New non-arithmetic ball quotients from elliptic curves on Abelian surfaces.
problem Constructing non-arithmetic ball quotients from Abelian surfaces.
method Branched covers of Abelian surface quotients by finite groups.
result Alternative construction of lattices from elliptic curves.
New abelian quotient found in symplectic derivation Lie algebra.
problem Understanding the structure of symplectic derivation Lie algebra.
method Computational approach to abelianization of the weight 12 part.
result 1-dimensional weight 12 part for g≥8. Study non-existence of complex ball quotients in Torelli locus.
problem Non-existence of totally geodesic complex ball quotients in Torelli locus.
method Analytic techniques.
result Analytic techniques used to study non-existence.
We study an integration theory in circle equivariant cohomology in order to prove a theorem relating the cohomology ring of a hyperkahler quotient to the cohomology ring of the quotient by a maximal abelian subgroup, analogous to a theorem of Martin for symplectic quotients. We discuss applications of this theorem to q…
The paper studies orbifold splice quotients and log covers of surface pairs.
problem Understanding orbifold splice quotients and log covers of surface pairs.
method Analyzes orbifold homology and constructs pairs with universal abelian log covers.
result Computes orbifold homology from resolutions and constructs orbifold splice quotients.
Classifies complex symplectic structures on Lie algebras with large abelian ideals.
problem Classifying complex symplectic structures on Lie algebras with large abelian ideals.
method Two constructions of complex symplectic structures on Lie algebras with large abelian ideals, considering compact quotients of Lie groups.
result Complete classification of complex symplectic structures on almost abelian Lie algebras.
Construct special Lagrangian fibrations on abelian varieties using retraction techniques.
problem Constructing special Lagrangian fibrations on abelian varieties.
method Explicit construction using special techniques in non-Archimedean geometry.
result Solved a conjecture of Kontsevich-Soibelman for finite quotients of abelian varieties.
This work shows non-abelian quotient groups in string link concordance.
problem Understanding the structure of string link concordance groups.
method Analyzing the quotient groups formed by string link concordance and pure braid groups.
result The quotient of each string link concordance group by its pure braid subgroup is non-abelian.
A homology cylinder over a compact manifold is a homology cobordism between two copies of the manifold together with a boundary parametrization. We study abelian quotients of the homology cobordism group of homology cylinders. For homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim that their …
Study non-arithmetic orbifold ball quotients using Jacobian of Bolza curve.
problem Identify and study non-arithmetic orbifold ball quotients.
method Use Jacobian of Bolza curve and birational transformations.
result Obtain orbifold ball quotient surfaces with interesting configurations.
Algorithm distinguishes Fuchsian groups with finite quotients.
problem Distinguishing between non-isomorphic Fuchsian groups.
method Develops an algorithm to create group extensions using finite quotients.
result Establishes an upperbound for the order of a distinguishing finite quotient.
It is shown that every abelian regular Lie group is a quotient of its Lie algebra via the exponential mapping.
It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "s…
A homology cylinder over a surface consists of a homology cobordism between two copies of the surface and markings of its boundary. The set of isomorphism classes of homology cylinders over a fixed surface has a natural monoid structure and it is known that this monoid can be seen as an enlargement of the mapping class…
The paper solves the conjugacy problem in a specific braid group quotient and finds infinite virtually cyclic subgroups.
problem Solving the conjugacy problem in the Artin braid group quotient Bn/[Pn,Pn]. method Using systems of equations over the integers derived from the action of Bn/[Pn,Pn] on the abelianization of Pn/[Pn,Pn]. Also, explicitly realizing infinite virtually cyclic subgroups. result Explicitly realized infinite virtually cyclic subgroups in Bn/[Pn,Pn]. The paper characterizes unit balls among Stein spaces with specific groups using Bergman-Einstein metrics.
problem Characterizing unit balls among Stein spaces with specific groups.
method Study of Bergman metric on finite ball quotients and its Kähler-Einstein property.
result The Bergman-Einstein metric exists only on the unit ball itself for finite ball quotients with trivial groups.
Complex projective manifolds without rational curves are quotients of Abelian varieties.
problem Characterizing complex projective manifolds without rational curves.
method Using conjectures about rational and entire curves on Calabi-Yau varieties.
result Non-hyperbolic complex projective manifolds contain the image of an Abelian variety.
Study locally conformally balanced metrics on specific Lie algebras.
problem Characterize and classify locally conformally balanced metrics on almost abelian Lie algebras.
method Characterizations and classifications based on specific properties of Lie algebras.
result Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.
Holomorphic curves found in compact quotients of SL(2,C).
problem Proving the existence of holomorphic curves in compact quotients of SL(2,C).
method Non-Abelian Hodge correspondence, WKB analysis, and Morgan-Shalen compactification.
result Every compact quotient of SL(2,C) contains a holomorphic curve of genus at least two.
Let K be a compact Lie group. We compute the abelianization of the Lie algebra of equivariant vector fields on a smooth K-manifold X. We also compute the abelianization of the Lie algebra of strata preserving smooth vector fields on the quotient X/K.
Quantum Kirwan maps between K-theories of G-varieties and GIT quotients.
problem Constructing maps between K-theories of G-varieties and their GIT quotients.
method Formal construction of maps in quantum K-theory, using equivariant and non-equivariant quantum K-theory.
result Presentation of quantum K-theory for smooth proper toric DM stacks.
Criterion for projectivisation on klt spaces, characterizing quotients and stability.
problem Characterizing finite quotients of projective spaces and Abelian varieties.
method Criterion based on reflexive sheaves and stability conditions.
result Characterization of finite quotients using Q-Chern class inequalities and stability condition. Study of quotient groups of mapping class groups by power subgroups.
problem Characterizing quotient groups of mapping class groups by power subgroups.
method Analyzing quotient groups Modgn/Modgn[p] for different values of p and n. result Found infinite normal subgroups and Kähler subgroups in specific quotient groups.
The note confirms a conjecture for specific Lie groups.
problem The conjecture about constant holomorphic sectional curvature in non-Kähler geometry.
method Compact quotients of Lie groups with specific properties.
result The conjecture is confirmed for almost abelian Lie algebras and those with certain abelian ideals.
We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space R^{4d} by the action of a compact Abelian group. These 4n-dimensional quotients carry a multi-Hamilitonian action of an n-torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration o…
In this paper we extend the results of Kirwan et alii on convexity properties of the moment map for Hamiltonian group actions, and on the connectedness of the fibers of the moment map, to the case of non-compact orbifolds. Our motivation is twofold. First, the category of orbifolds is important in symplectic geometry b…
The paper shows commensurators of certain subgroups are discrete.
problem Understanding commensurators of specific subgroups in Lie groups.
method Analyzing commensurators of thin normal subgroups with abelian quotients.
result The commensurator of certain subgroups is discrete.
Complexity of periodic graphs linked to Mahler measure.
problem Understanding growth rates of graph complexity.
method Defining graph complexity and linking it to Mahler measure.
result Lehmer's question about polynomial roots is equivalent to complexity growth of certain graphs.
New groups from surface braids help create complex geometric shapes.
problem Creating new geometric shapes from surface braids.
method Using finite quotients of surface braid groups to construct double Kodaira fibrations.
result New geometric shapes (double Kodaira fibrations) created using these groups.
New homomorphisms from homology cylinders to torsion modules via LMO functor.
problem Understanding the abelian quotients of homology cylinders.
method Constructing homomorphisms via LMO functor and analyzing their properties.
result Abelianization of Johnson kernel has torsion elements.
New criterion for complex hyperbolicity of quotient spaces.
problem Understanding complex hyperbolicity in quotient spaces of bounded symmetric domains.
method Developed a new criterion for complex hyperbolicity and used it to determine subvarieties of general type.
result Quotients X with subvarieties V of dimension p ≥ 1 are of general type if a specific condition is met.
Locally conformal SKT structures are introduced and studied on Lie groups and their compact quotients.
problem Existence and classification of LCSKT structures on Lie groups and their quotients.
method Introducing LCSKT structures and studying their properties on Lie groups and their quotients.
result Existence of non-trivial LCSKT structures on 6-dimensional nilpotent Lie algebras and almost abelian Lie algebras.
The paper classifies invariant structures on complex almost Abelian groups.
problem Investigating invariant geometric structures on almost Abelian Lie groups.
method Explicit formulas for Haar measures, modular function, and generator fields were derived.
result All invariant tensor fields have constant coefficients in the invariant frame.
Algebraic dimension is zero for generic complex structures on hypercomplex nilmanifolds.
problem Understanding the algebraic dimension of complex subvarieties in hypercomplex nilmanifolds.
method Analyzing the algebraic dimension of complex subvarieties of hypercomplex nilmanifolds using properties of hypercomplex structures and Lie algebras.
result For generic complex structures, the algebraic dimension of complex subvarieties in hypercomplex nilmanifolds is zero.
We construct lattices on six dimensional not completely solvable almost abelian Lie groups, for which the Mostow condition does not hold. For the corresponding compact quotients, we compute the de Rham cohomology (which does not agree in general with the Lie algebra one) and a minimal model. We show that some of these …
Study on curves on specific arithmetic quotients of hyperbolic 2-ball.
problem No complex curves of certain genus on these arithmetic quotients.
method Volume estimates and understanding special subvarieties.
result For large discriminants, no complex curves of fixed genus.
Let G be a complex reductive algebraic group (not necessarily connected), let K be a maximal compact subgroup, and let A be a finitely generated Abelian group. We prove that the conjugation orbit space Hom(A,K)/K is a strong deformation retract of the GIT quotient space Hom(A,G)//G. As a corollary, we determine necessa…
We establish a criterion for when an abelian extension of infinite-dimensional Lie algebras integrates to a corresponding Lie group extension G^ of G by A, where G is a connected, simply connected Lie group and A is a quotient of its Lie algebra by some discrete subgroup. When G is non-simply connected…
The set SL(n) of n-string links has a monoid structure, given by the stacking product. When considered up to concordance, SL(n) becomes a group, which is known to be abelian only if n=1. In this paper, we consider two families of equivalence relations which endow SL(n) with a group structure, namely the C_k-equivalence…
Let M be a symplectic manifold, equipped with a Hamiltonian action of a torus T. We give an explicit formula for the rational cohomology ring of the symplectic quotient M//T in terms of the cohomology ring of M and fixed point data. Under some restrictions, our formulas apply to integral cohomology. In certain …
Study of trigonal curves in abelian differentials with specific divisor properties.
problem Characterizing locally closed subspaces of abelian differentials.
method Using linear systems on Segre-Hirzebruch surfaces to describe orbifold structure and orbifold fundamental groups.
result Identified the orbifold fundamental group of a specific subspace as a quotient of the Artin group of type E8. The paper explores p-Kähler structures on Lie group quotients.
problem Existence of p-Kähler structures on compact quotients of Lie groups. method Analyzes p-Kähler structures on nilmanifolds and almost abelian solvmanifolds. result Proves that (n−2)-Kähler almost abelian solvmanifolds of complex dimension n≥3 are Kähler. In this paper we apply the hyper-Kähler quotient construction to Lie groups with a left invariant hyper-Kähler structure under the action of a closed abelian subgroup by left multiplication. This is motivated by the fact that some known hyper-Kähler metrics can be recovered in this way by considering different Lie grou…
Unified algebraic framework for virtual braid structures with strong structural consequences.
problem Unified algebraic framework for virtual braid structures with various types of crossings.
method Introducing the universal virtual braid group UVn(c) and proving its properties. result Strong structural consequences including residual finiteness, linearity, and solvability of conjugacy problems.
We study the word length entropy of automorphisms of residually nilpotent groups, and how the entropy of such group automorphisms relates to the entropy of induced automorphisms on various nilpotent quotients. We show that much like the structure of a nilpotent group is dictated to a large degree by its abelianization,…
Study Weyl-Einstein structures on conformal solvmanifolds, proving Einstein property and classifying metrics.
problem Characterize Weyl-Einstein structures on conformal solvmanifolds.
method Analyzing left-invariant metrics and using conformal Lie group structures.
result Every conformal solvmanifold with Weyl-Einstein structure is Einstein.
The study characterizes subgroups of braid groups and their finite quotients.
problem Understanding the relationship between braid groups and their congruence subgroups.
method Characterization and computation of finite quotients of braid groups.
result Explicit generators and free subgroups are found for certain braid groups.
Develops algorithm for finite generating set of liftable mapping class groups of regular abelian covers.
problem Finding finite generating sets for liftable mapping class groups of regular abelian covers.
method Algorithm based on a result providing generating sets for groups acting on graphs with finite quotients.
result Provides finite generating sets for LModp(Sg) for various regular abelian covers.