Nielsen equivalence in a class of random groups

We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples $$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{times}})$$ are not Nielsen-equivalent in $G$. The group $G$ 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 Out(Fn)Out(F_n)

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 $\sqrt2$, $\pi$ and $\zeta(3)$ 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 $\pi$, $e$ or $\zeta(3)$. 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 $\pi$, $\Pi_q$, has in general a linear complexity, except in the case $q=2$, when the complexity is quadratic. In particular, this implies the transcendence of $\Pi_2$ 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=(w_n), n\in \nit) consists of a sequence of first binary numbers of $\{P(n)\}$ i.e. $w_n=[2\{P(n)\}]$. Denote the number of different subwords of $w$ of length $k$ by $T(k)$ . \medskip {\bf Theorem.} {\it There exists a polynomial $Q(k)$, depending only on the power of the polynomial $P$, such that $T(k)=Q(k)$ for sufficiently great $k$.