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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.
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
 J. Reine Angew. Math
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Intersections of Magnus subgroups and embedding theorems for cyclically presented groups
 J. Pure Appl. Algebra
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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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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.
Parasurface groups 1
, 2009
"... A residually nilpotent group is kparafree if all of its lower central series quotients match those of a free group of rank k. Magnus proved that kparafree groups of rank k are themselves free. We mimic this theory with surface groups playing the role of free groups. Our main result shows that the ..."
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A residually nilpotent group is kparafree if all of its lower central series quotients match those of a free group of rank k. Magnus proved that kparafree groups of rank k are themselves free. We mimic this theory with surface groups playing the role of free groups. Our main result shows that the analog of Magnus ’ Theorem is false in this setting.
ONERELATOR GROUPS AND PROPER 3REALIZABILITY
, 910
"... Abstract. How different is the universal cover of a given finite 2complex from a 3manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group G is said to be properly 3realizable if there exists a compact 2polyhedron K with π1(K) ∼ = G whos ..."
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Abstract. How different is the universal cover of a given finite 2complex from a 3manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group G is said to be properly 3realizable if there exists a compact 2polyhedron K with π1(K) ∼ = G whose universal cover ˜K has the proper homotopy type of a PL 3manifold (with boundary). In this paper, we study the asymptotic behavior of finitely generated onerelator groups and show that those having finitely many ends are properly 3realizable, by describing what the fundamental progroup looks like, showing a property of onerelator groups which is stronger than the QSF property of Brick (from the proper homotopy viewpoint) and giving an alternative proof of the fact that onerelator groups are semistable at infinity. 1.