1,189 research outputs found
Nielsen equivalence in a class of random groups
We show that for every there exists a torsion-free one-ended
word-hyperbolic group of rank admitting generating -tuples
and such that the -tuples
are not
Nielsen-equivalent in . The group is produced via a probabilistic
construction.Comment: 34 pages, 2 figures; a revised final version, to appear in the
Journal of Topolog
Dual Garside structure and reducibility of braids
Benardete, Gutierrez and Nitecki showed an important result which relates the
geometrical properties of a braid, as a homeomorphism of the punctured disk, to
its algebraic Garside-theoretical properties. Namely, they showed that if a
braid sends a curve to another curve, then the image of this curve after each
factor of the left normal form of the braid (with the classical Garside
structure) is also standard. We provide a new simple, geometric proof of the
result by Benardete-Gutierrez-Nitecki, which can be easily adapted to the case
of the dual Garside structure of braid groups, with the appropriate definition
of standard curves in the dual setting. This yields a new algorithm for
determining the Nielsen-Thurston type of braids
On the geometry of a proposed curve complex analogue for
The group \Out of outer automorphisms of the free group has been an object
of active study for many years, yet its geometry is not well understood.
Recently, effort has been focused on finding a hyperbolic complex on which
\Out acts, in analogy with the curve complex for the mapping class group.
Here, we focus on one of these proposed analogues: the edge splitting complex
\ESC, equivalently known as the separating sphere complex. We characterize
geodesic paths in its 1-skeleton algebraically, and use our characterization to
find lower bounds on distances between points in this graph.
Our distance calculations allow us to find quasiflats of arbitrary dimension
in \ESC. This shows that \ESC: is not hyperbolic, has infinite asymptotic
dimension, and is such that every asymptotic cone is infinite dimensional.
These quasiflats contain an unbounded orbit of a reducible element of \Out.
As a consequence, there is no coarsely \Out-equivariant quasiisometry between
\ESC and other proposed curve complex analogues, including the regular free
splitting complex \FSC, the (nontrivial intersection) free factorization
complex \FFZC, and the free factor complex \FFC, leaving hope that some of
these complexes are hyperbolic.Comment: 23 pages, 6 figure
Subword complexity and Laurent series with coefficients in a finite field
Decimal expansions of classical constants such as , and
have long been a source of difficult questions. In the case of
Laurent series with coefficients in a finite field, where no carry-over
difficulties appear, the situation seems to be simplified and drastically
different. On the other hand, Carlitz introduced analogs of real numbers such
as , or . Hence, it became reasonable to enquire how
"complex" the Laurent representation of these "numbers" is. In this paper we
prove that the inverse of Carlitz's analog of , , has in general a
linear complexity, except in the case , when the complexity is quadratic.
In particular, this implies the transcendence of over \F_2(T). In the
second part, we consider the classes of Laurent series of at most polynomial
complexity and of zero entropy. We show that these satisfy some nice closure
properties
Inverse problems of symbolic dynamics
This paper reviews some results regarding symbolic dynamics, correspondence
between languages of dynamical systems and combinatorics. Sturmian sequences
provide a pattern for investigation of one-dimensional systems, in particular
interval exchange transformation. Rauzy graphs language can express many
important combinatorial and some dynamical properties. In this case
combinatorial properties are considered as being generated by substitutional
system, and dynamical properties are considered as criteria of superword being
generated by interval exchange transformation. As a consequence, one can get a
morphic word appearing in interval exchange transformation such that
frequencies of letters are algebraic numbers of an arbitrary degree.
Concerning multydimensional systems, our main result is the following. Let
P(n) be a polynomial, having an irrational coefficient of the highest degree. A
word (w=(w_n), n\in \nit) consists of a sequence of first binary numbers
of i.e. . Denote the number of different subwords
of of length by .
\medskip {\bf Theorem.} {\it There exists a polynomial , depending only
on the power of the polynomial , such that for sufficiently
great .
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