We build on the constructions in Brady (J Lond Math Soc 60(2):461–480, 1999) and Lodha (A hyperbolic group with a finitely presented subgroup that is not of type FP3, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, pp 67–81, 2017) to give infinite families of hyperbolic groups, each having a finitely presented subgroup that is not of type \(F_3\). By calculating

The orthoscheme complex of a graded poset is a metrization of its order complex such that the simplex of each maximal chain is isometric to the Euclidean simplex of vertices \(0, e_1,e_1+e_2,\ldots , e_1+e_2+ \cdots + e_n\). This notion was introduced by Brady and McCammond in geometric group theory, and has applications in discrete optimization and submodularity theory. We address the question of

We study Thurston’s Lipschitz and curve metrics, as well as the arc metric on the Teichmüller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston’s stretch maps and prove the following: (1) On the Teichmüller space of the torus

We provide an algorithm to determine whether a link L admits a crossing change that turns it into a split link, under some fairly mild hypotheses on L. The algorithm also provides a complete list of all such crossing changes. It can therefore also determine whether the unlinking number of L is 1.

Liouville’s theorem says that in dimension greater than two, all conformal maps are Möbius transformations. We prove an analogous statement about simplicial complexes, where two simplicial complexes are considered discretely conformally equivalent if they are combinatorially equivalent and the lengths of corresponding edges are related by scale factors associated with the vertices.

We show that an Anosov map has a geodesic axis on the curve graph of the torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any given Anosov map. The application of our result is threefold: (a) to determine

A complex K3 surface or an algebraic K3 surface in characteristics distinct from 2 cannot have more than 16 disjoint nodal curves.

Let X be a minimal projective threefold of general type over \(\mathbb {C}\) with only Gorenstein quotient singularities, and let \(\mathrm {Aut}_{\mathbb {Q}}(X)\) be the subgroup of automorphisms acting trivially on \(H^*(X,\mathbb {Q})\). In this paper, we show that if X is of maximal Albanese dimension, then \(|\mathrm {Aut}_{\mathbb {Q}}(X)|\le 6\). Moreover, if X is nonsingular and \(K_X\) is

While a generic smooth Ribaucour sphere congruence admits exactly two envelopes, a discrete R-congruence gives rise to a 2-parameter family of discrete enveloping surfaces. The main purpose of this paper is to gain geometric insights into this ambiguity. In particular, discrete R-congruences that are enveloped by discrete channel surfaces and discrete Legendre maps with one family of spherical curvature

We show that given a collection \(X=\{f_1\), ..., \(f_m\}\) of pure mapping classes on a surface S, there is an explicit constant N, depending only on X, such that their Nth powers \(\{f_1^N\), ..., \(f_m^N\}\) generate the expected right-angled Artin subgroup of MCG(S). Moreover, we show that these subgroups are undistorted, and that each element is pseudo-Anosov on the largest possible subsurface

Let \({\mathcal {O}}\) be a closed n-dimensional arithmetic (real or complex) hyperbolic orbifold. We show that the diameter of \({\mathcal {O}}\) is bounded above by $$\begin{aligned} \frac{c_1\log \mathrm {vol}({\mathcal {O}}) + c_2}{h({\mathcal {O}})}, \end{aligned}$$ where \(h({\mathcal {O}})\) is the Cheeger constant of \({\mathcal {O}}\), \(\mathrm {vol}({\mathcal {O}})\) is its volume, and constants

Let \(\Gamma \) be a finite simplicial graph such that the flag complex on \(\Gamma \) is a 2-dimensional triangulated disk. We show that with some assumptions, the Dehn function of the associated Bestvina–Brady group is either quadratic, cubic, or quartic. Furthermore, we can identify the Dehn function from the defining graph \(\Gamma \).

The goal of this paper is to construct distinct trisections of the same genus on a fixed 4-manifold. For every \(k \ge 2\), we exhibit infinitely many manifolds with \(2^{k}-1\) non-diffeomorphic (3k, k)-trisections. Here, the manifolds are spun Seifert fiber spaces and the trisections come from Meier’s spun trisections. The technique used to distinguish the trisections parallels a common setup for

We study, for the first time, the maximum modulus set of a quasiregular map. It is easy to see that these sets are necessarily closed, and contain at least one point of each modulus. Blumenthal showed that for entire maps these sets are either the whole plane, or a countable union of analytic curves. We show that in the quasiregular case, by way of contrast, any closed set containing at least one point

Hilbert volume is an invariant of real projective geometry. Polygons inscribed in polygons are considered for the real projective plane. In two-dimensions the correspondence between Fock–Goncharov and Cartesian coordinates is examined. Degeneration and Hilbert area of inscribed quadrilaterals are analyzed. A microlocal condition is developed for bounded Hilbert area under degeneration. The condition

A Correction to this paper has been published: https://doi.org/10.1007/s10711-015-0119-z

It is a well-known result that a stable curve of compact type over \({\mathbb {C}}\) having two components is hyperelliptic if and only if both components are hyperelliptic and the point of intersection is a Weierstrass point for each of them. With the use of admissible covers, we generalize this characterization in two ways: for stable curves of higher gonality having two smooth components and one

For any \(n\ge 3\), the classical Liouville theorem asserts that any local conformal diffeomorphism defined on a connected open subset of \({\mathbb {S}}^n\) can be uniquely extended to a M\(\ddot{\mathrm{o}}\)bius transformation. In this article, we formulate and prove a generalization of this theorem to general rigid geometric structures.

We prove some injectivity results: that a Coxeter monoid \(\mathbb {Z}\)-algebra (or 0-Hecke algebra) injects in the incidence \(\mathbb {Z}\)-algebra of the corresponding Bruhat poset, for any Coxeter group; that the Hecke algebra of a right-angled Coxeter group injects in the Coxeter monoid \(\mathbb {Z}[q,q^{-1}]\)-algebra (and then in the incidence \(\mathbb {Z}[q,q^{-1}]\)-algebra of the corresponding

The hexagon is the least-perimeter tile in the Euclidean plane. On hyperbolic surfaces, the isoperimetric problem differs for every given area. Cox conjectured that a regular k-gonal tile with 120-degree angles is isoperimetric for its area. We prove his conjecture and more.

In this paper we introduce flat grafting as a deformation of quadratic differentials on a surface of finite type that is analogous to the grafting map on hyperbolic surfaces. Flat grafting maps are generic in the strata structure and preserve parallel measured foliations. The 1-parameter family obtained by flat grafting allows us to explicitly describe a path connecting any pair of quadratic differentials

Let X be a proper geodesic metric space and let G be a group of isometries of X which acts geometrically. Cordes constructed the Morse boundary of X which generalizes the contracting boundary for CAT(0) spaces and the visual boundary for hyperbolic spaces. We characterize Morse elements in G by their fixed points on the Morse boundary \(\partial _MX\). The dynamics on the Morse boundary is very similar

In the theory of cluster algebras, a mutation loop induces discrete dynamical systems via its actions on the cluster \({\mathcal {A}}\)- and \({\mathcal {X}}\)-varieties. In this paper, we introduce a property of mutation loops, called the sign stability, with a focus on the asymptotic behavior of the iteration of the tropical \({\mathcal {X}}\)-transformation. The sign stability can be thought of

In the first page of the published article, the author name is mistakenly spelled out as Klamlein. The correct version of the author name is given below,

We investigate a metric structure on the Thurston boundary of Teichmüller space. To do this, we develop tools in sup metrics and apply Minsky’s theorem.

In this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.

We show a kind of Obata-type theorem on a compact Einstein n-manifold \((W, \bar{g})\) with smooth boundary \(\partial W\). Assume that the boundary \(\partial W\) is minimal in \((W, \bar{g})\). If \((\partial W, \bar{g}|_{\partial W})\) is not conformally diffeomorphic to \((S^{n-1}, g_S)\), then for any Einstein metric \(\check{g} \in [\bar{g}]\) with the minimal boundary condition, we have that

We study a class of continuous deformations of branched complex projective structures on closed surfaces of genus \(g\ge 2\), which preserve the holonomy representation of the structure and the order of the branch points. In the case of non-elementary holonomy we show that when the underlying complex structure is infinitesimally preserved the branch points are necessarily arranged on a canonical divisor

We investigate the geometry of word metrics on fundamental groups of manifolds associated with the generating sets consisting of elements represented by closed geodesics. We ask whether the diameter of such a metric is finite or infinite. The first answer we interpret as an abundance of closed geodesics, while the second one as their scarcity. We discuss examples for both cases.

Let X be a genus two double covering surface of the standard flat torus \(({\mathbb {C}}/{\mathbb {Z}}[i],dz)\) with two ramified values \(0\ mod\ {\mathbb {Z}}[i]\) and \(\lambda +i\mu \ mod\ {\mathbb {Z}}[i]\). We prove that if the equation \(l+m\lambda +n\mu =0\) has nonzero integer solutions then the Hausdorff dimension of the set of nonergodic directions of X is either 0 or 1/2. And the precise

We classify closed 3-braids which are L-space knots.

Nearly Euclidean Thurston (NET) maps are described by simple diagrams which admit a natural notion of size. Given a size bound C, there are finitely many diagrams of size at most C. Given a NET map F presented by a diagram of size at most C, the problem of determining whether F is equivalent to a rational function is, in theory, a finite computation. We give bounds for the size of this computation

In this paper, we establish two strengthened versions of Klamkin’s inequality for an n-dimensional simplex in Euclidean space \({E}^n\) and give some applications.

We apply the existence and special properties of Gauduchon metrics to give several applications. The first one is concerned with the implications of algebro-geometric nature under the existence of a Hermitian metric with nonnegative holomorphic sectional curvature. The second one is to show the non-existence of holomorphic sections on Hermitian vector bundles under certain conditions. The third one

In 1983 Culler and Shalen established a way to construct essential surfaces in a 3-manifold from ideal points of the \(\mathrm {SL}_2\)-character variety associated to the 3-manifold group. We present in this article an analogous construction of certain kinds of branched surfaces (which we call essential tribranched surfaces) from ideal points of the \(\mathrm {SL}_n\)-character variety for a natural

We study the expected volume of random polytopes generated by taking the convex hull of independent identically distributed points from a given distribution. We show that, for log-concave distributions supported on convex bodies, we need at least exponentially many (in dimension) samples for the expected volume to be significant, and that super-exponentially many samples suffice for \(\kappa \)-concave

Inspired by a recent work of Grove and Petersen (Alexandrov spaces with maximal radius, 2018), where the authors studied positively curved Alexandrov spaces with largest possible boundary, namely the round sphere, we study Alexandrov spaces with lower curvature bound 1 and with large boundary other than the sphere. In particular, we classify those spaces with radius equal to \(\pi /2\), and the intrinsic

The Hitchin component \(\mathrm {Hit}_n(S)\) of a closed surface S is a preferred component of the character variety \(\mathcal {X}_{\mathrm {PSL}_n(\mathbb {R})}(S)\) consisting of homomorphisms from the fundamental group \(\pi _1(S)\) to the Lie group \(\mathrm {PSL}_n(\mathbb {R})\) , whose elements enjoy remarkable geometric and dynamical properties. We consider a certain type of deformations of

Critical nets in \({\mathbb {R}}^k\) (sometimes called geodesic nets) are embedded graph with the property that their embedding is a critical point of the total (edge) length functional and under the constraint that certain 1-valent vertices have a fixed position. In contrast to what happens on generic manifolds, we show that, if the embedding is bounded and n is the number of 1-valent vertices, the

We study the homology of Riemannian manifolds of finite volume that are covered by an r-fold product \(({\mathbb {H}}^2)^r = {\mathbb {H}}^2 \times \cdots \times {\mathbb {H}}^2\) of hyperbolic planes. Using a variation of a method developed by Avramidi and Nguyen-Phan, we show that any such manifold M possesses, up to finite coverings, an arbitrarily large number of compact oriented flat totally geodesic

Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let \(\mathbf {G}\) be a semisimple algebraic \({\mathbb {R}}\)-group such that \(G=\mathbf {G}({\mathbb {R}})^{\circ }\) is of Hermitian type. If \(\Gamma \le L\) is a torsion-free lattice of a finite connected covering of \(\mathrm{PU}(1

We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through ramified coverings. These are among the main known examples of bundles with non-zero signature.

A pair \((\alpha , \beta )\) of simple closed geodesics on a closed and oriented hyperbolic surface \(M_g\) of genus g is called a filling pair if the complementary components of \(\alpha \cup \beta \) on \(M_g\) are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. In Aougab and Huang (Algebr Geom Topol 15:903–932, 2015), Aougab–Huang conjectured

In response to Sanki–Vadnere, we present a short proof of the following theorem: a pair of simple curves on a hyperbolic surface whose complementary regions are disks has length at least half the perimeter of the regular right-angled \((8g-4)\)-gon.

Let \(m \ge 6\) be an even integer. In this short note we prove that the Jacobian variety of a quasiplatonic Riemann surface with associated group of automorphisms isomorphic to \(C_2^2 \rtimes _2 C_m\) admits complex multiplication. We then extend this result to provide a criterion under which the Jacobian variety of a quasiplatonic Riemann surface admits complex multiplication.

We consider the evolution of a strictly convex hypersurface by a class of general curvature. We prove that given some Neumann boundary condition, the flow exists for all time and converges to a solution with prescribed general curvature that satisfies the Neumann boundary condition. Our method also works for the corresponding elliptic setting.

Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex in vertex groups, we prove that vertex groups are relatively quasiconvex if and only if all the vertex groups have finite relative height in the fundamental group.

We first extend the construction of the pressure metric to the deformation space of globally hyperbolic maximal Cauchy-compact anti-de Sitter structures. We show that, in contrast with the case of the Hitchin components, the pressure metric is degenerate and we characterize its degenerate locus. We then introduce a nowhere degenerate Riemannian metric adapting the work of Qiongling Li on the \(\mathrm

A long-standing conjecture asserts that any Anosov diffeomorphism of a closed manifold is finitely covered by a diffeomorphism which is topologically conjugate to a hyperbolic automorphism of a nilpotent manifold. In this paper, we show that any closed 4-manifold that carries a Thurston geometry and is not finitely covered by a product of two aspherical surfaces does not support (transitive) Anosov

Let \({\mathbb {F}}_{\Theta }=U/K_\Theta \) be a partial flag manifold, where \(K_\Theta \) is the centralizer of a torus in U. We study U-invariant almost Hermitian structures on \({\mathbb {F}}_{\Theta }\). The classification of these structures are naturally related with the system \(R_{\mathfrak {t}}\) of \({\mathfrak {t}}\)-roots associated to \({\mathbb {F}}_{\Theta }\). We introduced the notion

We give a complete description of all locally conformally Kähler structures with holomorphic Lee vector field on a compact complex manifold of Vaisman type. This provides in particular examples of such structures whose Lee vector field is not homothetic to the Lee vector field of a Vaisman structure. More generally, dropping the condition of being of Vaisman type, we show that on a compact complex

Weyl (Zur Infinitisimalgeometrie: Einordnung der projektiven und der konformen Auffasung, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Göttinger Akademie der Wissenschaften, Göttingen, 1921) demonstrated that for a connected manifold of dimension greater than 1, if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization

We provide sharp bounds for the squared norm of the second fundamental form of a wide class of Weingarten hypersurfaces in Euclidean space satisfying \(H_r = aH + b\), for constants \(a, b \in \mathbb {R}\), where \(H_r\) stands for the rth mean curvature and H the mean curvature of the hypersurface. Besides we are able to characterize those hypersurfaces for which these bounds are attained by showing

There is an isomorphism between the moduli spaces of \(\sigma \)-stable holomorphic triples and some of the critical submanifolds of the moduli space of k-Higgs bundles of rank three, whose elements \((E,\varphi ^k)\) correspond to variations of Hodge structure, VHS. There are special embeddings on the moduli spaces of k-Higgs bundles of rank three. The main objective here is to study the cohomology

In this paper, we construct the first families of distinct Lagrangian ribbon disks in the standard symplectic 4-ball which have the same boundary Legendrian knots, and are not smoothly isotopic or have non-homeomorphic exteriors.

Sharp inequalities of the parameterized functional \(U_j\) for Borel measures on the unit sphere in \({\mathbb {R}}^n\) are established. As two applications, some inequalities related to cone-volume measures and Schneider’s projection problem are obtained.

As in the case of irreducible holomorphic symplectic manifolds, the period domain Compl of compact complex tori of even dimension 2n contains twistor lines. These are special 2-spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irreducible holomorphic symplectic manifolds, we show that the periods of any two complex tori

We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the Kronheimer–Mrowka proof of the Thom conjecture.

A closed surface evolving under mean curvature flow becomes singular in finite time. Near the singularity, the surface resembles a self-shrinker, a surface that shrinks by dilations under mean curvature flow. If the singularity is modeled on a self-shrinker other than a round sphere or cylinder, then the singularity is unstable under perturbations of the flow. One can quantify this instability using

We study multi-moment maps on nearly Kähler six-manifolds with a two-torus symmetry. Critical points of these maps have non-trivial stabilisers. The configuration of fixed-points and one-dimensional orbits is worked out for generic six-manifolds equipped with an \(\mathrm {SU}(3)\)-structure admitting a two-torus symmetry. Projecting the subspaces obtained to the orbit space yields a trivalent graph