arXiv subjects
Publications and source records attributed to Yair N. Minsky.
Let phi be a pseudo-Anosov flow on a closed oriented atoroidal 3-manifold M. We show that if F is any taut foliation almost transverse to phi, then the action of pi_1(M) on the boundary of the flow space, together with a natural collection of explicitly described monotone maps, defines a universal circle for F in the sense of Thurston and Calegari-Dunfield.
Let $φ$ be a transitive pseudo-Anosov flow on an oriented, compact $3$-manifold $M$, possibly with toral boundary. We characterize the surfaces in $M$ that are (almost) transverse to $ϕ$. When $φ$ has no perfect fits (e.g. $φ$ is the suspension flow of a pseudo-Anosov homeomorphism), we prove that any Thurston-norm minimizing surface $S$ that pairs nonnegatively with the closed orbits of $φ$ is almost transverse to $φ$, up to isotopy. This answers a question of Cooper--Long--Reid. Our main tool is a correspondence between surfaces that are almost transverse to $φ$ and those that are relatively carried by any associated veering triangulation. The correspondence also allows us to investigate the uniqueness of almost transverse position, to extend Mosher's Transverse Surface Theorem to the case with boundary, and more generally to characterize when relative homology classes represent Birkhoff surfaces.
Let $Γ$ be a Zariski dense discrete subgroup of a connected semisimple real algebraic group $G$. Let $k=\operatorname{rank} G$. Let $ψ_Γ:\mathfrak{a} \to \mathbb{R}\cup \{-\infty\}$ be the growth indicator function of $Γ$, first introduced by Quint. In this paper, we obtain the following pointwise bound of $ψ_Γ$: for all $v\in \mathfrak{a}$, $$ ψ_Γ(v) \le \min_{1\le i\le k} δ_{α_i} α_i(v) $$ where $Δ=\{α_1, \cdots, α_k\}$ is the set of all simple roots of $(\mathfrak{g},\mathfrak{a})$ and $0<δ_{α_i}\le \infty$ is the critical exponent of $Γ$ associated to $α_i$. When $Γ$ is $Δ$-Anosov, there are precisely $k$-number of directions where the equality is achieved, and the following strict inequality holds for $k\ge 2$: for all $v\in \mathfrak{a}-\{0\}$, $$ψ_Γ(v) <\frac{1}{k}\sum_{i=1}^k δ_{α_i} α_i (v).$$ We discuss applications for self-joinings of convex cocompact subgroups in $\prod_{i=1}^k \operatorname{SO}(n_i,1)$ and Hitchin subgroups of $\operatorname{PSL}(d, \mathbb{R})$. In particular, for a Zariski dense Hitchin subgroup $Γ<\text{PSL}(d, \mathbb{R})$, we obtain that for any $ v=\operatorname{diag}(t_1, \cdots, t_d)\in \mathfrak{a}^+$, $$ψ_Γ(v) \le \min_{1\le i\le d-1} (t_i -t_{i+1}). $$
We show that every atoroidal endperiodic map of an infinite-type surface can be obtained from a depth one foliation in a fibered hyperbolic 3-manifold, reversing a well-known construction of Thurston. This can be done almost-transversely to the canonical suspension flow, and as a consequence we recover the Handel-Miller laminations of such a map directly from the fibered structure. We also generalize from the finite-genus case the relation between topological entropy, growth rates of periodic points, and growth rates of intersection numbers of curves. Fixing the manifold and varying the depth one foliations, we obtain a description of the Cantwell-Conlon foliation cones and a proof that the entropy function on these cones is continuous and convex.
For certain pseudo-Anosov flows $ϕ$ on closed $3$-manifolds, unpublished work of Agol--Guéritaud produces a veering triangulation $τ$ on the manifold $M$ obtained by deleting $ϕ$'s singular orbits. We show that $τ$ can be realized in $M$ so that its 2-skeleton is positively transverse to $ϕ$, and that the combinatorially defined flow graph $Φ$ embedded in $M$ uniformly codes $ϕ$'s orbits in a precise sense. Together with these facts we use a modified version of the veering polynomial, previously introduced by the authors, to compute the growth rates of $ϕ$'s closed orbits after cutting $M$ along certain transverse surfaces, thereby generalizing work of McMullen in the fibered setting. These results are new even in the case where the transverse surface represents a class in the boundary of a fibered cone of $M$. Our work can be used to study the flow $ϕ$ on the original closed manifold. Applications include counting growth rates of closed orbits after cutting along closed transverse surfaces, defining a continuous, convex entropy function on the `positive' cone in $H^1$ of the cut-open manifold, and answering a question of Leininger about the closure of the set of all stretch factors arising as monodromies within a single fibered cone of a $3$-manifold. This last application connects to the study of endperiodic automorphisms of infinite-type surfaces and the growth rates of their periodic points.
For a hyperbolic fibered 3-manifold M, we prove results that uniformly relate the structure of surface projections as one varies the fibrations of M. This extends our previous work from the fully-punctured to the general case.
We prove that the hierarchical hulls of finite sets of points in mapping class groups and Teichmüller spaces are stably approximated by a CAT(0) cube complexes, strengthening a result of Behrstock-Hagen-Sisto. As applications, we prove that mapping class groups are semihyperbolic and Teichmüller spaces are coarsely equivariantly bicombable, and both admit stable coarse barycenters. Our results apply to the broader class of "colorable" hierarchically hyperbolic spaces and groups.
We introduce a polynomial invariant $V_τ\in \mathbb{Z}[H_1(M)/\text{torsion}]$ associated to a veering triangulation $τ$ of a $3$-manifold $M$. In the special case where the triangulation is layered, i.e. comes from a fibration, $V_τ$ recovers the Teichmüller polynomial of the fibered faces canonically associated to $τ$. Via Dehn filling, this gives a combinatorial description of the Teichmüller polynomial for any hyperbolic fibered $3$-manifold. For a general veering triangulation $τ$, we show that the surfaces carried by $τ$ determine a cone in homology that is dual to its cone of positive closed transversals. Moreover, we prove that this is $\textit{equal}$ to the cone over a (generally non-fibered) face of the Thurston norm ball, and that $τ$ computes the norm on this cone in a precise sense. We also give a combinatorial description of $V_τ$ in terms of the $\textit{flow graph}$ for $τ$ and its Perron polynomial. This perspective allows us to characterize when a veering triangulation comes from a fibration, and more generally to compute the face of the Thurston norm determined by $τ$.
We prove that any mapping torus of a pseudo-Anosov mapping class with bounded normalized Weil-Petersson translation length contains a finite set of transverse and level closed curves, and drilling out this set of curves results in one of a finite number of cusped hyperbolic 3-manifolds. The number of manifolds in the finite list depends only on the bound for normalized translation length. We also prove a complementary result that explains the necessity of removing level curves by producing new estimates for the Weil-Petersson translation length of compositions of pseudo-Anosov mapping classes and arbitrary powers of a Dehn twist.
We study the connections between subsurface projections in curve and arc complexes in fibered 3-manifolds and Agol's veering triangulation. The main theme is that large-distance subsurfaces in fibers are associated to large simplicial regions in the veering triangulation, and this correspondence holds uniformly for all fibers in a given fibered face of the Thurston norm.
We give a generalization of Thurston's Bounded Image Theorem for skinning maps, which applies to pared 3-manifolds with incompressible boundary that are not necessarily acylindrical. Along the way we study properties of divergent sequences in the deformation space of such a manifold, establishing the existence of compact cores satisfying a certain notion of uniform geometry.
We show that if the totally geodesic boundary of a compact hyperbolic 3-manifold M has a large collar of depth d, then the diameter of the skinning map of M is no more than A exp(-d) for some A depending only on the genus and injectivity radius of the boundary of M.
We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed. This construction makes for some fun mathematical magic performances. More generally, we characterize the possible Boolean functions characterizing when the picture falls in terms of which nails get removed as all monotone Boolean functions. This construction requires an exponential number of twists in the worst case, but exponential complexity is almost always necessary for general functions.
We prove a continuity property for ending invariants of convergent sequences of Kleinian surface groups. We also analyze the bounded curve sets of such groups and show that their projections to non-annular subsurfaces lie a bounded Hausdorff distance from geodesics joining the projections of the ending invariants.
Thurston's Ending Lamination Conjecture states that a hyperbolic 3-manifold N with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups; the general case when N has incompressible ends relative to its cusps follows readily. The main ingredient is the establishment of a uniformly bilipschitz model for a Kleinian surface group. The first half of the proof appeared in math.GT/0302208, and a subsequent paper will establish the Ending Lamination Conjecture in general.
A translation surface on (S, Σ) gives rise to two transverse measured foliations \FF, \GG on S with singularities in Σ, and by integration, to a pair of cohomology classes [\FF], \, [\GG] \in H^1(S, Σ; \R). Given a measured foliation \FF, we characterize the set of cohomology classes \B for which there is a measured foliation \GG as above with \B = [\GG]. This extends previous results of Thurston and Sullivan. We apply this to two problems: unique ergodicity of interval exchanges and flows on the moduli space of translation surfaces. For a fixed permutation σ\in \mathcal{S}_d, the space \R^d_+ parametrizes the interval exchanges on d intervals with permutation σ. We describe lines \ell in \R^d_+ such that almost every point in \ell is uniquely ergodic. We also show that for σ(i) = d+1-i, for almost every s>0, the interval exchange transformation corresponding to σand (s, s^2, \ldots, s^d) is uniquely ergodic. As another application we show that when k=|Σ| \geq 2, the operation of `moving the singularities horizontally' is globally well-defined. We prove that there is a well-defined action of the group B \ltimes \R^{k-1} on the set of translation surfaces of type (S, Σ) without horizontal saddle connections. Here B \subset \SL(2,\R) is the subgroup of upper triangular matrices.
This note introduces and studies an open set of PSL(2,C) characters of a nonabelian free group, on which the action of the outer automorphism group is properly discontinuous, and which is strictly larger than the set of discrete, faithful convex-cocompact (i.e. Schottky) characters. This implies, in particular, that the outer automorphism group does not act ergodically on the set of characters with dense image. Hence there is a difference between the geometric (discrete vs. dense) decomposition of the characters, and a natural dynamical decomposition.
We study a notion of a Lipschitz, permutation-invariant "centroid" for triples of points in mapping class groups MCG(S), which satisfies a certain polynomial growth bound. A consequence (via work of Drutu-Sapir or Chatterji-Ruane) is the Rapid Decay Property for MCG(S).