100 research outputs found
Independence property and hyperbolic groups
We prove that existentially closed -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
Let be a non-trivial torsion free group and be an unknown. In this
paper we consider three equations (over ) of arbitrary length and show that
they have a solution (over ) 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
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 in the case of combinatorial asphericity of groups and
Let be a non-Abelian free group with basis , and be the normal
closures of sets and of words in the alphabet . As is
known, the group is torsion-free, but, in general, torsion in is possible.
In the paper of Hartley and Kuz'min (1991), it was proved that if
, and words and are not a proper power in ,
then is torsion-free.
In the present paper a sufficient condition for the absence of torsion in
is obtained, which allows to generalize the result of Hartley and
Kuz'min to arbitrary words and .Comment: arXiv admin note: text overlap with arXiv:1503.0619
Asphericity of symmetric presentations
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
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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