arXiv research

Thurston's spine dimension exceeds virtual cohomological dimension.

New contractible complex shows virtual cohomological dimension of RAAGs.

Constructs an explicit cycle in arithmetic group cohomology.

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.

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…

In this paper, we make use of the relations between the braid and mapping class groups of a compact, connected, non-orientable surface N without boundary and those of its orientable double covering S to study embeddings of these groups and their (virtual) cohomological dimensions. We first generalise results of Birman …

The article calculates the minimal model dimensions for classifying spaces of surface braid groups.

Study on dimensions of mapping class groups of non-orientable surfaces.

Study proves equality of LS-category and cohomological dimension for specific group homomorphisms.

Let G be the fundamental group of a connected, closed, orientable 3-manifold. We explicitly compute its virtually cyclic geometric dimension. Among the tools we use are the prime and JSJ decompositions of M, several push-out type constructions, as well as some Bredon cohomology computations.

Explicitly found generators of cohomology for SL_n(Z) using sharbly cycles and cosharbly cocycles.

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.

For a number ring $\mathcal{O}$, Borel and Serre proved that $\text{SL}_n(\mathcal{O})$ is a virtual duality group whose dualizing module is the Steinberg module. They also proved that $\text{GL}_n(\mathcal{O})$ is a virtual duality group. In contrast to $\text{SL}_n(\mathcal{O})$, we prove that the dualizing module of…

The paper is devoted to a detailed exposition of results outlined in author's 1983 note "On the virtual cohomology dimension of the Teichmüller modular group". The paper includes the full proofs and a discussion of the context which motivated both the results and the methods used. The reminiscences included into the la…

We prove that if $Γ$ is a lattice in the group of isometries of a symmetric space of non-compact type without euclidean factors, then the virtual cohomological dimension of $Γ$ equals its proper geometric dimension.

For any right-angled Coxeter group $Γ$ on $k$ generators, we construct proper actions of $Γ$ on $\mathrm{O}(p,q+1)$ by right and left multiplication, and on the Lie algebra $\mathfrak{o}(p,q+1)$ by affine transformations, for some $p,q\in\mathbb N$ with $p+q+1=k$. As a consequence, any virtually special group admits pr…

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…

We prove that if $Γ$ is a lattice in a classical simple Lie group $G$, then the symmetric space of $G$ is $Γ$-equivariantly homotopy equivalent to a proper cocompact $Γ$-CW complex of dimension the virtual cohomological dimension of $Γ$.

Study on Čech cohomology of Morse boundaries in hyperbolic manifolds.

The geometric dimension for proper actions $\underline{\mathrm{gd}}(G)$ of a group $G$ is the minimal dimension of a classifying space for proper actions $\underline{E}G$. We construct for every integer $r\geq 1$, an example of a virtually torsion-free Gromov-hyperbolic group $G$ such that for every group $Γ$ which con…

The paper proves nonvanishing cohomology for ball quotient fundamental groups.

Solves a problem related to classifying spaces for proper actions and Nielsen Realization.

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 …

Paper constructs hyperbolic Coxeter groups that virtually fiber over Z.

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…

Finite subgroups of good groups correspond to their profinite completions.

Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, ${\rm Out}(F_n)$, \dots) embeds via the natural restriction maps into the inverse limit of the secon…

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.

We find a presentation of symplectic Steinberg modules and show vanishing cohomology for certain groups.

Study shows mapping class group dimension for surfaces with punctures.

Unified algebraic framework for virtual braid structures with strong structural consequences.

The paper proves spaces associated to certain cubical presentations are aspherical.

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…

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 conically singular instantons over SU(3)-manifolds, proving existence and dimension formulas.

Study growth rates of automorphisms of special groups.

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 …

Let M be a surface, perhaps with boundary, and either compact, or with a finite number of points removed from the interior of the surface. We consider the inclusion i: F\_n(M) --\textgreater{} M^n of the nth configuration space F\_n(M) of M into the n-fold Cartesian product of M, as well as the induced homomorphism i\_…

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.

We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also define generalizations of virtual link diagrams and Gauss codes to represent virtua…

Proves graph 3-manifold groups have two specific properties.

Paper shows mapping class group-equivariant Teichmüller space deformation to Thurston spine.

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…

The automorphisms of free groups with boundaries form a family of groups A_{n,k} closely related to mapping class groups, with the standard automorphisms of free groups as A_{n,0} and (essentially) the symmetric automorphisms of free groups as A_{0,k}. We construct a contractible space L_{n,k} on which A_{n,k} acts wit…