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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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Cited by 41 (17 self)
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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.
Some Results on OneRelator Surface Groups
"... Introduction This short note was inspired by a question from Fico Gonzalez Acu~na: Question 1 If and are two closed curves (nonsimple, in general) on an orientable surface S, such that the normal closures of and in 1 (S) coincide, is freely homopotic to ? If S is noncompact, or has n ..."
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Cited by 5 (1 self)
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Introduction This short note was inspired by a question from Fico Gonzalez Acu~na: Question 1 If and are two closed curves (nonsimple, in general) on an orientable surface S, such that the normal closures of and in 1 (S) coincide, is freely homopotic to ? If S is noncompact, or has nonempty boundary, then 1 (S) is free, and the answer to Question 1 is yes, by an old result of Magnus [7] on onerelator groups. (Essentially, the de ning relator in a onerelator group on a given generating set is unique up to conjugacy and inversion.) We will show (see Theorem 3.4 below) that Question 1 also has an armative answer in the case of a closed surface S. In this case Question 1 can be interpreted in terms of onerelator surface groups, as introduced by Hempel [3]. Among other results, Hempel proved analogues for onerelator surface groups of two theorems from onerelator group theory: (i) a onerelator surface group is locally indicable if and only if the relator is not
RANDOM QUOTIENTS OF THE MODULAR GROUP ARE RIGID AND ESSENTIALLY INCOMPRESSIBLE
, 2006
"... Abstract. We show that for any positive integer m ≥ 1, mrelator quotients of the modular group M = PSL(2, Z) generically satisfy a very strong Mostowtype isomorphism rigidity. We also prove that such quotients are generically “essentially incompressible”. By this we mean that their “absolute Tinv ..."
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Cited by 4 (2 self)
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Abstract. We show that for any positive integer m ≥ 1, mrelator quotients of the modular group M = PSL(2, Z) generically satisfy a very strong Mostowtype isomorphism rigidity. We also prove that such quotients are generically “essentially incompressible”. By this we mean that their “absolute Tinvariant”, measuring the smallest size of any possible finite presentation of the group, is bounded below by a function which is almost linear in terms of the length of the given presentation. We compute the precise asymptotics of the number Im(n) of isomorphism types of mrelator quotients of M where all the defining relators are cyclically reduced words of length n in M. We obtain other algebraic results and show that such quotients are complete, Hopfian, coHopfian, oneended, wordhyperbolic groups. 1.
Minimal seifert manifolds for higher ribbon knots
 Aspherical LOTs and Knots Huck/Rosebrock September 30
, 1997
"... We show that a group presented by a labelled oriented tree presentation in which the tree has diameter at most three is an HNN extension of a nitely presented group. From results of Silver, it then follows that the corresponding higher dimensional ribbon knots admit minimal Seifert manifolds. ..."
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Cited by 1 (0 self)
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We show that a group presented by a labelled oriented tree presentation in which the tree has diameter at most three is an HNN extension of a nitely presented group. From results of Silver, it then follows that the corresponding higher dimensional ribbon knots admit minimal Seifert manifolds.
FREE SUBGROUPS OF SMALL CANCELLATION GROUPS
, 1971
"... A group is a small cancellation group if, roughly, it has a presentation 0 = (a,b,c,...; r=l{reR)) with the property that for any pair r, s of elements of R either r = s1 or there is very little free cancellation in forming the product rs. The classical ..."
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Cited by 1 (0 self)
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A group is a small cancellation group if, roughly, it has a presentation 0 = (a,b,c,...; r=l{reR)) with the property that for any pair r, s of elements of R either r = s1 or there is very little free cancellation in forming the product rs. The classical
Intersections of Magnus subgroups and embedding theorems for cyclically presented groups
, 2006
"... ..."
Contents
, 2003
"... 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 const1