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Generic properties of Whitehead’s algorithm and isomorphism rigidity of random onerelator groups
 Pacific J. Math
"... Abstract. We show that the “hard ” part of Whitehead’s algorithm for solving the automorphism problem in a fixed free group Fk terminates in linear time (in terms of the length of an input) on an exponentially generic set of input pairs and thus the algorithm has strongly lineartime genericcase co ..."
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Abstract. We show that the “hard ” part of Whitehead’s algorithm for solving the automorphism problem in a fixed free group Fk terminates in linear time (in terms of the length of an input) on an exponentially generic set of input pairs and thus the algorithm has strongly lineartime genericcase complexity. We also prove that the stabilizers of generic elements of Fk in Aut(Fk) are cyclic groups generated by inner automorphisms. We apply these results to onerelator groups and show that onerelator groups are generically complete groups, that is, they have trivial center and trivial outer automorphism group. We prove that the number In of isomorphism types of kgenerator onerelator groups with defining relators of length n satisfies c1 n (2k − 1)n ≤ In ≤ c2 n (2k − 1)n, where c1 = c1(k)> 0, c2 = c2(k)> 0 are some constants independent of n. Thus In grows in essentially the same manner as the number of cyclic words of length n.
Clusters, currents and Whitehead’s algorithm
, 2006
"... Abstract. Using geodesic currents, we provide a theoretical justification for some of the experimental results regarding the behavior of Whitehead’s algorithm on nonminimal inputs, that were obtained by Haralick, Miasnikov and Myasnikov via pattern recognition methods. In particular we prove that t ..."
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Cited by 15 (10 self)
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Abstract. Using geodesic currents, we provide a theoretical justification for some of the experimental results regarding the behavior of Whitehead’s algorithm on nonminimal inputs, that were obtained by Haralick, Miasnikov and Myasnikov via pattern recognition methods. In particular we prove that the images of “random ” elements of a free group F under the automorphisms of F form “clusters ” that share similar normalized Whitehead graphs and similar behavior with respect to Whitehead’s algorithm. 1.
A tighter bound for the number of words of minimum length in an automorphic orbit, preprint
, 2006
"... Abstract. Let u be a cyclic word in a free group Fn of finite rank n that has the minimum length over all cyclic words in its automorphic orbit, and let N(u) be the cardinality of the set {v: v  = u  and v = φ(u) for some φ ∈ AutFn}. In this paper, we prove that N(u) is bounded by a polynomial ..."
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Cited by 5 (0 self)
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Abstract. Let u be a cyclic word in a free group Fn of finite rank n that has the minimum length over all cyclic words in its automorphic orbit, and let N(u) be the cardinality of the set {v: v  = u  and v = φ(u) for some φ ∈ AutFn}. In this paper, we prove that N(u) is bounded by a polynomial function of degree 2n − 3 with respect to u  under the hypothesis that if two letters x, y occur in u, then the total number of x and x −1 occurring in u is not equal to the total number of y and y −1 occurring in u. We also prove that 2n − 3 is the sharp bound on the degree of polynomials bounding N(u). As a special case, we deal with N(u) in F2 under the same hypothesis. 1.
AN ALGORITHM THAT DECIDES TRANSLATION EQUIVALENCE IN A FREE GROUP OF RANK TWO
, 2006
"... Abstract. Let F2 be a free group of rank 2. We prove that there is an algorithm that decides whether or not, for given two elements u, v of F2, u and v are translation equivalent in F2, that is, whether or not u and v have the property that the cyclic length of φ(u) equals the cyclic length of φ(v) ..."
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Cited by 4 (1 self)
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Abstract. Let F2 be a free group of rank 2. We prove that there is an algorithm that decides whether or not, for given two elements u, v of F2, u and v are translation equivalent in F2, that is, whether or not u and v have the property that the cyclic length of φ(u) equals the cyclic length of φ(v) for every automorphism φ of F2. This gives an affirmative solution to problem F38a in the online version (http://www.grouptheory.info) of [1] for the case of F2. 1.
GROWING WORDS IN THE FREE GROUP ON TWO GENERATORS
, 909
"... Abstract. This paper is concerned with minimal length representatives of equivalence classes of F2 under Aut F2. We give a simple inequality characterizing words of minimal length in their equivalence class. We consider an operation that “grows ” words from other words, increasing the length, and we ..."
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Cited by 2 (1 self)
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Abstract. This paper is concerned with minimal length representatives of equivalence classes of F2 under Aut F2. We give a simple inequality characterizing words of minimal length in their equivalence class. We consider an operation that “grows ” words from other words, increasing the length, and we study root words — minimal words that cannot be grown from other words. Root words are “as minimal as possible ” in the sense that their characterization is the boundary case of the minimality inequality. The property of being a root word is respected by equivalence classes, and the length of each root word is divisible by 4. 1.
A Hybrid Search Algorithm for the Whitehead Minimization Problem
, 2006
"... The Whitehead Minimization problem is a problem of finding elements of the minimal length in the automorphic orbit of a given element of a free group. The classical algorithm of Whitehead that solves the problem depends exponentially on the group rank. Moreover, it can be easily shown that exponenti ..."
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Cited by 1 (0 self)
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The Whitehead Minimization problem is a problem of finding elements of the minimal length in the automorphic orbit of a given element of a free group. The classical algorithm of Whitehead that solves the problem depends exponentially on the group rank. Moreover, it can be easily shown that exponential blowout occurs when a word of minimal length has been reached and, therefore, is inevitable except for some trivial cases. In this paper we introduce a deterministic Hybrid search algorithm and its stochastic variation for solving the Whitehead minimization problem. Both algorithms use search heuristics that allow one to find a lengthreducing automorphism in polynomial time on most inputs and significantly improve the reduction procedure. The stochastic version of the algorithm employs a probabilistic system that decides in polynomial time whether or not a word is minimal. The stochastic algorithm is very robust. It has never happened that a nonminimal element has been claimed to be minimal.
A hybrid search algorithm for the Whitehead Minimization problem
, 2005
"... The Whitehead Minimization problem is a problem of finding elements of the minimal length in the automorphic orbit of a given element of a free group. The classical algorithm of Whitehead that solves the problem depends exponentially on the group rank. Moreover, it can be easily shown that exponenti ..."
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The Whitehead Minimization problem is a problem of finding elements of the minimal length in the automorphic orbit of a given element of a free group. The classical algorithm of Whitehead that solves the problem depends exponentially on the group rank. Moreover, it can be easily shown that exponential blowout occurs when a word of minimal length has been reached and, therefore, is inevitable except for some trivial cases. In this paper we introduce a deterministic Hybrid search algorithm and its stochastic variation for solving the Whitehead Minimization problem. Both algorithms use search heuristics that allow one to find a lengthreducing automorphism in polynomial time on most inputs and significantly improve the reduction procedure. The stochastic version of the algorithm employs a probabilistic system that decides in polynomial time whether or not a word is minimal. The stochastic algorithm is very robust. It has never happened that a nonminimal element has been claimed to be minimal. c © 2006 Elsevier Ltd. All rights reserved.
and
"... The Whitehead Minimization problem is a problem of finding elements of the minimal length in the automorphic orbit of a given element of a free group. The classical algorithm of Whitehead that solves the problem depends exponentially on the group rank. Moreover, it can be easily shown that exponenti ..."
Abstract
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The Whitehead Minimization problem is a problem of finding elements of the minimal length in the automorphic orbit of a given element of a free group. The classical algorithm of Whitehead that solves the problem depends exponentially on the group rank. Moreover, it can be easily shown that exponential blowout occurs when a word of minimal length has been reached and, therefore, is inevitable except for some trivial cases. In this paper we introduce a deterministic Hybrid search algorithm and its stochastic variation for solving the Whitehead minimization problem. Both algorithms use search heuristics that allow one to find a lengthreducing automorphism in polynomial time on most inputs and significantly improve the reduction procedure. The stochastic version of the algorithm employs a probabilistic system that decides in polynomial time whether or not a word is minimal. The stochastic algorithm is very robust. It has never happened that a nonminimal element has been claimed to be minimal.