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26
Genericcase complexity and decision problems in group theory, preprint
, 2003
"... Abstract. We give a precise definition of “genericcase complexity” and show that for a very large class of finitely generated groups the classical decision problems of group theory the word, conjugacy and membership problems all have lineartime genericcase complexity. We prove such theorems by ..."
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Cited by 48 (22 self)
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Abstract. We give a precise definition of “genericcase complexity” and show that for a very large class of finitely generated groups the classical decision problems of group theory the word, conjugacy and membership problems all have lineartime genericcase complexity. We prove such theorems by using the theory of random walks on regular graphs. Contents 1. Motivation
The Isomorphism Problem for Toral Relatively Hyperbolic Groups
"... We provide a solution to the isomorphism problem for torsionfree relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsionfree hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually ..."
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Cited by 21 (7 self)
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We provide a solution to the isomorphism problem for torsionfree relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsionfree hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic nmanifolds, for n ≥ 3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsionfree relatively hyperbolic group with abelian parabolics is
The Decidability of Distributed Decision Tasks
 In Proceedings of the twentyninth annual ACM symposium on Theory of computing
, 1997
"... ) Maurice Herlihy Computer Science Department Brown University, Providence RI 02912 herlihy@cs.brown.edu Sergio Rajsbaum y Instituto de Matem'aticas U.N.A.M., D.F. 04510, M'exico rajsbaum@servidor.unam.mx Abstract A task is a distributed coordination problem in which each process starts w ..."
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Cited by 15 (5 self)
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) Maurice Herlihy Computer Science Department Brown University, Providence RI 02912 herlihy@cs.brown.edu Sergio Rajsbaum y Instituto de Matem'aticas U.N.A.M., D.F. 04510, M'exico rajsbaum@servidor.unam.mx Abstract A task is a distributed coordination problem in which each process starts with a private input value taken from a finite set, communicates with the other processes by applying operations to shared objects, and eventually halts with a private output value, also taken from a finite set. A protocol is a distributed program that solves a task. A protocol is tresilient if it tolerates failures by t or fewer processes. A task is solvable in a given model of computation if it has a tresilient protocol in that model. A set of tasks is decidable in a given model of computation if there exists an effective procedure for deciding whether any task in that set has a tresilient protocol. This paper gives the first necessary and sufficient conditions for task decidability in ...
The groups of Richard Thompson and complexity
 International Conference on Semigroups and Groups in honor of the 65th birthday of Prof
, 2004
"... We prove new results about the remarkable infinite simple groups introduced by Richard Thompson in the 1960s. We define the groups as partial transformation groups and we give a faithful representation in the Cuntz C ⋆algebra. For the finitely presented simple group Tfin we show that the wordlengt ..."
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Cited by 14 (7 self)
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We prove new results about the remarkable infinite simple groups introduced by Richard Thompson in the 1960s. We define the groups as partial transformation groups and we give a faithful representation in the Cuntz C ⋆algebra. For the finitely presented simple group Tfin we show that the wordlength and the table size satisfy an n log n relation, just like the symmetric groups. We show that the word problem of Tfin belongs to the parallel complexity class AC 1 (a subclass of P). We show that the generalized word problem of Tfin is undecidable. We study the distortion functions of Tfin and we show that Tfin contains all finite direct products of finitely generated free groups as subgroups with linear distortion. As a consequence, up to polynomial equivalence of functions, the following three sets are the same: the set of distortions of Tfin, the set of all Dehn functions of finitely presented groups, and the set of time complexity functions of nondeterministic Turing machines. 1
HOMOLOGICAL CHARACTERIZATION OF THE UNKNOT
, 2003
"... Given a knot K in the 3sphere, let QK be its fundamental quandle as introduced by D. Joyce. Its first homology group is easily seen to be H1(QK) ∼ = Z. We prove that H2(QK) = 0 if and only if K is trivial, and H2(QK) ∼ = Z whenever K is nontrivial. An analogous result holds for links, thus c ..."
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Cited by 8 (2 self)
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Given a knot K in the 3sphere, let QK be its fundamental quandle as introduced by D. Joyce. Its first homology group is easily seen to be H1(QK) ∼ = Z. We prove that H2(QK) = 0 if and only if K is trivial, and H2(QK) ∼ = Z whenever K is nontrivial. An analogous result holds for links, thus characterizing trivial components. More detailed information can be derived from the conjugation quandle: let Qπ K be the conjugacy class of a meridian in the knot group π1(S3�K). We show that H2(Qπ K) ∼ = Zp, where p is the number of prime summands in a connected sum decomposition of K.
III, Structure and finiteness properties of subdirect products of groups
 Proc. London Math. Soc
"... Abstract. We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virt ..."
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Cited by 8 (5 self)
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Abstract. We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands. This leads to a characterization of the finitely presented subgroups of the direct product of 3 free or surface groups, and to a solution to the conjugacy problem for arbitrary finitely presented subgroups of direct products of surface groups. We obtain a formula for the first homology of a subdirect product of two free groups and use it to show there is no algorithm to determine the first homology of a finitely generated subgroup. A useful structure theory for subgroups of finite direct products of groups has yet to be developed. To begin to study such subgroups it is natural to assume one knows about the subgroups of the direct factors and to concentrate on subdirect products. Recall that G is termed a subdirect product of the groups A1,...,An if G ⊆ A1 × · · · × An is a subgroup that projects surjectively to each factor. Work by various authors has exposed the surprisingly rich structure to be found amongst the subdirect products of superficiallytame groups. For example, in contrast to the fact that subdirect products of abelian or nilpotent groups are again in the specified class, nonabelian free groups harbour a great diversity of subdirect products, including some with unsolvable decision problems [19]. This diversity has long been known, but it is only as a result of more recent work by BaumslagRoseblade [5] and BridsonHowieMillerShort [9] that it has been understood as a phenomenon that is intimately tied to the failure of various homological finiteness conditions.
Observations on coset enumeration
, 1998
"... Todd and Coxeter's method for enumerating cosets of finitely generated subgroups in finitely presented groups (abbreviated by Tc here) is one famous method from combinatorial group theory for studying the subgroup problem. Since prefix string rewriting is also an appropriate method to study this pro ..."
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Cited by 6 (1 self)
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Todd and Coxeter's method for enumerating cosets of finitely generated subgroups in finitely presented groups (abbreviated by Tc here) is one famous method from combinatorial group theory for studying the subgroup problem. Since prefix string rewriting is also an appropriate method to study this problem, prefix string rewriting methods have been compared to Tc. We recall and compare two of them briefly, one by Kuhn and Madlener [4] and one by Sims [15]. A new approach using prefix string rewriting in free groups is derived from the algebraic method presented by Reinert, Mora and Madlener in [14] which directly emulates Tc. It is extended to free monoids and an algebraic characterization for the "cosets" enumerated in this setting is provided.
Properties of Monoids That Are Presented By Finite Convergent StringRewriting Systems  a Survey
, 1997
"... In recent years a number of conditions has been established that a monoid must necessarily satisfy if it is to have a presentation through some finite convergent stringrewriting system. Here we give a survey on this development, explaining these necessary conditions in detail and describing the rela ..."
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Cited by 6 (5 self)
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In recent years a number of conditions has been established that a monoid must necessarily satisfy if it is to have a presentation through some finite convergent stringrewriting system. Here we give a survey on this development, explaining these necessary conditions in detail and describing the relationships between them. 1 Introduction Stringrewriting systems, also known as semiThue systems, have played a major role in the development of theoretical computer science. On the one hand, they give a calculus that is equivalent to that of the Turing machine (see, e.g., [Dav58]), and in this way they capture the notion of `effective computability' that is central to computer science. On the other hand, in the phrasestructure grammars introduced by N. Chomsky they are used as sets of productions, which form the essential part of these grammars [HoUl79]. In this way stringrewriting systems are at the very heart of formal language theory. Finally, they are also used in combinatorial semig...
Intersection Subgroups of Complex Hyperplane Arrangements
, 1999
"... Let A be a central arrangement of hyperplanes in C n , let M(A) be the complement of A, and let L(A) be the intersection lattice of A. For X in L(A) we set AX = fH 2 A;H ' Xg, and A=X = fH=X;H 2 AX g, and A X = fH"X;H 2 An AX g. We exhibit natural embeddings of M(AX ) in M(A) that give rise to ..."
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Cited by 4 (1 self)
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Let A be a central arrangement of hyperplanes in C n , let M(A) be the complement of A, and let L(A) be the intersection lattice of A. For X in L(A) we set AX = fH 2 A;H ' Xg, and A=X = fH=X;H 2 AX g, and A X = fH"X;H 2 An AX g. We exhibit natural embeddings of M(AX ) in M(A) that give rise to monomorphisms from 1 (M(AX )) to 1 (M(A)). We call the images of these monomorphisms intersection subgroups of type X and prove that they form a conjugacy class of subgroups of 1 (M(A)). Recall that X in L(A) is modular if X+Y is an element of L(A) for all Y in L(A). We call X in L(A) supersolvable if there exists a chain 0 ` X 1 ` : : : ` X d = X in L(A) such that X is modular and dimX = for all = 1; : : : ; d. Assume that X is supersolvable and view 1 (M(AX )) as an intersection subgroup of type X of 1 (M(A)). Recall that the commensurator of a subgroup S in a group G is the set of a in G such that S " aSa \Gamma1 has finite index in both S and aSa \Gamma1 . The main result o...