Study unipotent group actions on Kähler manifolds using moment maps.
problem Analyzing unipotent group actions on compact Kähler manifolds.
method Moment map techniques, stability conditions, symplectic reduction.
result Geometric quotients with compactifiable Kähler structures.
Constructs CAT(0) actions for certain groups without unipotent elements.
problem Understanding actions of certain groups on CAT(0) spaces.
method Constructs an isometric action of a group on a CAT(0) space.
result Fundamental groups of certain 3-manifolds do not admit faithful finite-dimensional unitary representations.
Reconfigures Milnor invariants using unipotent Magnus embeddings.
problem Computing and understanding Milnor invariants of links.
method Central group extensions and unipotent Magnus embeddings.
result First non-vanishing invariants and refinements of higher degree invariants computed.
Study of groups generated by unipotent maps in complex hyperbolic space.
problem Characterizing subgroups of PU(2,1) generated by unipotent maps.
method Provided coordinates and analyzed group actions on complex hyperbolic space.
result Disc Z parametrizes a family of discrete groups, proving Schwartz conjecture. New group preserves convex shape without repeating patterns.
problem No virtually abelian unipotent group preserves strictly convex domains.
method Examined Heisenberg group acting on convex shape.
result First example of non-virtually abelian unipotent group preserving strictly convex domain.
Effective Lie-Kolchin theorem for quasi-unipotent matrices with common eigenvector.
problem Establishing conditions for quasi-unipotent matrices to have a common eigenvector.
method Using the Jordan Canonical Form and properties of quasi-unipotent matrices.
result Quasi-unipotent matrices A and B have a common eigenvector under certain conditions. Linear groups without infinite order unipotents have good properties.
problem Properties of linear groups without unipotent elements of infinite order.
method Analyzing the structure and properties of linear groups without unipotent elements of infinite order.
result Linear groups without unipotent elements of infinite order have good properties, including centralisers virtually splitting and finitely generated abelian subgroups being undistorted.
Let M be a non-elementary convex cocompact hyperbolic 3 manifold and delta the critical exponent of its fundamental group. We prove that a one-dimensional unipotent flow for the frame bundle of M is ergodic for the Burger-Roblin measure provided that delta>1.
New CR representations are found and shown to be redundant.
problem Identifying and classifying CR representations of 3-manifolds.
method Experimental computation of limit sets and exact computations of triangle groups.
result Many CR representations are redundant and conjugate.
We study a natural map from representations of a free (resp. free abelian) group of rank g in GL_r(C), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (resp. complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor…
Study formalities on closed surfaces using connections.
problem Formalities of Goldman-Turaev Lie bialgebra on closed surfaces.
method Reformulated Kashiwara-Vergne groups and associators in higher genera using non-commutative connections.
result Determined pro-unipotent automorphism group of associated graded.
The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Gamma a closed subgroup …
For a compact 3-manifold M with arbitrary (possibly empty) boundary, we give a parametrization of the set of conjugacy classes of boundary-unipotent representations of the fundamental group of M into SL(n,C). Our parametrization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmueller spaces d…
We consider the discrete representations of 3-manifold groups into PU(2,1) that appear in the Falbel-Koseleff-Rouillier census, such that the peripheral subgroups have cyclic unipotent holonomy. We show that two of these representations have conjugate images, even though they represent different 3-manifold groups. Th…
Researchers provide coordinates for Lie group configurations, linking to Teichmüller space and knot representations.
problem Finding coordinates for Lie group configurations.
method Explicit Fock-Goncharov coordinates for triples and quadruples of affine flags in rank 2 Lie groups.
result Explicit coordinates on higher Teichmüller space and boundary-unipotent representations of 3-manifold groups.
We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot comp…
Study counts and parametrizes flag components in SO0(p,q) space.
problem Counting and characterizing flag components in SO0(p,q) space.
method Parametrization and computation of Plücker coordinates.
result Anosov subgroups are virtually isomorphic to surface or free groups.
The paper concerns a compactification of the isospectral varieties of nilpotent Toda lattices for real split simple Lie algebras. The compactification is obtained by taking the closure of unipotent group orbits in the flag manifolds. The unipotent group orbits are called the Peterson varieties and can be used in the co…
A graph connects Specht and web bases; matrix is unipotent with vanishing entries.
problem Comparing two bases of irreducible representations of the symmetric group.
method Graph theory and combinatorial analysis to describe relations between bases and prove properties of the transition matrix.
result The transition matrix between Specht and web bases is unipotent with additional vanishing entries.
Study Anosov representations of reducible suspensions of hyperbolic groups.
problem Characterize dynamical properties of reducible suspensions of Anosov representations.
method Analyzing linear representations of non-elementary hyperbolic groups, focusing on weak unipotent actions on subspaces.
result Characterize when reducible suspensions are discrete and faithful, quasi-isometrically embedded, and Anosov.
Establishes geometric properties of elements in the positive semigroup of a general real semisimple Lie group.
problem Generalizing Lusztig's total positivity to the setting of general real semisimple Lie groups.
method Classifying Lie groups admitting a positive structure and establishing key properties of unipotent positive semigroups.
result Establishes key geometric properties of elements in the positive semigroup.
Symplectic manifolds derived from Torelli groups, showing complex structure invariants without holomorphic structures.
problem Deriving symplectic structures from Torelli groups without holomorphic structures.
method Symplectization of mapping tori derived from Torelli group representatives.
result Symplectic manifolds with complex structure invariants but no holomorphic structures.
Proves polynomial error rate for equidistribution of unipotent flows.
problem Equidistribution of orbits of unipotent subgroups in arithmetic quotients.
method Uses Margulis function, incidence geometry tools, and spectral gap.
result Polynomial error rate for equidistribution theorems.
Investigates linearity of group amalgams and new examples of non-linear groups.
problem Linearity of group amalgams and examples of non-linear groups.
method Investigates linearity of amalgams of subgroups of algebraic groups.
result Establishes linearity of certain 'doubles' of linear groups and finds new non-linear examples.
Unipotent automorphisms on complex supermanifolds are studied for classification.
problem Classifying supermanifolds based on unipotent automorphisms.
method Analyzing even global degree increasing vector fields and their kernels.
result Classification of strictly t-nildominated supermanifolds up to degree t. The paper extends McShane identities to higher Teichmüller theory and convex real projective surfaces.
problem Deriving McShane identities for higher Teichmüller theory.
method By studying mapping class group invariant functions and generalizing horocycle lengths.
result Established McShane-type identities for various surface group representations.
In this note, we propose an approach to the study of the analogue for unipotent harmonic bundles of Schmid's Nilpotent Orbit Theorem. Using this approach, we construct harmonic metrics on unipotent bundles over quasi-compact Kähler manifolds with carefully controlled asymptotics near the compactifying divisor; such a m…
The paper studies groups formed by two parabolic maps and their properties.
problem Understanding groups generated by two parabolic maps in mSU(2,1). method Analyzes conditions for the group to be discrete and free, and calculates the diameter of a circle in the Heisenberg group.
result Conditions are provided to ensure the group is discrete and free.
New topological proof for a theorem about orbits in a specific group.
problem Characterizing orbits of dense subgroups in a specific group.
method Topological approach, using dynamics of unipotent flows.
result Orbits are either finite or dense, proving a theorem.
Classifies specific types of Lorentzian manifolds with unipotent holonomy.
problem Classifying closed conformally flat Lorentzian manifolds with unipotent holonomy.
method Analyzing the geometric properties and developing maps of the manifolds.
result Identifies four geometric types and two additional types of manifolds.
The paper proves orbit closure theorems for hyperbolic manifolds with Fuchsian ends.
problem Analyzing the orbit closures of unipotent flows on hyperbolic manifolds with specific ends.
method Establishing an analogue of Ratner's orbit closure theorem for unipotent subgroups in SO(d,1).
result Proves the closure of k-horospheres and (k+1)-geodesic planes are properly immersed submanifolds. Let X be a projective manifold, and D be a normal crossing divisor of X. By Jost-Zuo's theorem that if we have a reductive representation ρ of the fundamental group π1(X∗) with unipotent local monodromy, where X∗=X−D, then there exists a tame pluriharmonic metric h on the flat bundle V a…
This paper proves that there are no compact forms for a large class of homogeneous spaces admitting actions by higher-rank semisimple Lie groups. It builds on Zimmer's approach for studying such spaces using cocycle superrigidity. The proof involves cocycle superrigidity, measure rigidity for unipotent flows, technique…
New Ptolemy variety detects all SL(2,C) representations, independent of triangulation.
problem Finding all SL(2,C) representations in a 3-manifold, independent of triangulation.
method Defining a new Ptolemy variety that is independent of triangulation and detects all boundary-unipotent irreducible representations.
result The new Ptolemy variety detects all boundary-unipotent irreducible SL(2,C)-representations, solving the problem of detecting all representations independent of triangulation.
Polynomial error equidistribution for SL2 groups.
problem Equidistribution of orbits in arithmetic quotients.
method Margulis function, incidence geometry, spectral gap.
result Polynomial error rate for equidistribution.
In this paper we prove that finite index subgroups of genus 3 mapping class and Torelli groups that contain the group generated by Dehn twists on bounding simple closed curves are not Kahler. These results are deduced from explicit presentations of the unipotent (aka, Malcev) completion of genus 3 Torelli groups and of…
Refines mixed Hodge structures using non-abelian Hodge theory.
problem Mixed Hodge structures on Sullivan's minimal models.
method Non-abelian Hodge theory.
result Explicit representatives of real unipotent variations.
Let G be a simply connected solvable Lie group with a lattice Γ and N the nilradical of G. For a complex valued representation ρ:G→GL(Vρ) such that the restriction ρ∣N is unipotent, as an advanced variation of cohomology computation of solvmanifolds by using Lie algebra cohomology, we construct a…
We study the eta invariants of Dirac operators and the regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner M"uller and use relative traces to define these spectral invariants. We show the regularity of eta and zeta functions at s=0. The Selberg trace formula and the detai…
The Ptolemy coordinates for boundary-unipotent SL(n,C)-representations of a 3-manifold group were introduced in Garoufalidis-Thurston-Zickert inspired by the A-coordinates on higher Teichmüller space due to Fock and Goncharov. In this paper, we define the Ptolemy field of a (generic) PSL(2,\C)-representation and prove …
Let D be an irreducible lattice in a connected, semisimple Lie group G with finite center. Assume that the real rank of G is at least two, that G/D is not compact, and that G has more than one noncompact simple factor. We show that D has no orientation-preserving actions on the real line. (In algebraic terms, this mean…
Study on string links invariant under associator choice and Grothendieck--Teichmüller group action.
problem Independence of Kontsevich invariant under associator choice for 2-component string links.
method Investigation of Kontsevich invariant for 2-component string links and action of Grothendieck--Teichmüller group.
result Non-trivial action of Grothendieck--Teichmüller group on algebra of 2-component string links.
Study the asymptotics of holomorphic maps from polydisk to disk.
problem Understanding the asymptotics of holomorphic maps from polydisk to disk.
method Analyzing the conjugation action of a unipotent subgroup of PSL2(R).
result Results in the asymptotics of the translation flow on holomorphic maps.
Study balancing embedding of degenerating Abelian varieties.
problem Balancing embedding of degenerating Abelian varieties.
method Maximal unipotent family over punctured disc, base change, theta functions, Gromov-Hausdorff limit.
result Balanced embedding of degeneration into projective space.
New findings on cusped Borel Anosov representations and their properties.
problem Characterizing and understanding cusped Borel Anosov representations.
method Analyzing representations of lattices in PGL2(R) to PGLd(R). result Cusped Borel Anosov representations with specific properties are Hitchin representations.
We study the possibility of applying a finite-dimensionality argument in order to address parts of the Baum-Connes conjecture for finitely generated linear groups. This gives an alternative approach to the results of Guentner, Higson, and Weinberger concerning the Baum-Connes conjecture for linear groups. For any finit…
New groups discovered with unique properties in a specific space.
problem Finding new discrete subgroups with special properties in a mathematical space.
method Proved by showing groups play ping-pong on cones, related to crooked surfaces.
result Infinite family of discrete subgroups with remarkable properties in Sp4(R). We prove that the first complex homology of the Johnson subgroup of the Torelli group Tg is a non-trivial unipotent Tg-module for all g≥4 and give an explicit presentation of it as a $\Sym H_1(T_g,\C)$-module when g≥6. We do this by proving that, for a finitely generated group G satisfying an assumpti…