arXiv research

Criterion for subgroup separability in outer automorphism groups.

Study shows RAAG automorphisms and outer automorphisms are not relatively hyperbolic.

Outer automorphism group of hyperbolic groups is HHG under certain conditions.

Using the Guirardel-Levitt outer space of a free product, we prove that the outer automorphism group of the outer automorphism group of the universal Coxeter group of rank $n \geq 5$ is trivial, and that it is a cyclic group of order 2 if $n=4$. In addition we prove that the outer automorphism group of the automorphism…

Study on symmetric automorphisms of RAAGs, proving finiteness properties and contractibility.

Outer automorphisms of free products are represented by CTs.

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.

Study shows Morse elements are common in acylindrically hyperbolic groups.

Study reveals uniform difference in stretch factors between genus two handlebody group and outer automorphism group.

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is resudually finite. We…

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.

We correct the proof of Theorem 5 of the paper Homology stability for outer automorphism groups of free groups, by the first two authors.

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

The paper shows that for Coxeter groups, the commensurator of outer automorphisms is rigid.

Random walks on free groups reveal asymmetric expansion factors.

We investigate the combinatorial and geometric properties of automorphism groups of universal right-angled Coxeter groups, which are the automorphism groups of free products of copies of Z_2. It is currently an open question as to whether or not these automorphism groups have non-positive curvature. Analogous to Outer …

Study connects group invariants through outer automorphisms and polynomial relations.

The paper studies automorphisms of free groups with a North-South dynamics.

The paper shows symmetries of a geometric space for Coxeter groups.

Study growth rates of automorphisms of special groups.

Let $\mathcal{A} = {A_1, ..., A_k}$ be a system of free factors of $F_n$. The group of relative automorphisms $\mathrm{Aut}(F_n; \mathcal{A})$ is the group given by the automorphisms of $F_n$ that restricted to each $A_i$ are conjugations by elements in $F_n$. The group of relative outer automorphisms is defined as $\m…

For n>2, the action of the outer automorphism group of the rank n free group F_n on the SU(2)-character variety Hom(F_n,SU(2))/SU(2)$ is ergodic with respect to the Lebesgue measure class.

We define and discuss a notion called fibered commensurability of outer automorphisms of free groups. This notion lets us study symmetry of outer automorphisms. The notion of fibered commensurability is first defined by Calegari-Sun-Wang on mapping class groups. The Nielsen-Thurston type of mapping classes is a commens…

The study examines how automorphism growth rates of a group can be deduced from its simpler decompositions.

The study explores automorphisms and centralizers in free group outer automorphisms.

We show that the automorphism group of the disk complex is isomorphic to the handlebody group. Using this, we prove that the outer automorphism group of the handlebody group is trivial.

Outer space for RAAGs is a contractible finite-dimensional space for automorphisms.

Geometric model for a specific group in Artin groups.

H. Masur and J. Smillie proved precisely which singularity index lists arise from pseudo-Anosov mapping classes. In search of an analogous theorem for outer automorphisms of free groups, Handel and Mosher ask: Is each connected, simplicial, (2r-1)-vertex graph the ideal Whitehead graph of a fully irreducible outer auto…

We study the outer automorphism group of a right-angled Artin group $A_Γ$ with finite defining graph $Γ$. We construct a subnormal series for $Out(A_Γ)$ such that each consecutive quotient is either finite, free-abelian, $GL(n,\mathbb{Z})$, or a Fouxe-Rabinovitch group. The last two types act respectively on a symmetri…

The thesis shows how automorphisms of hyperbolic groups can be represented by train track maps.

Let Ng be the connected closed nonorientable surface of genus g >= 5 and Mod(Ng) denote the mapping class group of Ng. We prove that the outer automorphism group of Mod(Ng) is either trivial or Z if g is odd, and injects into the mapping class group of sphere with four holes if g is even.

Subsurface projection has become indispensable in studying the geometry of the mapping class group and the curve complex of a surface. When the subsurface is an annulus, this projection is sometimes called relative twisting. We give two alternate versions of relative twisting for the outer automorphism group of a free …

Let $φ\in \mbox{Out}(F_n)$ be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism $φ$ determines a free-by-cyclic group $Γ=F_n \rtimes_φ\mathbb Z,$ and a homomorphism $α\in H^1(Γ; \mathbb Z)$. By work of Neumann, Bieri-Neumann-Strebel and Dowdall-Kapovi…

We define lines of minima in the thick part of Outer space for the free group Fn with n>2 generators. We show that these lines of minima are contracting for the Lipschitz metric. Every fully irreducible outer automorphism of Fn defines such a line a minima. Now let G be a subgroup of the outer automorphism group of Fn …

Study knot invariants using automorphism groups of free nilpotent groups.

Let $A_1,...,A_k$ be a system of free factors of $F_n$. The group of relative automorphisms $Aut(F_n;A_1,...,A_k)$ is the group given by the automorphisms of $F_n$ that restricted to each $A_i$ are conjugations by elements in $F_n$. The group of relative outer automorphisms is defined as $Out(F_n;A_1,...,A_k) = Aut(F_n…

Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.

Free groups' automorphisms have bounded orbits.

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 also discuss some computational results on low index subgroups of Aut F_n and Out F…

Finite groups of RAAGs act on a contractible space without fixed points.

The study examines the stretch factors of outer automorphisms and their latent symmetry.

Study embeddings of free group products into automorphism groups.

The paper studies acylindrical actions on trees and proves acylindrical hyperbolicity of Baumslag-Solitar groups.

Let $G=G_1\ast\dots\ast G_k\ast F$ be a countable group which splits as a free product, where all groups $G_i$ are freely indecomposable and not isomorphic to $\mathbb{Z}$, and $F$ is a finitely generated free group. If for all $i\in\{1,\dots,k\}$, both $G_i$ and its outer automorphism group $\text{Out}(G_i)$ satisfy t…

Study mapping class group actions on surface fundamental groups.

We give a cohomological criterion for existence of outer automorphisms of a semisimple algebraic group over an arbitrary field. This criterion is then applied to the special case of groups of type D_2n over a global field, which completes some of the main results from the paper "Weakly commensurable arithmetic groups a…

The group of 2-by-2 matrices with integer entries and determinant $\pm > 1$ can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, name…