity theory is a part of computer science that emerged in the mid 1960’s and investigates the relation between (i) the time and memory of a computer necessary to solve a computational problem and (ii) the size of the input. For a word problem, the in- put size is simply the length of the finite sequence of generators. Early papers on the complexity of word problems for groups are [27, 28, 29, 144]. One of the early results in this context was that for every given n ≥ 0 there exist groups for which the word problem is decidable but does not belong to the n-th level of the Grze- gorczyk hierarchy (a hierarchy of decidable problems) [27]. On the other hand, for many prominent classes of groups the complexity of the word problem is quite low. For finitely generated linear groups Richard Jay Lipton and Yechezkel Zalcstein [108] proved that logarithmic working space (and hence polynomial time) suffices to solve the word problem. The class of automatic groups is another quite large class of groups including many classical group classes for which the word problem can be solved in quadratic time [57]. In contrast, groups with a difficult word prob- lem seem to be quite exotic objects. The vague statement that for most groups the word problem is easy can be even made precise: Mikhail Gromov stated in [68] that a randomly chosen (according to a certain probabilistic process) finitely presented group is hyperbolic with probability 1, a proof was given by Alexander Ol´shanski˘ı in [137]. Moreover, for every hyperbolic group the word problem can be solved in linear time. Hence, for most groups the word problem can be solved in linear time. One of the starting points for the work in this book was a question from the 2003 paper [94] by Ilya Kapovich, Alexei Myasnikov, Paul Eugene Schupp, and Vladimir Shpilrain: Is there a polynomial time algorithm for the word problem for the auto- morphism group of a finitely generated free group F? For every group G the set of all automorphisms of G together with the composition operation is a group, the automorphism group Aut(G) of G. Clearly, if G is finite, then also Aut(G) is finite. But if G is only finitely generated, then Aut(G) is not necessarily finitely gener- ated, see [105] for a counterexample. In a seminal paper [135] from 1924, Jakob Nielsen proved that the automorphism group of a finitely generated free group F is finitely generated (and in fact finitely presented). This makes it interesting to study the word problem for Aut(F). The straightforward algorithm for this problem has an exponential running time. Saul Schleimer [151] realized in 2006 that one can easily Preface ix reduce the word problem for Aut(F) to a variant for the word problem for F, where the input sequence is given in a succinct form by a so called straight-line program. A straight-line program can be seen as a hierarchical description of a string. From a formal language theory point of view it is a context-free grammar G that generates exactly one string that we denote with val(G). The length of this string can be ex- ponential in the size of the grammar G. In this sense, G can be seen as a succinct description of val(G). The variant of the word problem for a group G, where the input sequence is given by a straight-line program is the compressed word problem for G, which is the main topic of this book. Explicitly this problem was first studied in my paper [110] (a long version appeared as [112]). My motivation for this work came from two other papers: In 1994, Wojciech Plandowski [139] proved that one can decide in polynomial time whether two straight-line programs produce the same words; implicitly this result can be also found in papers by Kurt Mehlhorn, Raja- mani Sundar, and Christian Uhrig [127, 128] and Yoram Hirshfeld, Mark Jerrum and Faron Moller [79, 80] that were both published the same year as Plandowski’s work. One could also rephrase this result by saying that the compressed word prob- lem for a free monoid can be solved in polynomial time. My second motivation for studying the compressed word problem is the paper [15] by Martin Beaudry, Pierre McKenzie, Pierre P´eladeau and Denis Th´erien, where the authors studied the problem whether a given circuit over a finite monoid evaluates to a given monoid element. This problem can be seen as the compressed word problem for a finite monoid.2 Let me come back to the word problem for the automorphism group of a free group. In my paper [110] I proved that the compressed word problem for a finitely generatedfree groupcan be solved in polynomialtime; the proofbuilds on the above mentioned result by Plandowski. Using this result, Schleimer was able to show that the word problem for Aut(F) (with F a finitely generated free group) can be solved in polynomial time [151], which answered the question from [94] positively. After Schleimer’s work, it turned out that the compressed word problem has applications not only for the word problem of automorphism groups, but also for semidirect products and certain group extensions. This motivated the study of the compressed word problem as a topic of its own interest. The aim of this book is to give an extensive overview of known results on the compressed word problem. Some of the results in this book can be found in published articles while others ap- pear to be new. As a computer scientist with a strong interest in group theory, I tried to make this book accessible to both mathematicians and computer scientists. In particular, I tried to explain all necessary concepts from computer science (mainly computational complexity theory) and group theory. On the other hand, to keep the book focused on the compressed word problem I had to stop at a certain point. I hope that I could find the right balance. I want to thank my former supervisor Volker Diekert for his scientific influence.
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