We provide a complete description of realizable period representations for meromorphic differentials on Riemann surfaces with prescribed orders of zeros and poles, hyperelliptic structure and spin parity.

In Cartan’s PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra $\mathfrak {f}_4$. Cartan’s formula is written in the standard Cartesian coordinates in $\mathbb {R}^{15}$. In the present paper, we explain how to find analogous formulae for the flat models of any bracket generating distribution $\mathcal D$ whose symbol algebra $\mathfrak {n}({\mathcal D})$ is constant and 2-step graded, $\mathfrak {n}({\mathcal D})=\mathfrak {n}_{-2}\oplus \mathfrak {n}_{-1}$.

The formula is given in terms of a solution to a certain system of linear algebraic equations determined by two representations $(\rho ,\mathfrak {n}_{-1})$ and $(\tau ,\mathfrak {n}_{-2})$ of a Lie algebra $\mathfrak {n}_{00}$ contained in the $0$th order Tanaka prolongation $\mathfrak {n}_0$ of $\mathfrak {n}({\mathcal D})$.

Numerous examples are provided, with particular emphasis put on the distributions with symmetries being real forms of simple exceptional Lie algebras $\mathfrak {f}_4$ and $\mathfrak {e}_6$.

Let ${{\mathcal {H}}}$ be a stratum of translation surfaces with at least two singularities, let $m_{{{\mathcal {H}}}}$ denote the Masur-Veech measure on ${{\mathcal {H}}}$, and let $Z_0$ be a flow on $({{\mathcal {H}}}, m_{{{\mathcal {H}}}})$ obtained by integrating a Rel vector field. We prove that $Z_0$ is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector fields, for more general spaces $({\mathcal L}, m_{{\mathcal L}})$, where ${\mathcal L} \subset {{\mathcal {H}}}$ is an orbit-closure for the action of $G = \operatorname {SL}_2({\mathbb {R}})$ (i.e., an affine invariant subvariety) and $m_{{\mathcal L}}$ is the natural measure. These results are conditional on a forthcoming measure classification result of Brown, Eskin, Filip and Rodriguez-Hertz. We also prove that the entropy of $Z_0$ with respect to any of the measures $m_{{{\mathcal L}}}$ is zero.

We establish a link between the behavior of length functions on Teichmüller space and the geometry of certain anti-de Sitter $3$-manifolds. As an application, we give new purely anti-de Sitter proofs of results of Teichmüller theory such as (strict) convexity of length functions along shear paths and geometric bounds on their second variation along earthquakes. Along the way, we provide shear-bend coordinates for GHMC anti-de Sitter $3$-manifolds.

We formalize the concepts of holomorphic affine and projective structures along the leaves of holomorphic foliations by curves on complex manifolds. We show that many foliations admit such structures, we provide local normal forms for them at singular points of the foliation, and we prove some index formulae in the case where the ambient manifold is compact. As a consequence of these, we establish that a regular foliation of general type on a compact algebraic manifold of even dimension does not admit a foliated projective structure. Finally, we classify foliated affine and projective structures along regular foliations on compact complex surfaces.

In this paper, we focus on dynamical properties of (real) convex projective surfaces. Our main theorem provides an asymptotic formula for the number of free homotopy classes with roughly the same renormalized Hilbert length for two distinct convex real projective structures. The correlation number in this asymptotic formula is characterized in terms of their Manhattan curve. We show that the correlation number is not uniformly bounded away from zero on the space of pairs of hyperbolic surfaces, answering a question of Schwartz and Sharp. In contrast, we provide examples of diverging sequences, defined via cubic rays, along which the correlation number stays larger than a uniform strictly positive constant. In the last section, we extend the correlation theorem to Hitchin representations.

Recall that two geodesics in a negatively curved surface S are of the same type if their free homotopy classes differ by a homeomorphism of the surface. In this note we study the distribution in the unit tangent bundle of the geodesics of fixed type, proving that they are asymptotically equidistributed with respect to a certain measure ${\mathfrak {m}}^S$ on $T^1S$ . We study a few properties of this measure, showing for example that it distinguishes between hyperbolic surfaces.

Let G be a group, and let g be a nontrivial element in G. If some nonempty finite product of conjugates of g equals the identity, then g is called a generalized torsion element. We say that a knot K has generalized torsion if $G(K) = \pi _1(S^3 - K)$ admits such an element. For a $(2, 2q+1)$ -torus knot K, we demonstrate that there are infinitely many unknots $c_n$ in $S^3$ such that p-twisting K about $c_n$ yields a twist family $\{ K_{q, n, p}\}_{p \in \mathbb {Z}}$ in which $K_{q, n, p}$ is a hyperbolic knot with generalized torsion whenever $|p|> 3$ . This gives a new infinite class of hyperbolic knots having generalized torsion. In particular, each class contains knots with arbitrarily high genus. We also show that some twisted torus knots, including the $(-2, 3, 7)$ -pretzel knot, have generalized torsion. Because generalized torsion is an obstruction for having bi-order, these knots have non-bi-orderable knot groups.

Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all known examples of continuous functions on geodesic currents, such as nonpositively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length.

Let $\operatorname {\mathrm {{\rm G}}}(n)$ be equal to either $\operatorname {\mathrm {{\rm PO}}}(n,1),\operatorname {\mathrm {{\rm PU}}}(n,1)$ or $\operatorname {\mathrm {\textrm {PSp}}}(n,1)$ and let $\Gamma \leq \operatorname {\mathrm {{\rm G}}}(n)$ be a uniform lattice. Denote by $\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}}$ the hyperbolic space associated to $\operatorname {\mathrm {{\rm G}}}(n)$ , where $\operatorname {\mathrm {{\rm K}}}$ is a division algebra over the reals of dimension d. Assume $d(n-1) \geq 2$ .

In this article we generalise natural maps to measurable cocycles. Given a standard Borel probability $\Gamma $ -space $(X,\mu _X)$ , we assume that a measurable cocycle $\sigma :\Gamma \times X \rightarrow \operatorname {\mathrm {{\rm G}}}(m)$ admits an essentially unique boundary map $\phi :\partial _\infty \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \times X \rightarrow \partial _\infty \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ whose slices $\phi _x:\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ are atomless for almost every $x \in X$ . Then there exists a $\sigma $ -equivariant measurable map $F: \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \times X \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ whose slices $F_x:\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ are differentiable for almost every $x \in X$ and such that $\operatorname {\mathrm {\textrm {Jac}}}_a F_x \leq 1$ for every $a \in \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}}$ and almost every $x \in X$ . This allows us to define the natural volume $\operatorname {\mathrm {\textrm {NV}}}(\sigma )$ of the cocycle $\sigma $ . This number satisfies the inequality $\operatorname {\mathrm {\textrm {NV}}}(\sigma ) \leq \operatorname {\mathrm {\textrm {Vol}}}(\Gamma \backslash \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}})$ . Additionally, the equality holds if and only if $\sigma $ is cohomologous to the cocycle induced by the standard lattice embedding $i:\Gamma \rightarrow \operatorname {\mathrm {{\rm G}}}(n) \leq \operatorname {\mathrm {{\rm G}}}(m)$ , modulo possibly a compact subgroup of $\operatorname {\mathrm {{\rm G}}}(m)$ when $m>n$ .

Given a continuous map $f:M \rightarrow N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.

In this note we show that the expected value of the separating systole of a random surface of genus g with respect to Weil–Petersson volume behaves like $2\log g $ as the genus goes to infinity. This is in strong contrast to the behavior of the expected value of the systole which, by results of Mirzakhani and Petri, is independent of genus.

We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface $S$. Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden’s theorem.

A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with $2N$ crossings grows exponentially when $N$ grows, but the long-standing problem on the precise asymptotics is still out of reach.

We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as $N$ tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator.

The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinder. The proofs combine recent results on Masur–Veech volumes of moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics.

We construct prime amphicheiral knots that have free period 2. This settles an open question raised by the second-named author, who proved that amphicheiral hyperbolic knots cannot admit free periods and that prime amphicheiral knots cannot admit free periods of order > 2.

Under certain assumptions on CAT(0) spaces, we show that the geodesic flow is topologically mixing. In particular, the Bowen–Margulis’ measure finiteness assumption used by Ricks [Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces. Ergod. Th. & Dynam. Sys. 37 (2017), 939–970] is removed. We also construct examples of CAT(0) spaces that do not admit finite Bowen–Margulis measure.

We describe in this article the dynamics of a one-parameter family of affine interval exchange transformations. This amounts to studying the directional foliations of a particular dilatation surface introduced in Duryev et al [Affine surfaces and their Veech groups. Preprint, 2016, arXiv:1609.02130], the Disco surface. We show that this family displays various dynamical behaviours: it is generically dynamically trivial but for a Cantor set of parameters the leaves of the foliations accumulate to a (transversely) Cantor set. This study is achieved through analysis of the dynamics of the Veech group of this surface combined with a modified version of Rauzy induction in the context of affine interval exchange transformations.

The group of ${\mathcal{C}}^{1}$ -diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson’s groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin’s higher-dimensional generalizations $nV$ of Thompson’s group $V$ arise when we consider products of central ternary Cantor sets. We derive that the ${\mathcal{C}}^{2}$ -smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.

We determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible $\text{S}{{\text{L}}_{2}}(\mathbb{C})$-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We also give the set of the limits of the leading coeõcients in the higher dimensional Reidemeister torsion explicitly.

We show that Cannon–Thurston maps exist for degenerate free groups without parabolics, that is, for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon–Thurston maps for surface groups, we show that Cannon–Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon–Thurston maps for degenerate free groups without parabolics correspond to endpoints of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon–Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.