A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system (GIFS), based on a combinatorial structure we call a pre-tree, is introduced. For each GIFS, a family of tilings is constructed indexed by a parameter. For what we call a commensurate GIFS, our method is used to define what we refer to as balanced tilings. Under mild conditions on

We show that every plumbing of disc bundles over surfaces whose genera satisfy a simple inequality may be embedded as a convex submanifold in some closed hyperbolic four-manifold. In particular its interior has a geometrically finite hyperbolic structure that covers a closed hyperbolic four-manifold.

A Riemannian manifold is called almost positively curved if the set of points for which all 2-planes have positive sectional curvature is open and dense. We find three new examples of almost positively curved manifolds: \(Sp(3)/Sp(1)^2\), and two circle quotients of \(Sp(3)/Sp(1)^2\). We also show the quasi-positively curved metric of Tapp (J Differ Geom 65:273–287, 2003) on \(Sp(n+1)/Sp(n-1) Sp(1)\)

The Burau representation is a fundamental bridge between the braid group and diverse other topics in mathematics. A 1974 question of Birman asks for a description of the image; in this paper we give an approximate answer. Since a 1984 paper of Squier it has been known that the Burau representation preserves a certain Hermitian form. We show that the Burau image is dense in this unitary group relative

We provide criteria ensuring that a tunnel number one knot K is not determined by its double branched cover, in the sense that the double branched cover is also the double branched cover of a knot \(K'\) not equivalent to K.

We study linear series on a general curve of genus g, whose images are exceptional with respect to their secant planes. Each such exceptional secant plane is algebraically encoded by an included linear series, whose number of base points computes the incidence degree of the corresponding secant plane. With enumerative applications in mind, we construct a moduli scheme of inclusions of limit linear

Dilation surfaces are generalizations of translation surfaces where the geometric structure is modelled on the complex plane up to affine maps whose linear part is real. They are the geometric framework to study suspensions of affine interval exchange maps. However, though the \(SL(2,\mathbb {R})\)-action is ergodic in connected components of strata of translation surfaces, there may be mutually disjoint

We study three different topologies on the moduli space \(\mathcal {H}^\mathrm{loc}_m\) of equivariant local isometry classes of m-dimensional locally homogeneous Riemannian spaces. As an application, we provide the first examples of locally homogeneous spaces converging to a limit space in the pointed \(\mathcal {C}^{k,\alpha }\)-topology, for some \(k>1\), which do not admit any convergent subsequence

In this article we obtain the rigid isotopy classification of generic rational curves of degre 5 in \({\mathbb {R}}{\mathbb {P}}^{2}\). In order to study the rigid isotopy classes of nodal rational curves of degree 5 in \({\mathbb {R}}{\mathbb {P}}^{2}\), we associate to every real rational nodal quintic curve with a marked real nodal point a nodal trigonal curve in the Hirzebruch surface \(\Sigma

We study smooth rational closed embeddings of the real affine line into the real affine plane, that is algebraic rational maps from the real affine line to the real affine plane which induce smooth closed embeddings of the real euclidean line into the real euclidean plane. We consider these up to equivalence under the group of birational automorphisms of the real affine plane which are diffeomorphisms

Let X be a smooth complex projective curve, and let \(x\,\in \, X\) be a point. We compute the automorphism group of the moduli space of framed vector bundles on X of rank \(r\, \ge \, 2\) with a framing over x. It is shown that this automorphism group is generated by the following three: (1) pullbacks using automorphisms of the curve X that fix the marked point x, (2) tensorization with certain line

We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying comparable bounds. This kind of discs naturally come up in the context of the solution

We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author’s earlier result to the with-boundary setting. In the second part of the paper we comment on another direction for extending

We prove that any Fano manifold of coindex three admitting nef tangent bundle is homogeneous.

Given a simple closed curve \(\gamma \) on a connected, oriented, closed surface S of negative Euler characteristic, Mirzakhani showed that the set of points in the moduli space of hyperbolic structures on S having a simple closed geodesic of length L of the same topological type as \(\gamma \) equidistributes with respect to a natural probability measure as \(L \rightarrow \infty \). We prove several

This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasi-isometries and we compare the quasi-isometric classification with the bi-Lipschitz classification. On the other

Tête-à-tête graphs were introduced by N. A’Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed tête-à-tête graphs provide a generalization which define mixed tête-à-tête twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed tête-à-tête twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed

Mark and Paupert devised a general method for obtaining presentations for arithmetic non-cocompact lattices, \(\Gamma \), in isometry groups of negatively curved symmetric spaces. The method involves a classical theorem of Macbeath applied to a \(\Gamma \)-invariant covering by horoballs of the negatively curved symmetric space upon which \(\Gamma \) acts. In this paper, we will discuss the application

Let G be a compact semisimple linear Lie group. We study the action of \(\text {Aut}(F_r)\) on the space \(H_*(G^r; {\mathbb {Q}})\). We compute the image of this representation and prove that it only depends on the rank of \({\mathfrak {g}}\). We show that the kernel of this representation is always the Torrelli subgroup \(\text {IA}_r\) of \(\text {Aut}(F_r)\).

We define flag structures on a real three manifold M as the choice of two complex lines on the complexified tangent space at each point of M. We suppose that the plane field defined by the complex lines is a contact plane and construct an adapted connection on an appropriate principal bundle. This includes path geometries and CR structures as special cases. We prove that the null curvature models are

Let G be a connected Lie group and \(\varGamma \subset G\) a lattice. Connection curves of the homogeneous space \(M=G/\varGamma \) are the orbits of one parameter subgroups of G. To block a pair of points \(m_1,m_2 \in M\) is to find a finite set \(B \subset M{\setminus } \{m_1, m_2 \}\) such that every connecting curve joining \(m_1\) and \(m_2\) intersects B. The homogeneous space M is blockable

We construct in projective differential geometry of the real dimension 2 higher symmetry algebra of the symplectic Dirac operator acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds

We derive a lower bound on the size of finite non-cyclic quotients of the braid group that is superexponential in the number of strands. We also derive a similar lower bound for nontrivial finite quotients of the commutator subgroup of the braid group.

The purpose of this paper is to investigate relationship between the automorphism group of a rational surface and that of its Hilbert scheme of n points.

The action dimension of a discrete group G is the minimum dimension of a contractible manifold, which admits a proper G-action. In this paper, we study the action dimension of general Artin groups. The main result is that if an Artin group with the nerve L of dimension n for \(n \ne 2\) satisfies the \(K(\pi , 1)\)-Conjecture and the top cohomology group of L with \({\mathbb {Z}}\)-coefficients is

This paper is devoted to extend some Hu–Keel results on Mori dream spaces (MDS) beyond the projective setup. Namely, \(\mathbb {Q}\)-factorial algebraic varieties with finitely generated class group and Cox ring, here called weak Mori dream spaces (wMDS), are considered. Conditions guaranteeing the existence of a neat embedding of a (completion of a) wMDS into a complete toric variety are studied,

We use tools from combinatorial group theory in order to study actions of three types on groups acting on a curve, namely the automorphism group of a compact Riemann surface, the mapping class group acting on a surface (which now is allowed to have some points removed) and the absolute Galois group \(\mathrm {Gal}({\bar{{\mathbb {Q}}}}/{\mathbb {Q}})\) in the case of cyclic covers of the projective

We construct a natural prequantization space over a monotone product of a toric manifold and an arbitrary number of complex Grassmannians of 2-planes in even-dimensional complex spaces, and prove that the universal cover of the identity component of the contactomorphism group of its total space carries a nonzero homogeneous quasi-morphism. The construction uses Givental’s nonlinear Maslov index and

In this paper we give a complete description of the horofunction boundary of the infinite dimensional real hyperbolic space, and characterise its Busemann points.

A ruled real hypersurface in a complex space form is a real hypersurface having a codimension one foliation by totally geodesic complex hyperplanes of the ambient space. Our main purpose of this paper is to introduce a new viewpoint to investigate such hypersurfaces in complex hyperbolic space \(\mathbb {CH}^n\). As an application, we study minimal ruled real hypersurfaces in \(\mathbb {CH}^n\) and

Unless another thing is stated one works in the \(C^\infty \) category and manifolds have empty boundary. Let X and Y be vector fields on a manifold M. We say that Y tracks X if \([Y,X]=fX\) for some continuous function \(f:M\rightarrow \mathbb {R}\). A subset K of the zero set \({\mathsf {Z}} (X)\) is an essential block for X if it is non-empty, compact, open in \({\mathsf {Z}}(X)\) and its Poincaré-Hopf

A Morse 2-function is a generic smooth map f from a manifold M of arbitrary finite dimension to a surface B. Its critical set maps to an immersed collection of cusped arcs in B. The aim of this paper is to explain exactly when it is possible to move these arcs around in B by a homotopy of f and to give a library of examples when M is a closed 4-manifold. The last two sections give applications to the

It was shown by Seaman that if a compact, connected, oriented, riemannian 4-manifold (M, g) of positive sectional curvature admits a harmonic 2-form of constant length, then M has definite intersection form and such a harmonic form is unique up to constant multiples. In this paper, we show that such a manifold is diffeomorphic to \(\mathbb {CP}_{2}\) with a slightly weaker curvature hypothesis and

We prove that for any closed Lorentz 4-manifold (M, g) the isometry group \({\text {Isom}}(M,g)\) is Jordan. Namely, there exists a constant C (depending on M and g) such that any finite subgroup \(\Gamma \le {\text {Isom}}(M,g)\) has an abelian subgroup \(A\le \Gamma \) satisfying \([\Gamma :A]\le C\).

Let M be a complete Riemannian manifold and \(F\subset M\) a set with a nonempty interior. For every \(x\in M\), let \(D_x\) denote the function on \(F\times F\) defined by \(D_x(y,z)=d(x,y)-d(x,z)\) where d is the geodesic distance in M. The map \(x\mapsto D_x\) from M to the space of continuous functions on \(F\times F\), denoted by \({\mathcal {D}}_F\), is called a distance difference representation

We prove that a foliation on \(\mathbb {CP}^2\) of degree d with a singular point of type saddle-node with Milnor number \(d^2+d+1\) does not have invariant algebraic curves. We give a family of this kind of foliations. We also present a family of foliations of degree d with a unique nilpotent singularity without invariant algebraic curves for d odd greater than 1. Finally we prove that the space of

In this paper we start the study of configurations of flags in closed orbits of real forms using mainly tools of GIT. As an application, using cross ratio coordinates for generic configurations, we identify boundary unipotent representations of the fundamental group of the figure eight knot complement into real forms of \({{\,\mathrm{PGL}\,}}(4,{\mathbb {C}})\).

A Ricci flow (M, g(t)) on an n-dimensional Riemannian manifold M is an intrinsic geometric flow. A family of smoothly embedded submanifolds \((S(t), g_E)\) of a fixed Euclidean space \(E = \mathbb {R}^{n+k}\) is called an extrinsic representation in \(\mathbb {R}^{n+k}\) of (M, g(t)) if there exists a smooth one-parameter family of isometries \((S(t), g_E) \rightarrow (M, g(t))\). When does such a

Amos Nevo established the pointwise ergodic theorem in \(L^p\) for measure-preserving actions of \(\mathrm {PSL}_2(\mathbb {R})\) on probability spaces with respect to ball averages and every \(p>1\). This paper shows by explicit example that Nevo’s Theorem cannot be extended to \(p=1\).

We show that word hyperbolicity of automorphism groups of graph products \(G_\Gamma \) and of Coxeter groups \(W_\Gamma \) depends strongly on the shape of the defining graph \(\Gamma \). We also characterize those \(\mathrm{Aut}(G_\Gamma )\) and \(\mathrm{Aut}(W_\Gamma )\) in terms of \(\Gamma \) that are virtually free.

We start the study of reduced complex projective plane curves, whose Jacobian syzygy module has 3 generators. Among these curves one finds the nearly free curves introduced by the authors, and the plus-one generated line arrangements introduced by Takuro Abe. All the Thom–Sebastiani type plane curves, and more generally, any curve whose global Tjurina number is equal to a lower bound given by A. du

In this paper, we generalize a result of Satoh to show that for any odd natural n, the connected sum of the n-twist spun sphere of a knot K and an unknotted projective plane in the 4-sphere is equivalent to the same unknotted projective plane. We additionally provide a fix to a small error in Satoh’s proof of the case that K is a 2-bridge knot.

It is shown that various questions about the existence of simple closed curves in normal subgroups of surface groups are undecidable.

Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is \(\mathrm {CAT}(0)\) with a metric for which all vertex stars are convex. This strengthens and generalizes a result by Crowley. Further consequences of our work are: (1) All \(\mathrm {CAT}(0)\) cube complexes are collapsible. (2) Any triangulated

Let \(S_{g,n}\) be a surface of genus \(g > 1\) with \(n>0\) punctures equipped with a complete hyperbolic cusp metric. Then it can be uniquely realized as the boundary metric of an ideal Fuchsian polyhedron. In the present paper we give a new variational proof of this result. We also give an alternative proof of the existence and uniqueness of a hyperbolic polyhedral metric with prescribed curvature

The aim of this paper is to classify the cohomogeneity one conformal actions on the 3-dimensional Einstein universe \(\mathbb {E}{\mathrm {in}}^{1,2}\), up to orbit equivalence. In a recent paper (Hassani in C R Acad Sci Paris Ser I 355:1133–1137, 2017. https://doi.org/10.1016/j.crma.2017.10.003), we studied the unique (up to conjugacy) irreducible action of \({\mathrm {PSL}}(2,\mathbb {R})\) on \(\mathbb

We prove a partial converse to the main theorem of the author’s previous paper Proper affine actions: a sufficient criterion (submitted; available at arXiv:1612.08942). More precisely, let G be a semisimple real Lie group with a representation \(\rho \) on a finite-dimensional real vector space V, that does not satisfy the criterion from the previous paper. Assuming that \(\rho \) is irreducible and

A rigid set in a curve complex of a surface is a subcomplex such that every locally injective simplicial map from the set into the curve complex is induced by a homeomorphism of the surface. In this paper, we find finite rigid sets in the curve complexes of connected nonorientable surfaces of genus g with n holes for \(g+n \ne 4\).

Let S be a compact, orientable surface of hyperbolic type. Let \((k_+,k_-)\) be a pair of negative numbers and let \((g_+, g_-)\) be a pair of marked metrics over S of constant curvature equal to \(k_+\) and \(k_-\) respectively. Using a functional introduced by Bonsante, Mondello and Schlenker, we show that there exists a unique affine deformation \(\Gamma :=(\rho ,\tau )\) of a Fuchsian group such

We construct sequences of ‘expander manifolds’ and we use them to show that there is a complete connected 2-dimensional Riemannian manifold with discontinuous isoperimetric profile, answering a question of Nardulli and Pansu. Using expander manifolds in dimension 3 we show that for any \(\epsilon , M>0\) there is a Riemannian 3-sphere S of volume 1, such that any (not necessarily connected) surface

We introduce a new fundamental domain \(\mathscr {R}_n\) for a cusp stabilizer of a Hilbert modular group \(\Gamma \) over a real quadratic field \(K=\mathbb {Q}(\sqrt{n})\). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of \(\mathcal {H}^2\times \mathcal {H}^2\). The region \(\mathscr {R}_n\) is the product of \(\mathbb {R}^+\)

We determine the optimal horoball packings of the asymptotic or Koszul-type Coxeter simplex tilings of hyperbolic 5-space, where the symmetries of the packings are derived from Coxeter groups. The packing density \(\varTheta = \frac{5}{7 \zeta (3)} \approx 0.5942196502\ldots \) is optimal and realized in eleven cases in a commensurability class of arithmetic Coxeter tilings. For the optimal packing

Due to Janet–Cartan’s theorem, any analytic Riemannian manifolds can be locally isometrically embedded into a sufficiently high dimensional Euclidean space. However, for an individual Riemannian manifold (M, g), it is in general hard to determine the least dimensional Euclidean space into which (M, g) can be locally isometrically embedded, even in the case where (M, g) is homogeneous. In this paper

We determine the least-area unit-volume tetrahedral tile of Euclidean space, without the constraint of Gallagher et al. that the tiling uses only orientation-preserving images of the tile. The winner remains Sommerville’s type 4v.

Motivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we introduce the notion of generalized vector cross products on \({\mathbb {R}}^n\) and give their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of \(\mathrm {SU}(3)\)-structures in dimension 6 whose associated 3-form is Killing