Related papers: Growth of intersection numbers for free group auto…

We prove an asymptotic result on the maximum number of k-vertex subtrees in binary trees of given order. This problem turns out to be equivalent to determine the maximum number of k+2-cycles in n-vertex outerplanar graphs, thus we settle…

To each sequence $(a_n)$ of positive real numbers we associate a growing sequence $(T_n)$ of continuous trees built recursively by gluing at step $n$ a segment of length $a_n$ on a uniform point of the pre-existing tree, starting from a…

The main result of this paper is an algorithmic answer to the question raised in the title, up to replacing the given $\hat{\phi} \in Out(F_n)$ by a positive power. In order to provide this algorithm, it is shown that every polynomially…

Fix a graph $F$. We say that a graph is {\it $F$-free} if it does not contain $F$ as a subgraph. The {\it Tur\'an number} of $F$, denoted $\mathrm{ex}(n,F)$, is the maximum number of edges possible in an $n$-vertex $F$-free graph. The study…

The object of this expository work is to try to unveil the topological/geometric intuition behind the theory of free groups and their automorphism and outer automorphism groups. The method we follow is to focus on a series of problems in…

We quantitatively relate the Patterson-Sullivant currents and generic stretching factors for free group automorphisms to the asymmetric Lipschitz metric on Outer space and to Guirardel's intersection number.

We present a normal form for outer automorphisms $\phi$ of a non-abelian free group $F_N$ which grow quadratically (measured through the maximal growth of conjugacy classes in $F_N$ under iteration of $\phi$). In analogy to the known normal…

The period mapping assigns to each rank n, marked metric graph Gamma a positive definite quadratic form on H_1(Gamma). This defines maps Phi* and Phi on Culler--Vogtmann's outer space CV_n, and its Torelli space quotient T_n, respectively.…

Let $G$ be a group and let ${\mathcal G}$ be a free factor system of $G$, namely a free splitting of $G$ as $G=G_1*\dots*G_k*F_r$. In this paper, we study the set of train track points for ${\mathcal G}$-irreducible automorphisms $\phi$…

We study finite subgroups of outer automorphisms of free products. We give upper bounds for the orders of these finite subgroups as well as bounds for the orders of individual torsion outer automorphisms under some (necessary) conditions…

For $N\geq 4$, we show that there exist automorphisms of the free group $F_N$ which have a parabolic orbit in $\partial F_N$. In fact, we exhibit a technology for producing infinitely many such examples.

We prove various obstructions to the existence of regular maps (or coarse embeddings) between commonly studied spaces. For instance, there is no regular map (or coarse embedding) $\mathbb H^n\to\mathbb H^{n-1}\times Y$ for $n\geq 3$, or…

For two oriented simple closed curves on a compact orientable surface with a connected boundary we introduce a simple computation of a value in the first homology group of the surface, which detects in some cases that the geometric…

Let T1, T2,..., Tk be spanning trees in a graph G. If for any pair of vertices {u, v} of G, the paths between u and v in every Ti( 0 < i < k+1) do not contain common edges and common vertices, except the vertices u and v, then T1, T2,...,…

We present a coarse convexity result for the dynamics of free group automorphisms: Given an automorphism $\phi$ of a finitely generated free group $F$, we show that for all $x\in F$ and $0\leq i\leq N$, the length of $\phi^i(x)$ is bounded…

We study the asymptotic number of certain monotonically labeled increasing trees arising from a generalized evolution process. The main difference between the presented model and the classical model of binary increasing trees is that the…

A concrete family of automorphisms alpha_n of the free group F_n is exhibited, for any n > 2, and the following properties are proved: alpha_n is irreducible with irreducible powers, has trivial fixed subgroup, and has 2n-1 attractive as…

We prove that the group of automorphisms of the generic meet-tree expansion of an infinite non-unary free Fra\"{\i}ss\'{e} limit over a finite relational language is simple. As a prototypical case, the group of automorphism of the Rado…

Let $\phi$ be an automorphism of a free group $F_n$ of rank $n$, and let $M_{\phi}=F_n \rtimes_{\phi} \mathbb{Z}$ be the corresponding mapping torus of $\phi$. We study the group $Out(M_{\phi})$ under certain technical conditions on $\phi$.…

We construct integrals of motion (IM) for the sine-Gordon model with boundary at the free Fermion point which correctly determine the boundary S matrix. The algebra of these IM (``boundary quantum group'' at q=1) is a one-parameter family…