The paper proves rigidity for complex Kleinian groups.
problem Characterizing hyperconvex subgroups of complex Kleinian groups.
method Analyzing critical exponents and using representations of PSL(2, C).
result Uniform lattices in PSL(2, C) are the only (d-k)-hyperconvex subgroups with a specific critical exponent.
Study on hyperconvex representations of hyperbolic groups in complex flag manifolds.
problem Characterizing hyperconvex representations of hyperbolic groups.
method Geometric and topological analysis of representations in mPSL(d,C). result Virtual isomorphism to Kleinian groups and flag manifold Hausdorff dimension restriction.
Study on hyperconvex representations of surface groups and their geometric properties.
problem Understanding the geometry of hyperconvex representations of surface groups.
method Holomorphic extension of Ahlfors--Bers map and analysis of limit sets.
result Limit set has Hausdorff dimension 1 if and only if representation is in PSL(d,R).
We show a very general existence theorem to the complex Monge-Ampère type equation on hyperconvex domains.
Collar lemma proven for certain surface group representations.
problem Proving a collar lemma for specific surface group representations.
method Using partial hyperconvexity properties and Anosov representations.
result 'Positivity properties' hold for partially hyperconvex representations.
In this paper, we establish Basmajian's identity for (1,1,2)-hyperconvex Anosov representations from a free group into PGL(n,R). We then study our series identities on holomorphic families of Cantor non-conformal repellers associated to complex (1,1,2)-hyperconvex Anosov representations. We show that the series …
Study extends Hausdorff dimension Hessian results to new hyperconvex representations.
problem Extending classical results on Hausdorff dimension Hessian.
method Analyzes (1,1,2)-hyperconvex representations and small complex deformations.
result Positive definiteness of Hessian of Hausdorff dimension for co-compact Γ in PO(n,1).
In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that t…
Solves complex Monge-Ampère equation for measures with pluripolar parts.
problem Characterizing measures with complex Monge-Ampère equation solutions.
method Solves for measures with a pluripolar part in compact Kähler manifolds.
result Generalizes classical results in bounded hyperconvex domains.
We prove uniqueness of solutions to complex Monge-Ampère equations for small temperature.
problem Proving uniqueness of solutions to complex Monge-Ampère equations.
method Local and global analysis of bounded hyperconvex domains and compact complex manifolds.
result Uniqueness of solutions confirmed for small temperature parameters.
Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
Establishes a lower bound for Kähler hyperbolicity modulus in hyperconvex domains and bounded strongly pseudoconvex domains.
problem Kähler hyperbolicity modulus for simply-connected Kähler hyperbolic manifolds
method Computes the Kähler hyperbolicity modulus for bounded symmetric domains
result Establishes a lower bound for the Kähler hyperbolicity modulus in terms of the boundary behavior of the gradient length of a plurisubharmonic function
Inspired by the concept of hyperconvexity and its relation to curvature, we translate geometric properties of a metric space encoded by the curvature inequalities into the persistent homology induced by the Čech filtration of that space.
The paper connects Bergman-Calabi diastasis to Kähler metrics with constant holomorphic sectional curvature.
problem Characterizing domains with Bergman metrics of constant holomorphic sectional curvature.
method Using the Bergman-Calabi diastasis and its connection to the Bergman representative coordinate, the paper derives explicit formulas and proves properties of these metrics.
result Domains with Bergman metrics of constant holomorphic sectional curvature are hyperconvex or exhaustively biholomorphic to a ball.
Study confirms geodesic connectivity and rooftop envelopes in complex Monge-Ampère equation domains.
problem Confirming geodesic connectivity and rooftop envelopes in complex Monge-Ampère equation domains.
method Examined geodesics and plurisubharmonic envelopes within the Cegrell classes on bounded hyperconvex domains.
result Affirmative answer to a longstanding open question about geodesic connectivity and rooftop envelopes.
We show that the notion of 3-hyperconvexity on oriented flag manifolds defines a partial cyclic order. Using the notion of interval given by this partial cyclic order, we construct Schottky groups and show that they correspond to images of positive representations in the sense of Fock and Goncharov. We construct poly…
Establishes a lower bound for Kähler-Einstein distance on certain domains.
problem Finding a lower bound for Kähler-Einstein distance on specific types of domains.
method Proves an analog of the Hopf lemma for Riemannian manifolds with Ricci curvature bounded from below.
result Establishes a lower bound for the Kähler-Einstein distance on pseudoconvex domains with positive hyperconvexity index.
Maximal and Borel Anosov representations in Sp(4,R) are proven to be Hitchin.
problem Characterizing representations of surface groups into Sp(4,R) that are Borel Anosov and maximal. method Proving representations are Hitchin if they have maximal Toledo invariant and are Borel Anosov.
result Maximal and Borel Anosov representations in Sp(4,R) are Hitchin. In this paper, we show that the extremal length functions on Teichmüller space are log-plurisubharmonic. As a corollary, we obtain an alternative proof of L.Liu and W.Su's results on the plurisubharmonicity of extremal length functions. We also obtain alternative proofs of S.Krushkal's results that a function defined b…
In this paper, we compute the subgroup distortion of all finitely generated subgroups of all finitely generated 3-manifold groups, and the subgroup distortion in this case can only be linear, quadratic, exponential and double exponential. It turns out that the subgroup distortion of a subgroup of a 3-manifold group is …
Regular subgroups of SL3(R) are identified and ruled out.
problem Identifying and characterizing regular subgroups of SL3(R).
method Using Kapovich–Leeb–Porti and Guichard–Wienhard divergent subgroups criteria, and Oh's results.
result Regular subgroups of SL3(R) are precisely lattices in minimal horospherical subgroups.
For a bounded domain D and a real number p>0, we denote by Ap(D) the space of Lp integrable holomorphic functions on D, equipped with the Lp- pseudonorm. We prove that two bounded hyperconvex domains $D_1\subset \mc^n$ and $D_2\subset \mc^m$ are biholomorphic (in particular n=m) if there is a linear is…
Study on braid group quotients by congruence subgroups.
problem Understanding the image of congruence subgroups in GL(n,Z).
method Characterization through symplectic congruence subgroups.
result Open problem solved: image of congruence subgroups in GL(n,Z).
The paper explores geometric finiteness in mapping class groups and constructs new examples of these subgroups.
problem Understanding geometric finiteness in mapping class groups and constructing new examples.
method Examined several constructions of subgroups and determined conditions for geometric finiteness.
result Provides new examples of parabolically geometrically finite and reducibly geometrically finite subgroups.
Proposes a new method for finding non-redundant, standout subgroups in numeric datasets.
problem Mining large numbers of redundant subgroups in numeric datasets.
method Dispersion-aware problem formulation based on MDL principle for subgroup set discovery.
result Empirically demonstrates SSD++ returns outstanding subgroup lists.
Proves Congruence Subgroup Property for two types of groups.
problem Proving Congruence Subgroup Property for specific groups.
method Elementary proof of Johnson filtration and geometric subsurface inclusions.
result Proves Congruence Subgroup Property for nilpotent quotients and subsurface subgroups.
New method constructs non-quasiconvex subgroups in hyperbolic groups.
problem Creating non-quasiconvex subgroups in hyperbolic groups.
method Using Stallings-like techniques on right-angled Coxeter groups (RACGs).
result Explicit examples of non-quasiconvex subgroups constructed.
Characterizes knotted subgroups of Lie groups and provides examples.
problem Defining and understanding knotted subgroups of Lie groups.
method Geometric equivalence, one-parameter subgroups, infinitesimal elements, canonical forms, spectrum analysis.
result Completely classified knotted subgroups of SL(2,R) and SL(3,R).
Study subgroups of pro-p PD^3 groups, finding specific conditions.
problem Characterize subgroups of pro-p PD^3 groups. method Analyzes properties of subnormal and finitely presented subgroups.
result Conditions on subgroups of pro-p PD^3 groups. No hyperbolic group can have an infinite chain of free subgroups of fixed rank.
problem Infinite ascending chains of free subgroups in hyperbolic groups.
method Proof by contradiction and properties of hyperbolic groups.
result Hyperbolic groups do not contain strictly ascending chains of free quasiconvex subgroups of constant rank.
Robust subgroup discovery finds non-redundant, statistically significant subgroups.
problem Finding interpretable, robust subgroups from data.
method Formulated subgroup lists for univariate and multivariate targets, used MDL principle and greedy heuristic SSD++.
result SSD++ outperforms previous methods in quality and size of subgroup lists.
Sparse GFA identifies disease factors in FTD subgroups.
problem Heterogeneity in neurological disorders hinders understanding and treatment.
method Sparse Group Factor Analysis (GFA) with regularised horseshoe priors.
result Identified latent disease factors differentially expressed in FTD subgroups.
Let N be at least 4. We prove that every injective homomorphism from the Torelli subgroup into Out(FN) differs from the inclusion by a conjugation in Out(FN). This applies more generally to the following subgroups: every finite-index subgroup of Out(FN) (recovering a theorem of Farb and Handel); every subgro…
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.
For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor defined stability and proved stability is equivalent to convex cocompactness in mapping class …
A new algorithm COVA-FC improves subgroup-fair clustering efficiency.
problem Challenges in making cluster assignments independent of sensitive attributes in subgroups.
method Defining a subgroup-fairness gap, deriving a covariance-based surrogate, and introducing a continuous relaxation for efficient optimization.
result COVA-FC achieves competitive cost-fairness trade-offs and improves computational efficiency.
We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-end…
Study answers arithmeticity question for normal subgroup of lattices.
problem Arithmeticity of discrete subgroups of semisimple Lie groups with dense commensurators.
method Examined normal subgroups of lattices in semisimple Lie groups.
result Positive answer to Greenberg-Shalom's question for lattices.
We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin-Tits groups of spherical type a result that was previously known for bra…
Characterizes groups arising as fixed subgroups of RAAG automorphisms.
problem Identifying groups that can be fixed by finite-order automorphisms of RAAGs.
method Geometric characterisation using divisible cube complexes.
result Surface groups and commutator subgroups of RAAGs are fixed subgroups.
Researchers determine the rational abelianization of a subgroup of mapping class groups.
problem Understanding the structure of the Chillingworth subgroup of mapping class groups.
method Using Johnson homomorphism and Casson-Morita homomorphism, they compute the abelianization and order of related Euler classes.
result They find the rational abelianization of the Chillingworth subgroup as a full mapping class group module.
The paper disproves the existence of certain subgroups with nontrivial rational abelianization.
problem The existence of finite-index subgroups with nontrivial rational abelianization in handlebody groups.
method Proved that meridian multitwists vanish in H1(Γ;Q) and showed that H1(Γ;Q)=0 for specific subgroups. result No finite-index subgroups of the handlebody group contain nontrivial rational abelianization.
Enhances stability ranges for Torelli and congruence subgroup homologies.
problem Improving stability ranges for specific subgroup homologies.
method Analyzes H2(Torelli subgroup of Aut(Fn)'s), H2(Torelli subgroup of mapping class groups), and Hk(congruence subgroups of GL_n(R)'s).
result Improved central stability ranges for various subgroup homologies.
Example found of subgroup not a lattice in product of Lie groups
problem Finding irreducible discrete subgroups that are not lattices
method Produced an example in SL(2,R)imesSL(2,R) result Example of subgroup not a lattice in product of Lie groups
Study confined subgroups in groups with contracting elements, showing their growth rate is strictly greater than half of the ambient growth rate.
problem Understanding the growth rate of confined subgroups in groups with contracting elements.
method Through boundary actions, analyzing the Hopf decomposition and quotient growth.
result Confined subgroups have a growth rate strictly greater than half of the ambient growth rate.
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. New right-angled Artin subgroups found in Artin groups.
problem Finding large right-angled Artin subgroups in Artin groups.
method Examining centers of irreducible spherical special subgroups and their powers.
result Conjecture verified for certain classes of Artin groups, leading to hyperbolic surface subgroup conclusions.
Study extends Kobayashi's method to non-reductive subgroups for homogeneous spaces.
problem Existence of compact Clifford-Klein forms in homogeneous spaces.
method Extend Kobayashi's method to non-reductive subgroups and compare Cartan projections and non-compact dimensions.
result Examples of homogeneous spaces without compact Clifford-Klein forms.