arXiv research

Automorphisms of handlebodies arise naturally in the a classification of automorphisms of three-manifolds. Among automorphisms of handlebodies, there are certain automorphisms called irreducible (or generic), which are analogues of pseudo-Anosov automorphisms of surfaces. We show that irreducible automorphisms of handl…

Among (isotopy classes of) automorphisms of handlebodies those called irreducible (or generic) are the most interesting, analogues of pseudo-Anosov automorphisms of surfaces. We consider the problem of isotoping an irreducible automorphism so that it is most efficient (has minimal growth rate) in its isotopy class. We …

A surface automorphism is strongly irreducible if every essential simple closed curve in the surface has nontrivial geometric intersection with its image. We show that a three-manifold admits only finitely many inequivalent surface bundle structures with strongly irreducible monodromy.

By using a notion of a geometric Dehn twist in $\sharp_k(S^2 \times S^1)$, we prove that when projections of two $\mathbb{Z}$-splittings to the free factor complex are far enough from each other in the free factor complex, Dehn twist automorphisms corresponding to the $\mathbb{Z}$-splittings generate a free group of ra…

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

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…

Constructs pseudo-Anosov mappings related to toral automorphisms.

We finish proving that an irreducible automorphism f of a handlebody is efficient if, and only if, a certain standard pair of dual f--invariant laminations have the geometric tightness property. In a previous paper it was proved that this tightness property implies efficiency. We now prove the converse.

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

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

Classifies periodic diffeomorphisms on surfaces commuting with specific involutions.

Study on Hausdorff dimension of lamination endpoints for fully irreducible automorphisms.

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…

Study on automorphisms of K3 and Enriques surfaces, proving entropy gaps and achirality.

Decomposes axis bundles into cubist structures for fully irreducible outer automorphisms.

We classify isotopy classes of automorphisms (self-homeomorphisms) of 3-manifolds satisfying the Thurston Geometrization Conjecture. The classification is similar to the classification of automorphisms of surfaces developed by Nielsen and Thurston, except an automorphism of a reducible manifold must first be written as…

The paper studies the action of automorphisms on train tracks and finds that the set of minimally displaced points is co-compact.

Study shows Morse elements are common in acylindrically hyperbolic groups.

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

We analyze the mapping class group of extendible automorphisms of the exterior boundary W of a compression body of dimension 3 or 4, which extend over the compression body (Q,V), where V is the interior boundary. Those that extend as automorphisms of (Q,V) rel V are called discrepant automorphisms, forming the mapping …

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 classes of each $G_i$. We show that every finitely generated subgroup $H\subseteq Ou…

We provide an effective algorithm for determining whether an element of the outer automorphism group of a free group is fully irreducible. Our method produces a finite list which can be checked for periodic proper free factors.

We prove sharp limit theorems on random walks on graphs with values in finite groups. We then apply these results (together with some elementary algebraic geometry, number theory, and representation theory) to finite quotients of lattices in semisimple Lie groups (specifically SL(n,Z) and Sp(2n, Z) to show that a ``ran…

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 …

We let $\varphi$ be an ageometric fully irreducible outer automorphism so that its Handel-Mosher axis bundle consists of a single unique axis. We show that the centralizer $Cen(\langle\varphi\rangle)$ of the cyclic subgroup generated by $\varphi$ equals the stabilizer $\text{Stab}(Λ^+_\varphi)$ of the attracting lamina…

Rigidity of $L^2$-spaces on manifolds is characterized.

We provide an example in each rank of an ageometric fully irreducible outer automorphism whose ideal Whitehead graph has a cut vertex. Consequently, we show that there exist examples in each rank of Handel-Mosher axis bundles that are not just a single axis, as well as of "nongeneric" behavior in the sense of the "trai…

Study connects group invariants through outer automorphisms and polynomial relations.

In this paper we prove that a fully irreducible outer automorphism relative to a non-exceptional free factor system acts loxodromically on the relative free factor complex as defined by Handel and Mosher. We also prove a north-south dynamic result for the action of such outer automorphisms on the closure of relative ou…

We give upper bounds on the numbers of various classes of polynomials reducible over the integers and over integers modulo a prime and on the number of matrices in SL(n), GL(n) and Sp(2n) with reducible characteristic polynomials, and on polynomials with non-generic Galois groups. We use our result to show that a rando…

For a fully irreducible automorphism φof the free group F_k we compute the asymptotics of the intersection number n \mapsto i(T,T'φ^n) for trees T,T' in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T'φ^n for n large.

We show that K3 surfaces with non-symplectic automorphisms of prime order can be used to construct new compact irreducible G2-manifolds. This technique was carried out in detail by Kovalev and Lee for non-symplectic involutions. We use Chen-Ruan orbifold cohomology to determine the Hodge diamonds of certain complex thr…

In \cite{Ka14} we produced an algorithm for deciding whether or not an element $φ\in Out(F_N)$ is an iwip ("fully irreducible") automorphism. At several points that algorithm was rather inefficient as it involved some general enumeration procedures as well as running several abstract processes in parallel. In this pape…

We consider when automorphisms of a graph can be induced by homeomorphisms of embeddings of the graph in a $3$-manifold. In particular, we prove that every automorphism of a graph is induced by a homeomorphism of some embedding of the graph in a connected sum of one or more copies of $S^2\times S^1$, yet there exist au…

The paper proves geometrical finiteness for automorphism groups of K3 surfaces and related varieties.

We consider the group of isotopy classes of automorphisms of the 3-sphere that preserve a spatial graph or a handlebody-knot embedded in it. We prove that the group is finitely presented for an arbitrary spatial graph or a reducible handlebody-knot of genus two. We also prove that the groups for "most" irreducible genu…

We study the Lipschitz metric on Outer Space and prove that fully irreducible elements of Out(F_n) act by hyperbolic isometries with axes which are strongly contracting. As a corollary, we prove that the axes of fully irreducible automorphisms in the Cayley graph of Out(F_n) are stable, meaning that a quasi-geodesic wi…

The study proves a theorem about subword complexity for free group automorphisms.

For any finite collection $f_i$ of fully irreducible automorphisms of the free group $F_n$ we construct a connected $δ$-hyperbolic $Out(F_n)$-complex in which each $f_i$ has positive translation length.

For a finitely generated group $G$, we introduce an asymmetric pseudometric on projectivized deformation spaces of $G$-trees, using stretching factors of $G$-equivariant Lipschitz maps, that generalizes the Lipschitz metric on Outer space and is an analogue of the Thurston metric on Teichmüller space. We show that in t…

In this paper we define currents relative to a free factor system. We prove that a fully irreducible outer automorphism relative to a free factor system acts with uniform north-south dynamics on a subspace of the space of projective relative currents.

We show that for k at least 3, given any matrix in GL(k,Z), there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on Z^k is the given matrix.

Paper disproves Wright's periodic map conjecture.

In the present article, we provide examples of fake quadrics, that is, minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in $\PP ^3$ which are quotients of the bidisc by an irreducible lattice of automorphisms. Moreover, we list classes of arithmetic lattices over a real …

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…

If $G$ is a semisimple Lie group of real rank at least 2 and $Γ$ is an irreducible lattice in $G$, then every homomorphism from $Γ$ to the outer automorphism group of a finitely generated free group has finite image.

This paper uses Brin and Thickstun's theory of end reductions of non-compact 3-manifolds to study groups of covering translations of irreducible contractible open 3-manifolds W which are not homeomorphic to R^3. We associate to W an object S(W) called the simplicial complex of minimal R^2-irreducible end reductions of …

Groups with specific properties have similar cubulations and coarse median structures.