The paper studies Fox pairings of Poincaré duality groups using group cohomology.
problem Understanding Fox pairings of Poincaré duality groups.
method Using group cohomology, the paper computes cohomology groups of Fox pairings.
result The paper suggests fundamental and higher Fox pairings.
Study Coxeter groups over fusion rings and their geometric realisations.
problem Understanding Coxeter groups and their embeddings.
method Investigate faithful realisations and Vinberg systems.
result Induce embeddings of hyperplane complements.
The paper studies complex manifolds using groups Gnk and Γnk.
problem Understanding complex manifolds through algebraic structures.
method Constructing a moduli space and studying its fundamental group mapped to Gnk and Γnk. result The fundamental group of the moduli space is naturally mapped to Gnk and Γnk, advancing the Gnk and Γnk approach. Study of uncountable family of finitely generated groups.
problem Characterize a family of finitely generated residually finite groups.
method Examined groups of the form F2∗HF2 where F2 is a rank-2 free group and H is an infinitely generated subgroup. result Unveiled uncountable family of groups up to isomorphism.
Study of Sasaki groups vs Kähler groups, showing distinct behaviors.
problem Comparing fundamental groups of Sasaki and Kähler manifolds.
method Analysis of properties and behavior of Sasaki groups compared to Kähler groups.
result Sasaki groups exhibit distinct behavior from Kähler groups, not closed under direct products, and have upper bounds on dimension.
Study finite group actions on aspherical manifolds, proving rigidity and symmetry bounds.
problem Understanding actions of finite groups on aspherical manifolds.
method Analyzing the outer automorphism group and homeomorphism group of the fundamental group.
result Proves the homeomorphism group is Jordan and bounds the discrete degree of symmetry.
The paper studies orbifold braid groups and their properties.
problem Understanding orbifold braid groups and their subgroups.
method Detailed study of orbifold braid groups, proving injectivity and triviality of centers.
result Most orbifold braid groups have trivial centers.
Study of Dehn filling quotients in hierarchically hyperbolic groups.
problem Understanding the structure of Dehn filling quotients in specific groups.
method Introduced a construction for cusped spaces of relatively hyperbolic groups and used it to study Dehn-filling-like quotients.
result Infinite hyperbolic quotients of mapping class groups of punctured spheres and braid groups are found.
Study on braid groups' congruence subgroups and their crystallographic quotients.
problem Understanding congruence subgroups and crystallographic quotients of braid groups.
method Investigation of lower central series of congruence braid groups related to B3. result Quotients of congruence braid groups are almost crystallographic.
Finite 2-step nilpotent groups studied by Kim and Manturov.
problem Understanding the structure of groups defined by Kim and Manturov.
method Analyzing the groups Γn4 for all n≥6. result The groups are finite and 2-step nilpotent.
Study subgroup actions on mapping class groups using Heisenberg representations.
problem Untwisting representations of mapping class groups on Heisenberg subgroups.
method Restrict and analyze twisted representations of mapping class groups to Heisenberg subgroups.
result Untwisting representations on Torelli group for any Heisenberg representation.
Study of Torelli subgroups of automorphism groups of free groups.
problem Understanding the structure of Torelli subgroups of Out(Fn). method Adapting techniques from 3-manifolds and surface mapping class groups.
result These groups are finitely generated and satisfy a Birman exact sequence.
We study the fundamental groups of compact Sasakian manifolds, which we call Sasaki groups. It is shown that all known Ka¨hler groups are Sasaki, in particular, all finite groups are Sasaki. On the other hand, we show there exists many restrictions on the fundamental groups of compact Sasakian manifolds. We als…
The paper studies actions on Bass-Serre trees and identifies new C∗-simple groups.
problem Investigating actions of fundamental groups on Bass-Serre trees and their C∗-algebraic properties. method Analyzing boundary actions of fundamental groups of graphs of groups on their Bass-Serre trees.
result Identification of new families of C∗-simple groups, including tubular groups and certain graphs of groups. Introduces halo products and studies their geometric properties.
problem Understanding the large-scale geometry of halo groups.
method Introduces halo products and builds a geometric framework.
result Provides refined invariants distinguishing halo groups up to quasi-isometry.
The Baumslag-Solitar groups: BS(m,n)=<x,y| x y^{m} x^{-1} = y^{n}> are some of the simplest interesting infinite groups which are not lattices in Lie groups. They have been studied in depth from the point of view of combinatorial group theory. It is natural to ask if the geometric approach to the theory of infinite gro…
Characterizes a general range decreasing group homomorphism.
problem Understanding range decreasing group homomorphisms in the entire mapping group.
method Characterization of a general range decreasing group homomorphism.
result Computes a particular class of homomorphisms and identifies all range decreasing group homomorphisms on specific mapping groups.
Study Kähler groups from orbifold compactifications of curve moduli.
problem Shafarevich conjecture on holomorphic convexity.
method Quantum representations of orbifold compactifications and mapping class groups.
result Construct interesting Kähler groups and settle most conjectures.
Study on finiteness properties of handlebody mapping class groups.
problem Understanding finiteness properties of asymptotically rigid handlebody groups.
method Introduced asymptotically rigid mapping class groups and determined their finiteness properties based on the space of ends of handlebodies.
result Homology of these groups coincides with stable homology of handlebody groups in some cases.
Study homology groups of mapping and Torelli groups for surfaces with abelian covers.
problem Understanding the homology of mapping and Torelli groups for surfaces with specific topologies.
method Examined the first homology group of mapping and Torelli groups with coefficients in the first rational homology group of the universal abelian cover of the surface.
result For surfaces with one boundary component, the twisted homology groups are finite-dimensional, but for surfaces with one puncture, they are infinite-dimensional.
Study on totally symmetric sets with group applications.
problem Understanding totally symmetric sets and their group applications.
method Survey of existing theory and applications to various groups.
result Exploration of totally symmetric sets in multiple group contexts.
Study geometric actions of groups on horocyclic products.
problem Understanding geometric actions of groups on horocyclic products.
method Analyzing geometric actions of groups on horocyclic products of CAT(-κ) spaces.
result Groups acting on horocyclic products are either ascending HNN extensions of finitely-generated virtually nilpotent groups or not finitely presented.
Study left invariant Lorentzian metrics on 3D non-unimodular Lie groups.
problem Classify and analyze left invariant Lorentzian metrics on 3D non-unimodular Lie groups.
method Classify metrics up to automorphism, study curvature functions.
result Obtain Ricci operator, scalar curvature, and sectional curvatures as functions of metrics.
Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.
problem Understanding the stabilizers of complex hyperbolic triangle groups.
method Explicit generators and signatures of stabilizers computed for each group orbit of mirrors.
result Explicit generators and signatures of stabilizers for some triangle groups.
We study properties of a generalization of the Mahler measure to elements in group rings, in terms of the Lueck-Fuglede-Kadison determinant. Our main focus is the variation of the Mahler measure when the base group is changed. In particular, we study how to obtain the Mahler measure over an infinite group as limit of M…
Study on curvature in finitely generated groups, showing positive curvature in specific cases.
problem Understanding curvature in finitely generated groups.
method Analyzing dead-end elements and related elements to find curvature, studying effect of radius.
result Examples of positive curvature for arbitrary radius in lamplighter and Houghton's group.
Study of hyperelliptic mapping class groups with applications and profinite completions.
problem Understanding hyperelliptic mapping class groups and their properties.
method Defined and studied hyperelliptic mapping class groups, applied theory to counterexamples, and examined profinite completions.
result Found a counterexample to a conjecture about mapping class groups and extended congruence subgroup property.
Unified study of homological representations of mapping class groups.
problem Fundamental relationships between different homological representations.
method Analysis of twisted homology of configuration spaces.
result Unified account of non-degenerate pairings, embeddings, and isomorphisms.
The study shows acylindrical hyperbolicity for Artin groups not associated with joins or cones.
problem Proving acylindrical hyperbolicity for Artin groups of infinite type not associated with joins or cones.
method Developing and extending the clique-cube complex and action studies of Charney and Morris-Wright.
result Acylindrical hyperbolicity demonstrated for Artin groups of infinite type associated with graphs that are not cones.
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.
The study lists low-dimensional stratified groups and their properties.
problem Understanding the algebraic structure of stratified groups.
method Explicitly provided a list of low-dimensional stratified groups and their properties.
result All stratified groups in dimensions up to 7 and some free-nilpotent groups in dimensions up to 14 were studied.
Study Veech groups in fibered 3-manifolds, proving no parabolics for fibers.
problem Characterizing Veech groups in fibered 3-manifolds.
method Analyzing pseudo-Anosov monodromies and foliations, proving properties of Veech groups.
result Veech groups in fibers typically contain no parabolic elements.
Study covers of Chamanara surface with large Veech groups.
problem Characterize finite abelian covers with large Veech groups.
method Analyze Veech groups of finite abelian covers of the Chamanara surface.
result Every free group arises as the projective Veech group of a finite-area infinite translation surface.
Study deformation spaces of irregular isomonodromy systems on Riemann surfaces.
problem Understanding the topology of irregular isomonodromy systems.
method Define and study moduli spaces of deformations of irregular classes on Riemann surfaces.
result Generalize G-braid groups to study fundamental groups of deformation spaces.
Study of group orderability using tensor and exterior squares.
problem Circular orderability of groups and torsion elements.
method Analysis of tensor and exterior squares of groups.
result Circularly orderable groups have left-orderable tensor and exterior squares under certain conditions.
Study of pseudo-Riemannian manifolds with M{ö}bius group actions.
problem Characterizing pseudo-Riemannian manifolds with specific group actions.
method Analyzing the action of the conformal group on compact manifolds.
result If the non-compact semi-simple part of the conformal group is M{ö}bius, the manifold is conformally flat.
Study of rank 2 Kleinian groups with specific parabolic elements.
problem Characterizing discrete representations of rank 2 free groups into PSL(2,C). method Analyzing representations sending three disjoint curves to parabolics.
result Only maximal cusp groups of infinite covolume, and new finite covolume groups.
Study of groups acting on complex projective varieties.
problem Classifying groups of birational transformations on complex projective varieties.
method Free, properly discontinuous, cocompact action on open sets of complex projective varieties.
result Classification in dimension two.
Study on virtual singular braid groups with algebraic properties and homomorphisms.
problem Algebraic properties and homomorphisms of virtual singular braid groups.
method Numerical invariants, homomorphisms, semi-direct product decompositions, presentations, and quotients.
result Determined all group homomorphisms from VSGn to Sn and obtained corresponding semi-direct product decompositions. Study shows groups can act on torus without extending to 3-manifold.
problem Bordism problem for group actions on torus.
method Examples of groups acting on torus by diffeomorphisms isotopic to identity.
result Groups cannot be extended to action on bounding 3-manifold.
The hyperelliptic mapping class group has been studied in various contexts within topology and algebraic geometry. What makes this study tractable is that there is a surjective map from the hyperelliptic mapping class group to a mapping class group of a punctured sphere. The more general family of superelliptic mapping…
Study of generalized J-groups and their presentations.
problem Understanding the structure of generalized J-groups and their presentations.
method Determine finitely generated groups, classify up to reflection isomorphism, and derive explicit presentations.
result Generalized J-groups coincide with rank 2 complex reflection groups and their torsion quotients.
Study of modular representations in homology of congruence subgroups.
problem Understanding modular representations in homology of congruence subgroups.
method Analysis of sequences of modular representations of symplectic and special linear groups over finite fields.
result Established periodic representation stability in the sense of Church--Farb.
Study local control in a 7D quaternionic Heisenberg group.
problem Optimizing geodesics in a 7D quaternionic Heisenberg group.
method Matrix representation and analysis of sub-Riemannian structure symmetries.
result Impact of symmetries on geodesic optimality.
Study of orbifold mapping class groups via arc and curve actions.
problem Understanding the structure of orbifold mapping class groups.
method Defined orbifold mapping class groups and studied their actions on arcs and curves. Established a Birman exact sequence and derived finite presentations.
result Finite presentations of orbifold mapping class groups established.
Study numerical invariants for groups, computing for cyclic groups and surfaces.
problem Numerical invariants for groups and their computation.
method Computational and theoretical analysis of groups, including finite cyclic groups and nonorientable surfaces.
result Formula for the numerical invariant of free products of groups.
The paper studies twisted Alexander polynomials for knot groups in various extensions.
problem Understanding twisted Alexander polynomials in knot groups for different extensions.
method Developed mod p formula for twisted Alexander polynomials and studied central extensions.
result Established formulas for twisted Alexander polynomials in knot groups for various extensions.
Study finds all isometries for specific Lie groups.
problem Identifying isometry groups in nonunimodular Lie groups.
method Examined left-invariant Riemannian metrics on Lie groups of dimension four.
result Determined full group of isometries for each metric.