Related papers: Automorphisms of free groups have asymptotically p…

Given a free group $F_k$ of rank $k\ge 2$ with a fixed set of free generators we associate to any homomorphism $\phi$ from $F_k$ to a group $G$ with a left-invariant semi-norm a generic stretching factor, $\lambda(\phi)$, which is a…

If $N \subset \R$ is a separable II$_1$-factor, the space $\Hom(N,\R)$ of unitary equivalence classes of unital *-homomorphisms $N \to \R$ is shown to have a surprisingly rich structure. If $N$ is not hyperfinite, $\Hom(N,\R)$ is an…

A connected component of an affine algebraic group is called periodic if all its elements have finite order. We give a characterization of periodic components in terms of automorphisms with finite number of fixed points. It is also…

Let $\Phi:F\rightarrow F$ be an automorphism of the finite-rank free group $F$. Suppose that $G=F\rtimes_\Phi\mathbb Z$ is word-hyperbolic. Then $G$ acts freely and cocompactly on a CAT(0) cube complex.

We generalize the positive solution of the Frobenius conjecture and refinements thereof by studying the structure of groups that admit a fix-point-free automorphism satisfying an identity. We show, in particular, that for every polynomial…

This is a study of fusion ring automorphisms leaving only the trivial element fixed. We prove that a variety of classical results on fixed-point-free automorphisms of finite groups are true in the generality of fusion rings. As a result, we…

We show that every finite group occurs as the automorphism group of infinitely many finite (field) extensions of any given Hilbertian field. This extends and unifies previous results of M. Fried and Takahashi on the global field case.

Assume $(X, \omega)$ is a compact symplectic manifold with a Hamiltonian compact Lie group action and the zero in the Lie algebra is a regular value of the moment map $\mu$. We prove that a finite energy symplectic vortex exponentially…

We investigate the question of which growth rates are possible for the number of periodic points of a compact group automorphism. Our arguments involve a modification of Linnik's Theorem, concerning small prime numbers in arithmetic…

We consider complex dynamics of a critically finite holomorphic map from P^k to P^k, which has symmetries associated with the symmetric group S_{k+2} acting on P^k, for each k \ge 1. The Fatou set of each map of this family consists of…

For a rational map $\phi$ from a metric graph $\varGamma$ to a tropical projective space $\boldsymbol{TP^n}$ defined by a ratio of rational functions $f_1, \ldots, f_{n + 1}$, an automorphism $\sigma$ of $\varGamma$ induces a permutation of…

Using Frobenius normal forms of matrices over finite fields as well as the Burnside Basis Theorem, we give a direct proof of Horo\v{s}evski\u{i}'s result that every automorphism $\alpha$ of a finite nilpotent group has a cycle whose length…

We show that every homomorphism from the infinite-dimensional unitary or orthogonal group to a separable group is continuous.

The classical result by Dyer--Scott about fixed subgroups of finite order automorphisms of $F_n$ being free factors of $F_n$ is no longer true in $Z^m\times F_n$. Within this more general context, we prove a relaxed version in the spirit of…

For Hamiltonian flows we establish the existence of periodic orbits on a sequence of level sets approaching a Bott-nondegenerate symplectic extremum of the Hamiltonian. As a consequence, we show that a charge on a compact manifold with a…

We prove that the automorphism group of every infinitely-ended finitely generated group is acylindrically hyperbolic. In particular $\mathrm{Aut}(\mathbb{F}_n)$ is acylindrically hyperbolic for every $n\ge 2$. More generally, if $G$ is a…

Let $G=G_1 \ast \ldots \ast G_k \ast F_N$ be a free product of finitely presented groups, where $F_N$ is a free group of rank $N \in \mathbb{N}$. Let $\mathrm{Out}(G,\mathcal{G})$ be the subgroup of $\mathrm{Out}(G)$ preserving the set of…

We introduce the notion of the $k$-closure of a group of automorphisms of a locally finite tree, and give several examples of the construction. We show that the $k$-closure satisfies a new property of automorphism groups of trees that…

We analyze the structure of the \emph{frequency space} $Q(F)$ of a nonabelian free group $F=F(a_1,...,a_k)$ consisting of all shift-invariant Borel probability measures on $\partial F$ and construct a natural action of $Out(F)$ on $Q(F)$.…

The special linear groups, the mapping class groups of surfaces, the outer autormorphism groups of free groups appear in numerous domains. Their analogies, developped in particular in K. Vogtmann's work, have been written about a lot. In…