Thurston's spine dimension exceeds virtual cohomological dimension.
problem Understanding the dimension of Thurston's spine in moduli space.
method Analyzing closed hyperbolic surfaces and their systoles.
result The dimension of the set of surfaces is at least \(5-\varepsilon\) for large genus.
New contractible complex shows virtual cohomological dimension of RAAGs.
problem Determining virtual cohomological dimension of RAAGs.
method Equivariant deformation retraction of spine to contractible cube complex.
result Dimension of new complex realises virtual cohomological dimension of RAAGs.
The paper studies embeddings and cohomological dimensions of braid and mapping class groups of surfaces.
problem Understanding embeddings and cohomological dimensions of braid and mapping class groups of surfaces.
method Utilizing the relationship between braid and mapping class groups of a surface and its orientable double covering, the paper generalizes results and computes cohomological dimensions.
result Upper bounds for the virtual cohomological dimension of MCG(N ; k) are given if the genus of N is greater than or equal to 2.
Constructs an explicit cycle in arithmetic group cohomology.
problem Cohomology of SLn(Z) at virtual cohomological dimension. method Geometric rigidity of Voronoi tessellations and abstract framework for polyhedral tessellations.
result Explicit canonical cycle in top-dimensional homology of Voronoi complex.
Let Mod_g be the mapping class group of a genus g >= 2 surface. The group Mod_g has virtual cohomological dimension 4g-5. In this note we use a theorem of Broaddus and the combinatorics of chord diagrams to prove that H^{4g-5}(Mod_g; Q) = 0.
The paper discusses results on Teichmüller modular groups' cohomology.
problem Virtual cohomology dimension of Teichmüller modular groups.
method Use of simply-connectedness of the Hatcher-Thurston complex.
result Full proofs and context discussion of results.
In dimension 3 and above, Bredon cohomology gives an acurate purely algebraic description of the minimal dimension of the classifying space for actions of a group with stabilisers in any given family of subgroups. For some Coxeter groups and the family of virtually cyclic subgroups we show that the Bredon cohomological…
Let G be a group that admits a cocompact classifying space for proper actions X. We derive a formula for the Bredon cohomological dimension for proper actions of G in terms of the relative cohomology with compact support of certain pairs of subcomplexes of X. We use this formula to compute the Bredon cohomologi…
Example groups show outer automorphism groups can have lower cohomological dimension than expected.
problem Outer automorphism groups of certain groups have lower cohomological dimension than expected.
method Constructing specific groups to demonstrate the phenomenon.
result Outer automorphism groups can have cohomological dimension lower than their geometric dimension.
Computes virtually cyclic dimension for 3-manifold groups.
problem Computing the virtually cyclic geometric dimension of 3-manifold groups.
method Prime and JSJ decompositions of M, push-out type constructions, Bredon cohomology computations.
result Explicit computation of virtually cyclic geometric dimension for 3-manifold groups.
The article calculates the minimal model dimensions for classifying spaces of surface braid groups.
problem Classifying spaces for families of virtually abelian subgroups of surface braid groups.
method Analyzes the pure and full braid groups of surfaces with at least one boundary component or one puncture, proving analogous results for amenable subgroups.
result Minimal dimensions of models for classifying spaces are equal to the virtual cohomological dimension plus the rank of virtually abelian subgroups.
Constructs proper affine actions for right-angled Coxeter groups.
problem Proper affine actions for right-angled Coxeter groups.
method Constructs proper actions of right-angled Coxeter groups on O(p,q+1) and its Lie algebra by affine transformations.
result Any virtually special group admits proper affine actions on some R^n.
Lattices in Lie groups have a geometric dimension equal to their cohomological dimension.
problem Understanding the geometric structure of lattices in Lie groups.
method Proved homotopy equivalence to a CW complex.
result Symmetric space dimension equals virtual cohomological dimension of the lattice.
Study on dimensions of mapping class groups of non-orientable surfaces.
problem Determining dimensions of mapping class groups for non-orientable surfaces.
method Analyzing cohomological, geometric, and virtual dimensions.
result Equal dimensions for Ng when geq4,5. Study proves equality of LS-category and cohomological dimension for specific group homomorphisms.
problem Proving equality of LS-category and cohomological dimension for specific group homomorphisms.
method Analyzing epimorphisms and homomorphisms between specific types of almost nilpotent and virtually nilpotent groups.
result Equality of LS-category and cohomological dimension for specified group homomorphisms.
Explicitly found generators of cohomology for SL_n(Z) using sharbly cycles and cosharbly cocycles.
problem Finding explicit generators for the cohomology of SL_n(Z).
method Using sharbly cycles and cosharbly cocycles, and applying Borel-Serre duality.
result Explicitly found generators of H_t(SL_n(Z),St) in terms of sharbly cycles and cosharbly cocycles.
The dualizing module of GL_n(O) varies, affecting cohomology vanishing and nonvanishing.
problem Understanding the dualizing module of GL_n(O) and its impact on cohomology.
method Analyzing the Steinberg module and a variant for GL_n(O), proving vanishing and nonvanishing theorems.
result The dualizing module of GL_n(O) is not always the Steinberg module, but a variant that accounts for orientation.
We show that the mapping class group of a closed surface admits a cocompact classifying space for proper actions of dimension equal to its virtual cohomological dimension.
We show certain symmetry of the dimensions of cohomologies of the funda- mental groups of compact Sasakian manifolds by using the Hodge theory of twisted basic cohomology. As applications, we show that the polycyclic fundamental groups of compact Sasakian manifolds are virtually nilpotent and Sasakian solvmanifolds are…
Study on Čech cohomology of Morse boundaries in hyperbolic manifolds.
problem Computing Čech cohomology groups of Morse boundaries.
method Analyzing cusped hyperbolic n-manifolds and relatively hyperbolic groups.
result Reduced Čech cohomology vanishes in dimensions ≤ n-3 and does not vanish in dimension n-2.
The paper proves a conjecture about braid groups of surfaces and their virtual cohomological dimensions.
problem Investigating the braid groups of surfaces and their homomorphisms.
method Analyzing the inclusion of configuration spaces and induced homomorphisms.
result Proves the conjecture for S^2 and RP^2, determining their virtual cohomological dimensions.
Proves lattice dimensions match in certain groups.
problem Understanding dimensions of lattices in Lie groups.
method Analyzes virtual cohomological and proper geometric dimensions.
result Virtual cohomological dimension equals proper geometric dimension for specified lattices.
The paper proves nonvanishing cohomology for ball quotient fundamental groups.
problem Proving nonvanishing cohomology for ball quotient fundamental groups.
method Using arithmetic lattices and profinite completions, the paper constructs an open subgroup with nontrivial cohomology.
result The virtual cohomological dimension of the fundamental group is at least 2n. Solves a problem related to classifying spaces for proper actions and Nielsen Realization.
problem Whether a cocompact proper topological Γ-manifold is equivariantly homotopy equivalent to the classifying space for proper actions.
method Using Poincaré models and assuming a zero-dimensional singular set, the problem is solved in the Poincaré category.
result New results about Brown's problem are obtained under certain conditions on the underlying group.
We initiate the study of holomorphically convex groups: groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers. If G is a holomorphically convex group of cohomological dimension two, we show that G is isomorphic to the fundamental group …
New result on embedding cohomology of hyperbolic groups.
problem Embedding cohomology of hyperbolic groups into their virtually free subgroups.
method Probabilistic argument and inverse limit of cohomologies.
result Second bounded cohomology of acylindrically hyperbolic groups embeds into cohomologies of virtually free subgroups.
Paper constructs hyperbolic Coxeter groups that virtually fiber over Z.
problem Creating hyperbolic Coxeter groups that virtually fiber.
method Iterative procedure combining Jankiewicz-Norin-Wise results and Osajda's construction.
result Proves construction of groups with increasing vcd with each iteration.
Finite subgroups of good groups correspond to their profinite completions.
problem Characterizing finite subgroups of profinite completions of good groups.
method Proving bijective correspondence between conjugacy classes of finite p-subgroups in G and G^, and analyzing centralizers and normalizers. result Bijective correspondence between conjugacy classes of finite p-subgroups in G and G^. We use virtual neighborhood technique to establish GW-invariants, Quantum cohomology, equivariant GW-invariants, equivariant quantum cohomology and Floer cohomology for general symplectic manifold. We also establish GW-invariants for a family of symplectic manifolds. As a consequence, we prove Arnold conjecture for non…
We show that, if g is more than or equal to 2, the virtual cohomological dimension of the mapping class group of a 3-dimensional handlebody of genus g is equal to 4g-5 and the Euler number of it is equal to 0.
Study cohomology of surfaces with punctures and boundaries, proving bounds on rational cohomology.
problem Understanding cohomology of surfaces with punctures and boundaries.
method Two proofs showing congruence subgroups have enormous rational cohomology.
result Bounds on cohomology are super-exponential in number of punctures and boundary components.
We find a presentation of symplectic Steinberg modules and show vanishing cohomology for certain groups.
problem Cohomology vanishing for specific groups and modules.
method Presented a symplectic Steinberg module and used it to prove cohomology vanishing.
result Cohomology of Sp2n(Z) vanishes in a specific degree for n≥2. Study shows mapping class group dimension for surfaces with punctures.
problem Determining the geometric dimension of mapping class groups of surfaces with punctures.
method Proved cocompact classifying space for proper actions with dimension equal to virtual cohomological dimension.
result Proper geometric dimension of mapping class groups of orientable surfaces with punctures.
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.
The paper proves spaces associated to certain cubical presentations are aspherical.
problem Proving the asphericity of cubical presentations.
method Using coned-off spaces and properties of fundamental groups.
result The fundamental groups of cubical presentations have aspherical associated spaces.
We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virt…
Study conically singular instantons over SU(3)-manifolds, proving existence and dimension formulas.
problem Existence and dimension of moduli spaces of conically singular instantons.
method Develop Fredholm deformation theory, investigate cokernel of instanton operator.
result Formula for virtual dimension of moduli space of conically singular instantons with structure group P(U(n)).
We consider aspherical manifolds with torsion-free virtually polycyclic fundamental groups, constructed by Baues. We prove that if those manifolds are cohomologically symplectic then they are symplectic. As a corollary we show that cohomologically symplectic solvmanifolds are symplectic.
Study growth rates of automorphisms of special groups.
problem Understanding the growth rates of automorphisms of special groups.
method Analyzing outer automorphisms of virtually special groups, showing polynomial or exponential growth, and constructing Nielsen-Thurston decompositions.
result Outer automorphism groups of virtually special groups are boundary amenable, have finite virtual cohomological dimension, and satisfy the Tits alternative.
Lower bounds of betti numbers for homology groups of racks and quandles will be given using the quotient homomorphism to the orbit quandles. Exact sequences relating various types of homology groups are analyzed. Geometric methods of proving non-triviality of cohomology groups are also given, using virtual knots. The r…
We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and …
Extends virtual link theory to arbitrary dimensions.
problem Defining and studying virtual links in higher dimensions.
method Geometric and combinatorial approaches, including diagrams and Gauss codes.
result Many classical link invariants extend to virtual links.
We show that the group cohomology of torsion-free virtually polycyclic groups and the continuous cohomology of simply connected solvable Lie groups can be computed by the rational cohomology of algebraic groups. Our results are generalizations of certian results on the cohomology of solvmanifolds and infra-solvmanifold…
Study on cohomology of SL_n(Z) for n>=3, proving vanishing of certain cohomology groups.
problem Determine the cohomology of SL_n(Z) for n>=3.
method Construct a partial resolution of the Steinberg module to show vanishing of specific cohomology groups.
result Vanishing of codimension-2 rational cohomology group H^{{n \choose 2} -2} for n >= 3.
Study calculates virtually-cyclic dimension for specific mapping class groups.
problem Calculating virtually-cyclic dimension for mapping class groups of specific surfaces.
method Analyzes mapping class groups of punctured spheres with up to six punctures.
result Determines virtually-cyclic dimension for specific surfaces.
Proves graph 3-manifold groups have two specific properties.
problem Understanding fundamental groups of graph 3-manifolds.
method Constructing sequences of covers to prove properties.
result Graph 3-manifold groups are virtually poly-free and in Lex family.
Paper shows mapping class group-equivariant Teichmüller space deformation to Thurston spine.
problem Mapping Teichmüller space to Thurston spine.
method Equivariant deformation retraction of Teichmüller space onto a cell complex.
result Thurston spine contains points corresponding to hyperbolic surfaces with shortest geodesics forming polygons.
We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module St_n(K) is generated by integral apartments if and only if the ideal class group cl(O) is trivial. We deduce this int…