Related papers: The Recognition Theorem for Out(F_n)

Several known results, by Rivin, Calegari-Maher and Sisto, show that an element $\phi_n\in Out(F_r)$, obtained after $n$ steps of a simple random walk on $Out(F_r)$, is fully irreducible with probability tending to 1 as $n\to\infty$. In…

We prove that a "random" free group outer automorphism is an ageometric fully irreducible outer automorphism whose ideal Whitehead graph is a union of triangles. In particular, we show that its attracting (and repelling) tree is a…

We relate the McMullen polynomial of a free-by-cyclic group to its Alexander polynomial. To do so, we introduce the notion of an orientable fully irreducible outer automorphism $\varphi$ and use it to characterize when the homological…

We introduce a new class of automorphisms $\varphi$ of the non-abelian free group $F_N$ of finite rank $N \geq 2$ which contains all iwips (= fully irreducible automorphisms), but also any automorphism induced by a pseudo-Anosov…

An automorphism of a group is called outer if it is not an inner automorphism. Let $G$ be a finite $p$-group. Then for every outer $p$-automorphism $\phi$ of $G$ the subgroup $C_G(\phi)=\{x\in G \;|\; x^\phi=x\}$ has order $p$ if and only…

In this article we raise some new questions about positive definite functions on free groups, and explain how these are related to more well-known questions. The article is intended as a survey of known results that also offers some new…

We prove that, for n=3 and 4, the minimal nonabelian finite factor group of the outer automorphism group Out F_n of a free group of rank n is the linear group PSL_n(Z_2) (conjecturally, this may remain true for arbitrary rank n > 2). We…

We prove that the conjugacy problem in Out(Fm) is solvable for the class of outer automorphisms whose restrictions to their polynomial subgroups are of finite order. To do this, we first investigate the structure of suspensions of free…

We prove that the isomorphism problem for finitely generated fully residually free groups is decidable. We also show that each finitely generated fully residually free group G has a decomposition that is invariant under automorphisms of G,…

We classify abelian subgroups of Out(F_n) up to finite index in an algorithmic and computationally friendly way. A process called disintegration is used to canonically decompose a single rotationless element \phi into a composition of…

For any subgroup H of Out(F_n), either H has a finite index subgroup that fixes the conjugacy class of some proper, nontrivial free factor of F_n, or H contains a fully irreducible element phi, meaning that no positive power of phi fixes…

Given a countable group $G$ splitting as a free product $G=G_1\ast\dots\ast G_k\ast F_N$, we establish classification results for subgroups of the group $Out(G,\mathcal{F})$ of all outer automorphisms of $G$ that preserve the conjugacy…

We produce an algorithm that, given $\phi\in Out(F_N)$, where $N\ge 2$, decides wether or not $\phi$ is an iwip ("fully irreducible") automorphism.

For an automorphism $\phi$ of a free group $F_n$ of rank $n$, Bestvina and Handel showed that the rank $rk Fix(\phi)$ of the fixed subgroup is not greater than $n$ (the so-called Scott conjecture). Soon after Bestvina and Handel's…

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism F, we denote by k(X)^F its field of invariants, i.e. the set of rational functions f on X such that f(F)=f. Let n(F)…

We consider the action of an irreducible outer automorphism $\phi$ on the closure of Culler--Vogtmann Outer space. This action has north-south dynamics and so, under iteration, points converge exponentially to $[T^\phi_+]$. For each $N \geq…

In this work, we formally prove that, under certain conditions, if a neural network is invariant to a finite group then its weights recover the Fourier transform on that group. This provides a mathematical explanation for the emergence of…

We give a concise presentation for the group of pure symmetric outer automorphisms of a given splitting of a free product $G_{1}\ast\dots\ast G_{n}$. These are the (outer) automorphisms which preserve the conjugacy classes of the free…

This is the introduction to a series of four papers that develop a decomposition theory for subgroups of Out(F_n) which generalizes the theory for elements of Out(F_n) found in the work of Bestvina, Feighn, and Handel on the Tits…

We generalize the combinatorial approaches of Rapaport and Higgins--Lyndon to the Whitehead algorithm. We show that for every automorphism $\varphi$ of a free group $F$ and every word $u\in F$ there exists a finite multiset of words…