A Coxeter $n$–orbifold is an $n$–dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order $m$, whose neighborhood is locally modeled on ${ℝ}^n$ modulo the dihedral group of order $\mathfrak{2}m$ generated by two reflections. For $n\ge \mathfrak{3}$, we study the deformation space of real projective structures on a compact Coxeter $n$–orbifold $Q$ admitting a hyperbolic structure. Let $e_{+}\left(Q\right)$ be the number of ridges of order greater than or equal to $\mathfrak{3}$. A neighborhood of the hyperbolic structure in the deformation space is a cell of dimension $e_{+}\left(Q\right)-n$ if $n=\mathfrak{3}$ and $Q$ is weakly orderable, ie the faces of $Q$ can be ordered so that each face contains at most $\mathfrak{3}$ edges of order $\mathfrak{2}$ in faces of higher indices, or $Q$ is based on a truncation polytope.

The purpose of this paper is to generalize a theorem of Segal proving that the space of holomorphic maps from a Riemann surface to a complex projective space is homology equivalent to the corresponding space of continuous maps through a range of dimensions increasing with degree. We will address if a similar result holds when other almost-complex structures are put on a projective space. For any compatible almost-complex structure $J$ on $ℂP^{\mathfrak{2}}$, we prove that the inclusion map from the space of $J$–holomorphic maps to the space of continuous maps induces a homology surjection through a range of dimensions tending to infinity with degree. The proof involves comparing the scanning map of topological chiral homology with analytic gluing maps for $J$–holomorphic curves . This is an extension of the author’s work regarding genus-zero case.

We classify the locally finite ergodic invariant measures of certain infinite interval exchange transformations (IETs). These transformations naturally arise from return maps of the straight-line flow on certain translation surfaces, and the study of the invariant measures for these IETs is equivalent to the study of invariant measures for the straight-line flow in some direction on these translation surfaces. For the surfaces and directions to which our methods apply, we can characterize the locally finite ergodic invariant measures of the straight-line flow in a set of directions of Hausdorff dimension larger than $\frac{\mathfrak{1}}{\mathfrak{2}}$. We promote this characterization to a classification in some cases. For instance, when the surfaces admit a cocompact action by a nilpotent group, we prove each ergodic invariant measure for the straight-line flow is a Maharam measure, and we describe precisely which Maharam measures arise. When the surfaces under consideration are of finite area, the straight-line flows in the directions we understand are uniquely ergodic. Our methods apply to translation surfaces admitting multitwists in a pair of cylinder decompositions in nonparallel directions.

Nous contrôlons deux invariants globaux des variétés hyperboliques à bouts cuspidaux : la longueur de la plus courte géodésique fermée (la systole), et le rayon de la plus grande boule plongée (le rayon maximal). Nous majorons la systole en fonction de la dimension et du volume simplicial. Nous minorons le rayon maximal par une constante positive indépendante de la dimension. Ces bornes sont optimales en dimension $\mathfrak{3}$. Cela donne une nouvelle caractérisation de la variété de Gieseking.

We bound two global invariants of cusped hyperbolic manifolds: the length of the shortest closed geodesic (the systole), and the radius of the biggest embedded ball (the inradius). We give an upper bound for the systole, expressed in terms of the dimension and simplicial volume. We find a positive lower bound on the inradius independent of the dimension. These bounds are sharp in dimension $\mathfrak{3}$, realized by the Gieseking manifold. They provide a new characterization of this manifold.

We propose a program to study groups acting faithfully on $S^{\mathfrak{1}}$ in terms of numbers of pairwise transverse dense invariant laminations. We give some examples of groups that admit a small number of invariant laminations as an introduction to such groups. The main focus of the present paper is to characterize Fuchsian groups in this scheme. We prove a group acting on $S^{\mathfrak{1}}$ is conjugate to a Fuchsian group if and only if it admits three very full laminations with a variation on the transversality condition. Some partial results toward a similar characterization of hyperbolic $\mathfrak{3}$–manifold groups that fiber over the circle have been obtained. This work was motivated by the universal circle theory for tautly foliated $\mathfrak{3}$–manifolds developed by Thurston, Calegari and Dunfield.

Let $M$ and $N$ be two closed $C^{\infty }$ manifolds and let ${Diff}_c\left(M\right)$ denote the group of $C^{\infty }$ diffeomorphisms isotopic to the identity. We prove that any (discrete) group homomorphism between ${Diff}_c\left(M\right)$ and ${Diff}_c\left(N\right)$ is continuous. We also show that a nontrivial group homomorphism $\Phi :{Diff}_c\left(M\right)\to {Diff}_c\left(N\right)$ implies that $dim\left(M\right)\le dim\left(N\right)$ and give a classification of such homomorphisms when $dim\left(M\right)=dim\left(N\right)$.

We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring Möbius invariance, the definition of discrete conformal maps as circumcircle-preserving piecewise projective maps, and two variational principles. We show how literally the same theory can be reinterpreted to address the problem of constructing an ideal hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations. It also shows how the definitions of discrete conformality considered here are closely related to the established definition of discrete conformality in terms of circle packings.

For $k\ge \mathfrak{1}$, let ${\mathcal{ℐ}}_g^{\mathfrak{1}}\left(k\right)$ be the $k^{\text{ th}}$ term in the Johnson filtration of the mapping class group of a genus $g$ surface with one boundary component. We prove that for all $k\ge \mathfrak{1}$, there exists some $G_k\ge \mathfrak{0}$ such that ${\mathcal{ℐ}}_g^{\mathfrak{1}}\left(k\right)$ is generated by elements which are supported on subsurfaces whose genus is at most $G_k$. We also prove similar theorems for the Johnson filtration of $Aut\left(F_n\right)$ and for certain mod-$p$ analogues of the Johnson filtrations of both the mapping class group and of $Aut\left(F_n\right)$. The main tools used in the proofs are the related theories of FI–modules (due to the first author with Ellenberg and Farb) and central stability (due to the second author), both of which concern the representation theory of the symmetric groups over $\mathbb{Z}$.

We study quotients $\Gamma \setminus {ℍ}^n$ of the $n$–fold product of the upper half-plane $ℍ$ by irreducible and torsion-free lattices $\Gamma <{PSL}_{\mathfrak{2}}{\left(ℝ\right)}^n$ with the same Betti numbers as the $n$–fold product ${\left({\mathbb{P}}^{\mathfrak{1}}\right)}^n$ of projective lines. Such varieties are called fake products of projective lines or fake ${\left({\mathbb{P}}^{\mathfrak{1}}\right)}^n$. These are higher-dimensional analogs of fake quadrics. In this paper we show that the number of fake ${\left({\mathbb{P}}^{\mathfrak{1}}\right)}^n$ is finite (independently of $n$), we give examples of fake ${\left({\mathbb{P}}^{\mathfrak{1}}\right)}^{\mathfrak{4}}$ and show that for $n>\mathfrak{4}$ there are no fake ${\left({\mathbb{P}}^{\mathfrak{1}}\right)}^n$ of the form $\Gamma \setminus {ℍ}^n$ with $\Gamma$ contained in the norm-one group of a maximal order of a quaternion algebra over a real number field.

For any closed oriented hyperbolic $\mathfrak{3}$–manifold $M$ and any closed oriented $\mathfrak{3}$–manifold $N$, we show that $M$ admits a finite cover $M^{\prime }$ such that there exists a degree-$\mathfrak{2}$ map $f:M^{\prime }\to N$, ie $M$ virtually $\mathfrak{2}$–dominates $N$.

Scattering rigidity of a Riemannian manifold allows one to recognize the metric of a manifold with boundary by looking at the directions of geodesics at the boundary. Lens rigidity allows one to recognize the metric of a manifold with boundary from the same information plus the length of geodesics. There are a variety of results about lens rigidity but very little is known for scattering rigidity. We will discuss the subtle difference between these two types of rigidities and prove that they are equivalent for two-dimensional simple manifolds with boundaries. In particular, this implies that two-dimensional simple manifolds (such as the flat disk) are scattering rigid since they are lens/boundary rigid.

We classify thick subcategories of the $\infty$–categories of perfect modules over ring spectra which arise as functions on even periodic derived stacks satisfying affineness and regularity conditions. For example, we show that the thick subcategories of perfect modules over $TMF$ are in natural bijection with the subsets of the underlying space of the moduli stack of elliptic curves which are closed under specialization.

We prove that any translating soliton for the mean curvature flow that is noncollapsed and uniformly 2–convex must be the rotationally symmetric bowl soliton. In particular, this proves a conjecture of White and Wang in the 2–convex case in arbitrary dimension.