Related papers: Growth of automorphisms of virtually special group…

We initiate the study of outer automorphism groups of special groups $G$, in the Haglund-Wise sense. We show that ${\rm Out}(G)$ is infinite if and only if $G$ splits over a co-abelian subgroup of a centraliser and there exists an…

Let $G$ be a (compact) special group in the sense of Haglund and Wise. We show that ${\rm Out}(G)$ is finitely generated, and provide a virtual generating set consisting of Dehn twists and ``pseudo-twists''. We exhibit instances where Dehn…

We study the outer automorphism group of a right-angled Artin group A_G in the case where the defining graph G is connected and triangle-free. We give an algebraic description of Out(A_G) in terms of maximal join subgraphs in G and prove…

We prove that an outer automorphism of the free group is exponentially growing if and only if it induces an outer automorphism of infinite order of free Burnside groups with sufficiently large odd exponent.

Using the canonical JSJ splitting, we describe the outer automorphism group $\Out(G)$ of a one-ended word hyperbolic group $G$. In particular, we discuss to what extent $\Out(G)$ is virtually a direct product of mapping class groups and a…

We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the…

In this article, we study the outer automorphism group of a group G decomposed as a finite graph of group with finite edge groups and finitely generated vertex groups with at most one end. We show that Out(G) is essentially obtained by…

For a finite alphabet $\mathcal{A}$ and shift $X\subseteq\mathcal{A}^{\mathbb{Z}}$ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group ${\rm Aut}(X)$. For such systems, we…

In this article, we further explore the nature of a connection between the groups of automorphisms of full shift spaces and the groups of outer automorphisms of the Higman--Thompson groups $\{G_{n,r}\}$. We show that the quotient of the…

Given a finitely generated semigroup S of the (normed) set of linear maps of a vector space V into itself, we find sufficient conditions for the exponential growth of the number N(k) of elements of the semigroup contained in the sphere of…

We compute explicitly the automorphism and outer automorphism group of all large-type free-of-infinity Artin groups. Our strategy involves reconstructing the associated Deligne complexes in a purely algebraic manner, i.e. in a way that is…

We study the dilatation of outer automorphisms of right-angled Artin groups. Given a right-angled Artin group defined on a simplicial graph: $A(\Gamma) = \langle V | E \rangle$ and an automorphism $\phi \in Out(A(\Gamma))$ there is a…

We prove that the outer automorphism group of a one-ended hyperbolic group is virtually a hierarchically hyperbolic group (HHG), under mild orientability conditions on the associated JSJ decomposition. This is done by proving that a…

Let $G$ be a finitely generated group with an automorphism $\varphi\in{\rm Aut}(G)$, or an outer automorphism $\phi\in{\rm Out}(G)$. Suppose that $G$ decomposes into simpler pieces on which the growth behaviour of $\varphi$ and $\phi$ is…

A fully irreducible outer automorphism phi of the free group F_n of rank n has an expansion factor which often differs from the expansion factor of the inverse of phi. Nevertheless, we prove that the ratio between the logarithms of the…

We study the outer automorphism group of a right-angled Artin group $A_\Gamma$ with finite defining graph $\Gamma$. We construct a subnormal series for $Out(A_\Gamma)$ such that each consecutive quotient is either finite, free-abelian,…

We determine the twisted conjugacy growth function for automorphisms on generalised Heisenberg groups. In particular we demonstrate for these groups that up to a natural equivalence this function is either given by $n^k$ or $n^k\ln(n)$ for…

The group of automorphisms of a symbolic dynamical system is countable, but often very large. For example, for a mixing subshift of finite type, the automorphism group contains isomorphic copies of the free group on two generators and the…

For any pseudo-Anosov automorphism on an orientable closed surface, an inquality is established bounding certain growth of virtual homological eigenvalues with the Weil--Petersson translation length. The new inquality fits nicely with other…

We will survey the work on the topology of $Out(F_n)$ in the last 20 years or so. Much of the development is driven by the tantalizing analogy with mapping class groups. Unfortunately, $Out(F_n)$ is more complicated and less well-behaved.…