Saul Schleimer
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.
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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 handlebodies exist and develop methods for constructing a range of examples.
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.
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 describe a property, called tightness, of certain invariant laminations, which we conjecture characterizes this efficiency. We obtain partial results towards proving the conjecture. For example, we prove it for genus two handlebodies. We also show that tightness always implies ef…
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 rank $2$. Moreover, every element from this free group is either conjugate to a power of one of the Dehn twists or it is a fully irreducible outer automorphism of the free group. We also prove that, when projected to the intersection graph, the same group of Dehn twists produce ato…
We let $φ$ 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 $Cen(\langleφ\rangle)$ of the cyclic subgroup generated by $φ$ equals the stabilizer $\text{Stab}(Λ^+_φ)$ of the attracting lamination $Λ^+_φ$ and is isomorphic to $\mathbb Z$. We further show, via an analogous result about the commensurator, that the normalizer $N(\langleφ\rangle)$ of $\langle φ\rangle$ is isomorphic to either $\mathbb Z$ or $\mathbb Z_2 * \mathbb Z_2$.
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 a suitable composition of two automorphisms, each of which fits into our classification. Given an automorphism, the goal is to show, loosely speaking, either that it is periodic, or that it can be decomposed on a surface invariant up to isotopy, or that it has a "dynamically nic…
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 paper we refine the algorithm from \cite{Ka14} by eliminating these inefficient features, and also by eliminating any use of mapping class groups algorithms. Our main result is to produce, for any fixed $N\ge 3$, an algorithm which, given a topological representative $f$ of an elemen…
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-Kapovich-Leininger, $α$ has an open cone neighborhood $\mathcal A$ in $H^1(Γ;\mathbb R)$ whose integral points correspond to other fibrations of $Γ$ whose associated outer automorphisms are themselves representable by expanding irreducible train-track maps. In this paper, we define an …
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 ``random'' element in one of these lattices has irreducible characteristic polynomials (over the integers. The term ``random'' can be defined in at least two ways (in terms of height and also in terms of word length in terms of a generating set) -- we show the result using both defini…
We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed.
On the action of relatively irreducible automorphisms on their train tracks
本文研究相对𝒢的指数型𝒢-不可约自同构的轨道轨点集（即最小位移集Min(φ)），证明其在φ生成的循环子群作用下余紧，得到相关辅助结果，并推广了自由群自同构中心化子的经典定理。
Colding and Gabai have given an effective version of Li's theorem that non-Haken hyperbolic 3-manifolds have finitely many irreducible Heegaard splittings. As a corollary of their work, we show that Haken hyperbolic 3-manifolds have a finite collection of strongly irreducible Heegaard surfaces $S_i$ and incompressible surfaces $K_j$ such that any strongly irreducible Heegaard surface is a Haken sum $S_i + \sum_j n_j K_j$, up to one-sided associates of the Heegaard surfaces.
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 random (in the appropriate sense) element of the mapping class group of a closed surface is pseudo-Anosov, and that a random automorphism of a free group is strongly irreducible (aka irreducible with irreducible powers). We also give a necessary condition for all powers of an algebrai…
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 automorphism in Out(F_r)? We answer this question in the negative by exhibiting, for each r, examples of connected (2r-1)-vertex graphs that are not the ideal Whitehead graph of any fully irreducible outer automorphism in Out(F_r)? In the course of our proof we also develop machinery…
On a classification of irreducible periodic diffeomorphisms on surfaces which commute with certain involution
本文对曲面上与特定对合（其商空间同胚于$T^{2}$）交换的不可约周期自同构进行了共轭分类，推广了Ishizaka的超椭圆周期自同构分类结果。
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.
Gap theorems and achirality for automorphisms of K3 surfaces and Enriques surfaces
本文研究K3和恩里克斯面自同构群的熵范数极值定理，并探讨其在 genus-1 纤维化下的非手性性质。
For every irreducible automorphism $ϕ\in\text{SL}_3({\mathbb Z})$ of the $3$-torus, for which the product of the expanding eigenvalues is positive, we construct a pseudo-Anosov mapping $f$ of an associated surface, semi-conjugate and almost-isomorphic to $ϕ$, whose stretch factor is the product of the expanding eigenvalues of $ϕ$. This shows that any norm-$1$ cubic Pisot number occurs as the stretch factor of a pseudo-Anosov mapping, proving a conjecture of Fried in degree $3$. A similar construction works for the $4$-torus on condition that $ϕ$ has exactly two eigenvalues outside the unit cir…
On the Hausdorff dimension and attracting laminations for fully irreducible automorphisms of free groups
研究自由群自同构相关抽象代数层叠端点集的豪斯多夫维数，对指数增长及环面完全不可约外自同构，其吸引层叠或终结层叠端点集的豪斯多夫维和填充维数为\(0\)，上盒维数\(>0\)。
Uniform difference between the length spectra of Out(F2) and the genus two handlebody group
本文研究了双曲面二柄体群与 Out(F2) 之间长度谱的均匀差异，揭示了伪 Anosov 映射的伸缩因子与完全不可约外自同构之间的关系。
Latent symmetry of graphs and stretch factors in Out(Fr)
本文通过图的潜在对称性，为自由群外自同构的拉伸因子提供下界，并分类了所有不可约单折叠列车轨道映射。
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