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Zariski closures and subgroup separability
The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We...
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Co-rank and Betti number of a group
For a finitely generated group, we study the relations between its rank, the maximal rank of its free quotient, called co-rank (inner rank, cut...
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Free groups with involution satisfying a *-group identity
Let G be a nonabelian free group with involution *. In the present note, we show that G satisfies a *-group identity if and only if * is the...
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Filtrations of free groups as intersections
For several natural filtrations of a free group S , we express the n -th term of the filtration as the intersection of all kernels of homomorphisms...
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On the abelianizations of congruence subgroups of Aut(F 2)
Let F n be the free group of rank n , and let Aut + ( F n ) be its special automorphism group. For an epimorphism π : F n → G of the free group F ...
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The complex of partial bases for F n and finite generation of the Torelli subgroup of Aut (F n )
We study the complex of partial bases of a free group, which is an analogue for Aut( F n ) of the curve complex for the map** class group. We prove...
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An algorithm to identify automorphisms which arise from self-induced interval exchange transformations
We give an algorithm to determine if the dynamical system generated by a positive automorphism of the free group can also be generated by a...
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Free lattice-ordered groups and the space of left orderings
For a left-orderable group G , let LO( G ) denote its space of left orderings, and F( G ) the free lattice-ordered group over G . This paper establishes a...
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Regular Sets and Counting in Free Groups
In this paper we study asymptotic behavior of regular subsets in a free group F of finite rank, compare their sizes at infinity, and develop...
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The monomorphism problem in free groups
Let F be a free group of finite rank. We say that the monomorphism problem in F is decidable if there is an algorithm such that, for any two elements u ...
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Subgroups of Small Index in Aut(F n ) and Kazhdan’s Property (T)
We introduce a series of interesting subgroups of finite index in Aut(F n ). One of them has index 42 in...
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Twisting Out Fully Irreducible Automorphisms
By a theorem of Thurston, in the subgroup of the map** class group generated by Dehn twists around two curves which fill, every element not...
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A measurable-group-theoretic solution to von Neumann’s problem
We give a positive answer, in the measurable-group-theory context, to von Neumann’s problem of knowing whether a non-amenable countable discrete...
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Counting Growth Types of Automorphisms of Free Groups
Given an automorphism of a free group Fn, we consider the following invariants: e is the number of exponential strata (an upper bound for the number...
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Noncommutative Geometry and Path Integrals
We argue that there should exist a “noncommutative Fourier transform” which should identify functions of noncommutative variables (say, of matrices...