We study those fully irreducible outer automorphisms phi of a finite rank free group F_r which are ``parageometric'', meaning that the attracting fixed point of phi in the boundary of outer space is a geometric R-tree with respect to the action of F_r, but phi itself is not a geometric outer automorphism in that it is not represented by a homemorphism of a surface. Our main result shows that the expansion factor of phi is strictly larger than the expansion factor of the inverse of phi. As corollaries (proved independently by Guirardel), the inverse of a parageometric outer automorphism is neither geometric nor parageometric, and a fully irreducible outer automorphism phi is geometric if and only if its attracting and repelling fixed points in the boundary of outer space are geometric R-trees.
We work with attracting subshifts generated by substitutions which are also irreducible parageometric automorphisms of free groups. For such a dynamical system, we construct a tree substitution to approximate the repelling real tree of the automorphism. We produce images of this tree inside the Rauzy fractal when the substitution is irreducible Pisot. We describe the contour of this tree and compute an interval exchange transformation of the circle covering the original substitution.
Using abelian coverings of Salvetti complexes, embeddings of outer automorphism groups of right-angled Artin groups (RAAGs) into outer automorphism groups of their particular characteristic subgroups are constructed. Virtual embeddings of outer automorphism groups of finitely generated groups having the unique root property into outer automorphism groups of their particular subgroups are also given. These results provide us with rich examples of (virtual) embeddings between outer automorphism groups of RAAGs. Finite subgroups of pure (outer) automorphism groups of RAAGs are also investigated. …
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 $ϕ$ of $G$ the subgroup $C_G(ϕ)=\{x\in G \;|\; x^ϕ=x\}$ has order $p$ if and only if $G$ is of order at most $p^2$.
In this paper, we study some properties of the outer automorphism group of free Burnside groups of large odd exponent. In particular, we prove that it contains free and free abelian subgroups.
Bumagin-Wise posed the question of whether every countable group can be realised as the outer automorphism group of a finitely generated, residually finite group. We give a partial answer to this problem for recursively presentable groups.
We introduce a homology theory for subspace arrangements, and use it to extract a new system of numerical invariants from the Bieri-Neumann-Strebel invariant of a group. We use these to characterize when the set of basis conjugating outer automorphisms (a.k.a. the pure symmetric outer automorphism group) of a right-angled Artin group is itself a right-angled Artin group.
The main result of this paper is a complete classification of the outer automorphism groups of two-generator, one-relator groups with torsion. To this classification we apply recent algorithmic results of Dahmani--Guirardel, which yields an algorithm to compute the isomorphism class of the outer automorphism group of a given two-generator, one-relator group with torsion.
Let $Γ$ be a crystallographic group of dimension $n,$ i.e. a discrete, cocompact subgroup of $\operatorname{Isom}(\mathbb{R}^n)$ = $O(n)\ltimes\mathbb{R}^n.$ For any $n\geq 2,$ we construct a crystallographic group with a trivial center and a trivial outer automorphism group.
The so-called Tits class, associated to an adjoint absolutely almost simple algebraic group, provides a cohomological obstruction for this group to admit an outer automorphism. If the group has inner type, this obstruction is the only one. In this paper, we prove this is not the case for classical groups of outer type, except for groups of type $^2\mathsf{A}_n$ with $n$ even, or $n=5$. More precisely, we prove a descent theorem for exponent $2$ and degree $6$ algebras with unitary involution, which shows that their automorphism groups have outer automorphisms. In all other relevant classical t…
In 1911, Burnside asked whether or not there exist groups that have an outer automorphism which preserves conjugacy classes. Two years later he answered his own question by constructing a family of such groups. Using the small group library in MAGMA we determine all of the groups of order n < 512 that possess such an automorphism. Our investigations led to the discovery of four new infinite families of such groups, all of which are 2-groups of coclass 4.
It is shown that for graph groups (right-angled Artin groups) the conjugacy problem as well as a restricted version of the simultaneous conjugacy problem can be solved in polynomial time even if input words are represented in a compressed form. As a consequence it follows that the word problem for the outer automorphism group of a graph group can be solved in polynomial time.
For a classical group $G$ of type $\mathsf D_n$ over a field $k$ of characteristic different from $2$, we show the existence of a finitely generated regular extension $R$ of $k$ such that $G$ admits outer automorphisms over $R$. Using this result and a construction of groups of type $\mathsf A$ from groups of type $\mathsf D$, we construct new examples of groups of type $^2\mathsf A_n$ with $n\equiv 3\bmod 4$ and the first examples of type $^2\mathsf A_n$ with $n\equiv 1\bmod 4$ $(n\geq 5)$ that are not $R$-trivial, hence not rational (nor stably rational).
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 expansion factors of phi and its inverse is bounded above by a constant depending only on the rank n. We also prove a more general theorem applying to an arbitrary outer automorphism of F_n and its inverse, and their entire spectrum of expansion factors.
Bestvina, Feighn and Handel proved that every subgroup of the outer automorphism group, $\textrm{Out}(F_n)$, of the free group of rank $n$ is either virtually finitely generated abelian or contains a nonabelian free group. In this note we consider the more general situation of the outer automorphism group $\textrm{Out}(G)$ of a finitely generated free-by-finite group $G$. We show that $\textrm{Out}(G)$ is translation discrete and that every subgroup of $\textrm{Out}(G)$ is either virtually finitely generated abelian or contains a nonabelian free group.
The main theorem of this document emulates, in the context of Out(F_r) theory, a mapping class group theorem (by H. Masur and J. Smillie) that determines precisely which index lists arise from pseudo-Anosov mapping classes. Since the ideal Whitehead graph gives a finer invariant in the analogous setting of a fully irreducible outer automorphism, we instead focus on determining which of the 21 connected, loop-free, 5-vertex graphs are ideal Whitehead graphs of ageometric, fully irreducible outer automorphisms of the free group of rank 3. Our main theorem accomplishes this by showing that there …
We consider the action of an irreducible outer automorphism $ϕ$ on the closure of Culler--Vogtmann Outer space. This action has north-south dynamics and so, under iteration, points converge exponentially to $[T^ϕ_+]$. For each $N \geq 3$, we give a family of outer automorphisms $ϕ_k \in \textrm{Out}(\mathbb{F}_N)$ such that as, $k$ goes to infinity, the rate of convergence of $ϕ_k$ goes to infinity while the rate of convergence of $ϕ_k^{-1}$ goes to one. Even if we only require the rate of convergence of $ϕ_k$ to remain bounded away from one, no such family can be constructed when $N < 3$. Thi…
For any finitely generated group $G$, two complexity functions $α_G$ and $β_G$ are defined to measure the maximal possible gap between the norm of an automorphism (respectively outer automorphism) of $G$ and the norm of its inverse. Restricting attention to free groups $F_r$, the exact asymptotic behaviour of $α_2$ and $β_2$ is computed. For rank $r\geqslant 3$, polynomial lower bounds are provided for $α_r$ and $β_r$, and the existence of a polynomial upper bound is proved for $β_r$.
We give a condition on the defining graph of a right-angled Artin group which implies its automorphism group is virtually indicable, that is, it has a finite-index subgroup that admits a homomorphism onto $\Z$. We use this as part of a criterion that determines precisely when the outer automorphism group of a right-angled Artin group defined on a graph with no separating intersection of links has property (T). As a consequence we also obtain a similar criterion for graphs in which each equivalence class under the domination relation of Servatius generates an abelian group.
We generalise the notion of a separating intersection of links (SIL) to give necessary and sufficient criteria on the defining graph $Γ$ of a right-angled Coxeter group $W_Γ$ so that its outer automorphism group is large: that is, it contains a finite index subgroup that admits the free group $F_2$ as a quotient. When $Out(W_Γ)$ is not large, we show it is virtually abelian. We also show that the same dichotomy holds for the outer automorphism groups of graph products of finite abelian groups. As a consequence, these groups have property (T) if and only if they are finite, or equivalently $Γ$ …
We give simple examples of Kazhdan groups with infinite outer automorphism groups. This answers a question of Paulin, independently answered by Ollivier and Wise by completely different methods. As arithmetic lattices in (non-semisimple) Lie groups, our examples are in addition finitely presented. We also use results of Abels about compact presentability of p-adic groups to exhibit a finitely presented non-Hopfian Kazhdan group. This answers a question of Ollivier and Wise.
We show that the outer automorphism groups of graph products of finitely generated abelian groups satisfy the Tits alternative, are residually finite, their so-called Torelli subgroups are finitely generated, and they satisfy a dichotomy between being virtually nilpotent and containing a non-abelian free subgroup that is determined by a graphical condition on the underlying labelled graph. Graph products of finitely generated abelian groups simultaneously generalize right-angled Artin groups (RAAGs) and right-angled Coxter groups (RACGs), providing a common framework for studying these groups.…
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.
We compute the virtual cohomological dimension (VCD) of the group of partially symmetric outer automorphisms of a free group. We use this to obtain new upper and lower bounds on the VCD of the outer automorphism group of a two-dimensional right-angled Artin group. In the case of a right-angled group with defining graph a tree, the bounds agree.
We construct the first known examples of infinite subgroups of the outer automorphism group of Out(A_Gamma), for certain right-angled Artin groups A_Gamma. This is achieved by introducing a new class of graphs, called focused graphs, whose properties allow us to exhibit (infinite) projective linear groups as subgroups of Out(Out(A_Gamma)). This demonstrates a marked departure from the known behavior of Out(Out(A_Gamma)) when A_Gamma is free or free abelian, as in these cases Out(Out(A_Gamma)) has order at most 4. We also disprove a previous conjecture of the second author, producing new exampl…
We use the Bass--Jiang group for automorphism-induced HNN-extensions to build a framework for the construction of tractable groups with pathological outer automorphism groups. We apply this framework to a strong form of a question of Bumagin--Wise on the outer automorphism groups of finitely presented, residually finite groups.
We study the dilatation of outer automorphisms of right-angled Artin groups. Given a right-angled Artin group defined on a simplicial graph: $A(Γ) = \langle V | E \rangle$ and an automorphism $ϕ\in Out(A(Γ))$ there is a natural measure of how fast the length of a word of $A(Γ)$ grows after $n$ iterations of $ϕ$ as a function of $n$, which we call the dilatation of $w$ under $ϕ$. We define the dilatation of $ϕ$ as the supremum over dilatations of all $w \in A(Γ)$. Assuming that $ϕ$ is a pure and square map, we show that if the dilatation of $ϕ$ is positive, then either there exists a free abeli…
We study subgroups and quotients of outer automorphism groups of right-angled Artin groups (RAAGs). We prove that for all RAAGS, the outer automorphism group is residually finite and, for a large class of RAAGs, it satisfies the Tits alternative. We also investigate which of these automorphism groups contain non-abelian solvable subgroups.
We study the loxodromic elements for the action of $Out(F_n)$ on the free splitting complex of the rank $n$ free group $F_n$. We prove that each outer automorphism is either loxodromic, or has bounded orbits without any periodic point, or has a periodic point; and we prove that all three possibilities can occur. We also prove that two loxodromic elements are either co-axial or independent, meaning that their attracting/repelling fixed point pairs on the Gromov boundary of the free splitting complex are either equal or disjoint as sets. Each of the alternatives in these results is also characte…
Among the simplest invariants of the sporadic finite simple groups are their outer automorphism groups. For 12 of the 26 possible isomorphism types of a sporadic simple group G, the outer automorphism group Out(G) has order 2, and in the remaining 14 cases, Out(G) is trivial. Historically the suspicion of the existence of a sporadic group was followed in fairly short order by the calculation of a good upper bound on the size of its outer automorphism group. In a few cases establishing the existence of certain outer automorphisms, like the existence of the groups themselves, presented difficult…
For each countable group $Q$ we produce a short exact sequence $1\to N \to G \to Q\to 1$ where $G$ is f.g. and has a graphical $\frac16$ presentation and $N$ is f.g. and satisfies property $T$. As a consequence we produce a group $N$ with property $T$ such that $\Out(N)$ is infinite. Using the tools developed we are also able to produce examples of nonHopfian and non-coHopfian groups with property $T$. One of our main tools is the use of random groups to achieve certain properties.
Similarly to the action of $Out(F_N)$ on Outer Space, the outer automorphism group of a Generalized Baumslag Solitar group acts on a deformation space endowed with the Lipschitz metric and the action of any fully irreducible automorphism with a train track representative is hyperbolic. Inspired by previous work by Algom-Kfir, we prove that the axis of such an automorphism is strongly contracting.
We consider the outer automorphism group Out(A_Gamma) of the right-angled Artin group A_Gamma of a random graph Gamma on n vertices in the Erdos--Renyi model. We show that the functions (log(n)+log(log(n)))/n and 1-(log(n)+log(log(n)))/n bound the range of edge probability functions for which Out(A_Gamma) is finite: if the probability of an edge in Gamma is strictly between these functions as n grows, then asymptotically Out(A_Gamma) is almost surely finite, and if the edge probability is strictly outside of both of these functions, then asymptotically Out(A_Gamma) is almost surely infinite. T…
We explore the existence of homomorphisms between outer automorphism groups of free groups Out(F_n) \to Out(F_m). We prove that if n > 8 is even and n \neq m \leq 2n, or n is odd and n \neq m \leq 2n - 2, then all such homomorphisms have finite image; in fact they factor through det: Out(F_n) \to Z/2. In contrast, if m = r^n(n - 1) + 1 with r coprime to (n - 1), then there exists an embedding Out(F_n) \to Out(F_m). In order to prove this last statement, we determine when the action of Out(F_n) by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with a…
We discuss the possibility of lifting finite subgroups, and in particular finite cyclic subgroups, with respect to the canonical projections between automorphism and outer automorphism groups of free groups, surface groups and their abelianizations.
An infinite linearly ordered set (S,<=) is called doubly homogeneous if its automorphism group A(S) acts 2-transitively on it. We show that any group G arises as outer automorphism group G cong Out(A(S)) of the automorphism group A(S), for some doubly homogeneous chain (S,<=).
Fix an equilateral triangle group $T_i=\langle a, b; a^i, b^i, (ab)^i\rangle$ with $i\geq6$ arbitrary. Our main result is: for every presentation $\mathcal{P}$ of every countable group $Q$ there exists an HNN-extension $T_{\mathcal{P}}$ of $T_i$ such that $\operatorname{Out}(T_{\mathcal{P}})\cong Q$. We construct the HNN-extensions explicitly, and examples are given. The class of groups constructed have nice categorical and residual properties. In order to prove our main result we give a method for recognising malnormal subgroups of small cancellation groups, and we introduce the concept of "m…
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 nongeometric $\mathbb R$-tree all of whose branch points are trivalent
Handel and Mosher define the axis bundle for a fully irreducible outer automorphism in "Axes in Outer Space." In this paper we give a necessary and sufficient condition for the axis bundle to consist of a unique periodic fold line. As a consequence, we give a setting, and means for identifying in this setting, when two elements of an outer automorphism group $Out(F_r)$ have conjugate powers.
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.
