100 research outputs found

    Independence property and hyperbolic groups

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    We prove that existentially closed CSACSA-groups have the independence property. This is done by showing that there exist words having the independence property relatively to the class of torsion-free hyperbolic groups.Comment: v3: 10 pages (11pt), a few typos corrected, minor rearrangements (e.g. Fact 2.3 and Lemma 2.5); v2: 8 pages (10pt), a false statement in the proof of Fact 2.4 is replaced with a true one; v1: 8 page

    On certain equations of arbitrary length over torsion-free groups

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    Let GG be a non-trivial torsion free group and tt be an unknown. In this paper we consider three equations (over GG) of arbitrary length and show that they have a solution (over GG) provided two relations among their coefficients hold. Such equations appear for all lengths greater than or equal to eight and the results presented in this article can substantially simplify their solution.Comment: arXiv admin note: substantial text overlap with arXiv:1903.0650

    Groups of Fibonacci type revisited

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    This article concerns a class of groups of Fibonacci type introduced by Johnson and Mawdesley that includes Conway?s Fibonacci groups, the Sieradski groups, and the Gilbert-Howie groups. This class of groups provides an interesting focus for developing the theory of cyclically presented groups and, following questions by Bardakov and Vesnin and by Cavicchioli, Hegenbarth, and Repov?s, they have enjoyed renewed interest in recent years. We survey results concerning their algebraic properties, such as isomorphisms within the class, the classification of the finite groups, small cancellation properties, abelianizations, asphericity, connections with Labelled Oriented Graph groups, and the semigroups of Fibonacci type. Further, we present a new method of proving the classification of the finite groups that deals with all but three groups

    On the torsion in a group F/[M,N]\bf F/[M,N] in the case of combinatorial asphericity of groups F/M\bf F/M and F/N\bf F/N

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    Let FF be a non-Abelian free group with basis AA, MM and NN be the normal closures of sets RMR_M and RNR_N of words in the alphabet A±1A^{\pm 1}. As is known, the group F/[N,N]F/[N, N] is torsion-free, but, in general, torsion in F/[M,N]F/[M, N] is possible. In the paper of Hartley and Kuz'min (1991), it was proved that if RM={v}R_M=\{v\}, RN={w}R_N=\{w\} and words vv and ww are not a proper power in FF, then F/[M,N]F/[M,N] is torsion-free. In the present paper a sufficient condition for the absence of torsion in F/[M,N]F/[M,N] is obtained, which allows to generalize the result of Hartley and Kuz'min to arbitrary words vv and ww.Comment: arXiv admin note: text overlap with arXiv:1503.0619

    Asphericity of symmetric presentations

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    Using the notion of relative presentation due to Bogley and Pride, we give a new proof of a theorem of Prishchepov on the asphericity of certain symmetric presentations of groups. Then we obtain further results and applications to topology of low-dimensional manifolds

    Finitely generated infinite simple groups of infinite square width and vanishing stable commutator length

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    It is shown that there exist finitely generated infinite simple groups of infinite commutator width and infinite square width on which there exists no stably unbounded conjugation-invariant norm, and in particular stable commutator length vanishes. Moreover, a recursive presentation of such a group with decidable word and conjugacy problems is constructed.Comment: v4: 41 pages, 6 figures rescaled at 120%; references updated, typos corrected, other minor corrections. v3: minor changes to the title, text and figures. v2: 41 pages, 6 figures; correction: Ore's conjecture was proved in 2008; 2 references added. v1: 40 pages, 6 figure
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