arXiv research

The paper studies Fox pairings of Poincaré duality groups using group cohomology.

The paper shows plentiful non-homotopy finite Poincaré duality spaces.

New proof of surface group theorem for 2D Poincaré duality groups.

We describe classes of potential structures (covector fields) on Minkowski space that admit subgroups of the Poincaré group. We describe also seven classes of Maxwell spaces that admit subgroups of the Poincaré group.

A combination of Bestvina--Brady Morse theory and an acyclic reflection group trick produces a torsion-free finitely presented Q-Poincaré duality group which is not the fundamental group of an aspherical closed ANR Q-homology manifold. The acyclic construction suggests asking which Q-Poincaré duality groups act freely …

In this paper, we show that if G is a finite p-group (p prime) acting by automorphisms on a $δ$-hyperbolic Poincare Duality group, then the fixed subgroup is a Poincare Duality group over Z/p. We also provide examples to show that the fixed subgroup might not even be a Duality group over Z.

The paper removes a condition for the Nielsen realisation problem using equivariant Poincaré duality.

Proves 3D Poincaré duality groups without property (T)

The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.

Researchers describe and compare decompositions of Poincaré duality pairs.

Growth functions of Coxeter groups and the Poincare series of Kleinian and Fuchsian singularities are $2$-tangle $q$-fractions.

We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n-1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action of G on X d…

Study subgroups of pro-$p$ PD^3 groups, finding specific conditions.

Equivalence proven between equivariant K-theory and K-homology for certain matrix group actions.

Reduces field theories using Poisson-Poincaré method.

We recall the construction of non-formal deformation quantization of the Poincare Group ISO(1,1) on its coadjoint orbit and exhibit the associated non-formal star-exponentials.

Explains the Borromean rings, icosahedron, and Poincaré homology sphere.

New spaces help connect manifold structures on equivariant Poincaré spaces.

In this note, we fill in a gap in the literature by proving that the Teichmueller modular groups (mapping class groups) are not Poincare duality groups and the complexes of curves of surfaces have infinite homotopy type (i.e. are not homotopy equivalent to a finite CW-complex).

We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.

In this paper we address the relation between the orbifold fundamental group and the topology of the underlying space. In particular, under the assumption that the orbifold fundamental group is equal to the fundamental group of the underlying space, we prove Poincaré Duality for orbifolds of dimension 4 and 5.

For G an almost-connected Lie group, we study G-equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G-invariant Riemannian metrics of positive scalar curvature. We prove a rigidity result for almost-complex manifolds, generalising Hattori's result…

We study the cohomology properties of the singular foliation $\F$ determined by an action $Φ\colon G \times M\to M$ where the abelian Lie group $G$ preserves a riemannian metric on the compact manifold $M$. More precisely, we prove that the basic intersection cohomology $\lau{\IH}{*}{\per{p}}{\mf}$ is finite dimensiona…

This paper addresses questions of quasi-isometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the Bass-Serre tree of the graph of groups has finite depth. The main example of a finite depth graph of groups is one whose vertex and edge groups are coarse Poinca…

Uniform Poincaré inequalities established for various metric spaces.

4-manifolds with specific groups have unique homotopy types.

For Poincare series of binary polyhedral groups and Coxeter polynomials there are obtained statements close to the Euclid algorithm and orthogonal polynomials theory: generalized Ebeling formula, decompositions into ramified continued fractions, Christoffel-Darboux identity, combinatorial formula. Known results about t…

We give a group cohomological description of the Čech cohomology of the Bowditch boundary of a relatively hyperbolic group pair, generalizing a result of Bestvina-Mess about hyperbolic groups. In case of a relatively hyperbolic Poincaré duality group pair, we show the Bowditch boundary is a homology manifold. For a thr…

We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincaré profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini--Schramm--Timár. In this paper we focus on properties of the Poincaré profiles of groups with …

Develops parametrised Poincaré duality for equivariant fixed points.

Introduces Epstein-Poincaré surfaces for G-oper, generalizing classical construction.

Study shows Bergman kernels match averages on quotient spaces, proving non-vanishing of Poincaré series.

Profinite rigidity studied for algebraic fibring of groups.

PD_3-groups split as HNN extensions, revealing homology class properties.

Souriau studies Gibbs states for symplectic manifolds with group actions.

Extends Euler class result to symplectic group.

The authors previously described an algebraic analogue of the JSJ-decomposition of a 3-manifold. This analogue is defined for any finitely presented, one-ended group. We study this analogue in the special case of Poincaré duality pairs.

Homotopy classification for certain 4-manifolds with dihedral fundamental groups.

On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concrete realizations of the cohomology groups decompose into orthogonal subspaces corresponding to cohomology coming from the interior and bound…

Classifies mapping tori of specific groups, generalizing known results.

For any compact and connected Lie group $G$ and any free abelian or free nilpotent group $Γ$ , we determine the cohomology of the path component of the trivial representation of the representation space (character variety) $Rep(Γ,G)_1$, with coefficients in a field $F$ with ${char} (F)$ either 0 or relatively prime to …

Study vector fields on non-compact manifolds with group action.

Two quasi-morphisms on disk symplectomorphisms linked to Poincaré's translation number.

Poincaré's Polyhedron Theorem is a widely known valuable tool in constructing manifolds endowed with a prescribed geometric structure. It is one of the few criteria providing discreteness of groups of isometries. This work contains a version of Poincaré's Polyhedron Theorem that is applicable to constructing fibre bund…

Poincar{é} and Sobolev inequalities for differential forms on Heisenberg balls, involving Rumin's differentials, are given. Furthermore, a global homotopy of Rumin's complex which improves differentiability of Rumin forms is provided on any bounded geometry contact manifold.

In this paper, we prove interior Poincar{é} and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L 1 norm. Unlike for L p , p > 1, the estimates are doomed to fail in top degree. The singular integ…

We prove that the basic intersection cohomology $IH^*_{\overline{p}}(M / \mathcal{F})$, where $\mathcal{F}$ is the singular foliation determined by an isometric action of a Lie group $G$ on the compact manifold $M$, verifies the Poincaré Duality Property.

Develops higher-order Euler-Poincaré field equations for principal G-bundles.