The study proves a local rigidity theorem for hyperbolic lattices using real projective structures.
problem Local rigidity of hyperbolic lattices.
method Calculation of Zariski tangent spaces and real projective structures.
result Weil's local rigidity theorem for uniform hyperbolic lattices proved.
Study on cubic curves and their tangents, finding Zariski pairs.
problem Understanding the topology of cubic curves and their tangents.
method Topological analysis of subarrangements of k-Artal arrangements. result Existence of Zariski pairs for k=3,4,5,6. The paper examines the smooth structure at singular points of diffeological spaces.
problem Understanding the smooth structure at singular points of diffeological spaces.
method Analyzes the derivatives along lines through the origin in the pencil diffeological space.
result The Zariski cotangent space at the origin is determined by derivatives along lines and forms a σ-continuous family. Study on deformations of trivial character in SL_2(C) for groups.
problem Deforming the trivial character in SL_2(C) for groups.
method Analyzing maps satisfying the parallelogram identity on groups and relating them to deformations of the trivial character.
result The trivial character is a smooth point if and only if H_1(G,C) has dimension less than 2.
In this paper we aim at the description of foliations having tangent sheaf TF with c1(TF)=c2(TF)=0 on non-uniruled projective manifolds. We prove that the universal covering of the ambient manifold splits as a product, and that the Zariski closure of a general leaf of F is an…
The paper proves properties of a specific Seiberg-Witten equation over 3-manifolds.
problem Analyzing the Sp(1)-Seiberg-Witten equation over 3-manifolds. method Analyzes the Sp(1)-Seiberg-Witten equation over closed hyperbolic 3-manifolds and S1imesΣ. result The canonical irreducible solution is infinitesimally rigid under certain conditions.
New topological invariant distinguishes real line arrangements with same combinatorics.
problem Determining topological properties from combinatorial data of real line arrangements.
method Introducing chamber weight invariant based on dual configuration points in real projective plane.
result New Zariski pairs of 13, 15, and 17 lines with different topological embeddings.
A new invariant distinguishes algebraic curves with similar combinatorics but different topologies.
problem Distinguishing algebraic curves with similar combinatorial structures but different topologies.
method Constructing a topological invariant based on the linking number of knot theory, using modifications of braid monodromy and connected numbers.
result The invariant distinguishes several Zariski pairs, including the well-known Artal pair and a quartic with bitangents.
We prove that every Bers slice of quasi-Fuchsian space is Zariski dense in the character variety.
Paper finds surface groups can deform in reductive symmetric spaces.
problem Finding deformations of discontinuous groups in reductive symmetric spaces.
method Analyzing Zariski dense surface subgroups and their deformations.
result Surface groups of high genus can deform in reductive symmetric spaces.
Classifies Zariski closures of positive representations in Lie groups.
problem Classifying Zariski closures of positive representations in Lie groups.
method Classifies the Lie algebra of the Zariski closure of a discrete subgroup with specific properties.
result Obtains a new proof of Guichard's classification of Zariski closures of Hitchin representations.
We show that a surface group contained in a reductive real algebraic group can be deformed to become Zariski dense, unless its Zariski closure acts transitively on a Hermitian symmetric space of tube type. This is a kind of converse to a rigidity result of Burger, Iozzi and Wienhard.
Study conic line arrangements of degree 7, finding their topology and connected components.
problem Understanding the topology of conic line arrangements of degree 7.
method Identifying a π1-equivalent Zariski pair to prove the existence of a conic line arrangement with specific combinatorics. result Determine the number of connected components of conic line arrangements of degree 7.
The study finds conditions for certain groups to be dense in a specific mathematical space.
problem Conditions for linear reflection groups to be dense in a projective space.
method Analyzes necessary and sufficient conditions for Zariski-density, applies to Coxeter groups and surface subgroups.
result Establishes conditions for Zariski-dense subgroups in SLn(Z) for various n. Proves unique maps from certain spaces to others.
problem Uniqueness of equivariant harmonic maps into specific spaces.
method Analyzes maps into irreducible symmetric spaces and Euclidean buildings.
result Proves uniqueness of maps for certain actions.
We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dens…
Hyperbolic groups' infinite orbits spread evenly in spaces.
problem Equidistribution of hyperbolic groups in homogeneous spaces.
method Averaging measures along spheres in Cayley graphs converges to Haar measure.
result Infinite orbits of hyperbolic groups equidistribute in homogeneous spaces.
New theorem about limit points in symmetric spaces.
problem Understanding limit points in symmetric spaces.
method Analyzing Zariski dense discrete subgroups and convex cocompact groups.
result Every limit point of a convex cocompact subgroup is conical.
The study explores deformations of discrete subgroups in non-compact homogeneous spaces.
problem Addressing the proper discontinuity of discrete subgroups in non-compact homogeneous spaces.
method Classification results for deformations of standard discontinuous groups in pseudo-Riemannian homogeneous spaces.
result Conditions for local rigidity and Zariski-dense deformations in standard quotients.
In this note we introduce the notion of a smooth structure on a conical pseudomanifold M in terms of C∞-rings of smooth functions on M. For a finitely generated smooth structure C∞(M) we introduce the notion of the Nash tangent bundle, the Zariski tangent bundle, the tangent bundle of M, and the …
The study explores deformations of standard locally homogeneous spaces.
problem Understanding how discrete subgroups can be deformed while preserving proper discontinuity.
method Classification results for standard quotients, including local rigidity, deformation criteria, and Zariski-closure conditions.
result Conditions for local rigidity, deformation into nonstandard quotients, and maximal Zariski-closure of discontinuous groups.
New domains of discontinuity found for Anosov representations.
problem Understanding Anosov representations acting on homogeneous spaces.
method Constructing open domains of discontinuity for Anosov representations acting on specific homogeneous spaces.
result Describes the largest possible open domains of discontinuity for Zariski dense Anosov representations.
Maximal representations in infinite dimensional Hermitian spaces are studied with boundary maps.
problem Characterizing maximal representations in infinite dimensional Hermitian symmetric spaces.
method Definition of Toledo number, study of boundary maps, geometric constructions.
result Existence and non-existence conditions for maximal representations.
The study finds criteria for discreteness in quaternionic hyperbolic space.
problem Discreteness of subgroups in quaternionic hyperbolic space.
method Using test maps to establish discreteness criteria for Zariski-dense subgroups.
result Discreteness criteria for quaternionic hyperbolic subgroups.
The study shows how discrete subgroups' critical exponents relate to their Zariski density in certain groups.
problem Understanding the density of discrete subgroups in semisimple Lie groups.
method Critical exponents and unitary representations.
result Discrete subgroups with critical exponents greater than a certain value are Zariski dense.
Study parametrized Kähler class for cocycles on Hermitian symmetric spaces.
problem Understanding the cohomology of measurable cocycles on Hermitian symmetric spaces.
method Define and analyze parametrized Kähler class to determine cocycles up to cohomology.
result Parametrized Kähler class completely determines the cocycle up to cohomology.
Let G be a simply connected, solvable Lie group and Γ a lattice in G. The deformation space D(Γ,G) is the orbit space associated to the action of $\Aut(G)$ on the space X(Γ,G) of all lattice embeddings of Γ into G. Our main result generalises the classical rigidity theorems of Mal'tsev…
Study trisections on rational elliptic surfaces to find new Zariski pairs.
problem Constructing trisections and related plane curves on rational elliptic surfaces.
method Utilized Mumford representations of semi-reduced divisors to construct trisections and plane curves.
result Existence of a family of Zariski pairs degenerating to the same conic-line arrangement.
Discrete hyperbolic isometries proven via test maps.
problem Proving discreteness of hyperbolic isometries.
method Using test maps to show discreteness of subgroups.
result Zariski dense subgroups are discrete under certain conditions.
The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.
New lattices in higher dimensions have dense surface subgroups.
problem Finding dense subgroups in higher-dimensional arithmetic lattices.
method Exhibited nonuniform arithmetic lattices in SO(n,1).
result Contain Zariski-dense surface subgroups.
Discrete subgroups of quaternionic hyperbolic isometries are proven under certain conditions.
problem Proving discreteness of subgroups of quaternionic hyperbolic isometries.
method Proving discreteness for Zariski dense subgroups under specific conditions involving loxodromic elements and their two-generator subgroups.
result Zariski dense subgroups of mSp(n,1) are discrete under given conditions. Deforms surface groups to be Zariski dense in SL(n,R)
problem Finding Zariski dense surface groups in SL(n,R)
method Deforming K-integral representations of surface groups result Generalizes Long and Thistlethwaite's method to SL(n,R)
Two unique conic-line arrangements with degree 9 are found.
problem Identifying Zariski pairs of conic-line arrangements.
method Using connected numbers to distinguish topologies.
result Found two pairs of conic-line arrangements with a unique conic.
The paper finds dense subgroups in certain Lie groups.
problem Finding dense subgroups in Lie groups.
method Constructing dense surface subgroups in specific Lie groups.
result Uniform lattices contain infinitely many dense Hitchin representations.
The article contains a survey of results on length-commensurable and isospectral locally symmetric spaces and related problems in the theory of semi-simple algebraic groups.
In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose co…
The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.
problem Finding free semigroups with critical exponents arbitrarily close to a subgroup's in dense subgroups of Lie groups.
method Analyzing Zariski dense discrete subgroups of Lie groups, showing the existence of free semigroups with critical exponents arbitrarily close to the subgroup's.
result The existence of free semigroups with critical exponents arbitrarily close to the subgroup's in dense subgroups of Lie groups.
The paper extends character varieties to marked surfaces and discovers their triangular decompositions.
problem Character varieties on marked surfaces and their properties.
method Triangulations of marked surfaces and cohomological groups.
result Stated character varieties admit triangular decompositions.
Survey on minimal rational curves and their geometric structures.
problem Germ-equivalence problem of minimal rational curves on uniruled projective manifolds.
method Analysis of isotrivial families of projective varieties and G-structures.
result Natural G-structure on Zariski-open subset of uniruled projective manifolds.
Paper defines new stability and metrics for complex spaces.
problem Stability and metrics for complex spaces with big cohomology classes.
method Introduces slope stability and Hermitian-Einstein metrics for big cohomology classes.
result Kobayashi Hitchin correspondence and Bogomolov Gieseker inequality proved.
Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.
problem Characterizing dense subgroups of algebraic groups.
method Bi-Lipschitz rigidity theorem for Zariski dense discrete subgroups.
result No C1-smooth slim limit set for higher rank semisimple algebraic groups. Generic Hitchin representations generate dense subgroups.
problem Understanding dense subgroups in SL_n(R) representations.
method Using a theorem by Rapinchuk, Benyash-Krivetz, and Chernousov.
result Generic Hitchin representations are strongly dense.
Maximal representations in symplectic lattices proven for most cases.
problem Understanding maximal representations in symplectic lattices.
method Analyzing mapping class group orbits and continuous deformations of maximal diagonal representations.
result Proof of maximal representations in most lattices of Sp(2n,R).
We study the Abel-Jacobi map for bisections of a certain rational elliptic surface. As an application, we construct examples of Zariski N-plets for conic arrangements.
Paper develops techniques for singular metrics on vector bundles.
problem Developing techniques for singular metrics on vector bundles.
method Introducing non-pluripolar products and defining I-good singularities. result Derives a Chern--Weil type formula for Hermitian vector bundles with I-good singularities. Study TQFT representations for surfaces with boundary, showing irreducibility at prime roots of unity and Zariski density for transcendental parameters.
problem Understanding TQFT representations of mapping class groups with boundary.
method Examined TQFT representations for surfaces with boundary associated with SU(2) gauge group or $U_q(\Sl(2))$ quantum group. result At prime roots of unity, representations are irreducible; for transcendental parameters, the image of mapping class groups is Zariski dense.
Character variety of Whitehead link described in detail.
problem Character variety of Whitehead link.
method Nice description of a Zariski open subset.
result Zariski open subset of the character variety described.