arXiv research

The paper studies actions on Bass-Serre trees and identifies new $C^*$-simple groups.

The study counts ideal points in 2-bridge knot complements using knot diagrams.

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…

New groups prevent certain geometric actions on spaces.

We prove some general results about quasi-actions on trees and define Property (QFA), which is analogous to Serre's Property (FA), but in the coarse setting. This property is shown to hold for a class of groups, including $SL(n,\Z)$ for $n\geq 3$. We also give a way of thinking about Property (QFA) by breaking it down …

We show that for groups acting acylindrically on simplicial trees the $K$- and $L$-theoretic Farrell-Jones Conjecture relative to the family of subgroups consisting of virtually cyclic subgroups and all subconjugates of vertex stabilisers holds. As an application, for amalgamated free products acting acylindrically on …

The paper studies acylindrical actions on trees and proves acylindrical hyperbolicity of Baumslag-Solitar groups.

Constructs Serre spectral sequence for bounded cohomology.

Study links using Soergel bimodules and Serre duality.

Extends topological groupoids and studies their properties.

Polynomial maps are shown to be Serre fibrations under specific conditions.

The paper proves a Serre-Swan Theorem for coisotropic algebras.

Describes the relationship between two spectral sequences and their joint refinement.

This is an account of the theory of JSJ decompositions of finitely generated groups, as developed in the last twenty years or so. We give a simple general definition of JSJ decompositions (or rather of their Bass-Serre trees), as maximal universally elliptic trees. In general, there is no preferred JSJ decomposition, a…

In geometric group theory one uses group actions on spaces to gain information about groups. One natural space to use is the Cayley graph of a group. The Cayley graph arguments that one encounters tend to require local finiteness, and hence finite generation of the group. In this paper, I take the theory of intersectio…

Extends Serre-Swan theorem to all finitely generated modules over smooth functions.

Proves a vanishing property for symplectic manifold cohomology.

We prove that the full twist is a Serre functor in the homotopy category of type A Soergel bimodules. As a consequence, we relate the top and bottom Hochschild degrees in Khovanov-Rozansky homology, categorifying a theorem of Kálmán.

The Serre-Swan theorem provides the link between projective modules of finite rank and vector bundles over compact manifolds, and plays a prominent role in non-commutative geometry. Its extension to non-compact manifolds is discussed.

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. An extension of the Whitney's theorem to the case when all fibers are homeomorphic to some fixed compact two-dimensional manifold was proved by the…

In this paper, we generalize the notion of Serre fibration to the Morita category of topological groupoids and derive the associated long exact sequence of homotopy groups. We use this results for calculation of homotopy groups of various groupoids, such as the foliation groupoid of a Riemannian foliation.

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. Results of this paper extend Whitney theorem to the case when all fibers are homeomorphic to a given compact two-dimensional manifold.

New topology shows Morse boundaries are topologically invariant.

Study on twisted Dolbeault cohomology in Kähler foliations.

Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact étale Lie groupoid $M$, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on $M$. This result can be seen as an …

The abstract discusses a spectral sequence for Lie algebroids.

Study non-vanishing $\ell^2$-Betti numbers for specific groups.

We prove properties of the Schweitzer complex and its cohomologies.

We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper CAT(0) spaces, elements acting hyperbolically on the Bass-Serr…

Generalizes jet differential bounds and proves asymptotic Serre duality.

Develops differential K-theory for noncommutative algebras.

We give an example of a subgroup of SL(2,C) which is a strictly ascending HNN extension of a non-abelian finitely generated free group F. In particular, we exhibit a free group F in SL(2,C) of rank 6 which is conjugate to a proper subgroup of itself. This answers positively a question of Drutu and Sapir. The main ingre…

Extends equivariant contact structure results to mod p L-spaces.

The present paper is a short survey on the mathematical basics of Classical Field Theory including the Serre-Swan' theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spin^C-structures, Dirac operators, exterior algebra bundles and Connes' differential algebras in the commutative case…

We study the behaviour of analytic torsion under smooth fibrations. Namely, let F \to E \to^{f} B be a smooth fiber bundle of connected closed oriented smooth manifolds and let $V$ be a flat vector bundle over $E$. Assume that $E$ and $B$ come with Riemannian metrics and $V$ comes with a unimodular (not necessarily fla…

Reduces Lie (bi-)algebroids and Dirac manifolds using constraint vector bundles.

In this paper, we study strongly quasiconvex subgroups in a finitely generated $3$--manifold group $π_1(M)$. We prove that if $M$ is a compact, orientable $3$--manifold that does not have a summand supporting the Sol geometry in its sphere-disc decomposition then a finitely generated subgroup $H \le π_1(M)$ has finite …

The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In t…

The Serre-Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. I…

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

The paper certifies projective rigidity for once-punctured torus bundles using twisted Alexander polynomials.

We completely describe the finitely generated pro-$p$ subgroups of the profinite completion of the fundamental group of an arbitrary $3$-manifold. We also prove a pro-$p$ analogue of the main theorem of Bass--Serre theory for finitely generated pro-$p$ groups.

This paper introduces persistent equivariant cohomology and applies it to circle actions.

For a transversal pair of closed Lagrangian submanifolds L, L' of a symplectic manifold M so that $π_{1}(L)=π_{1}(L')=0=c_{1}|_{π_{2}(M)}=ω|_{π_{2}(M)}$ and a generic almost complex structure J we construct an invariant with a high homotopical content which consists in the pages of order $\geq 2$ of a spectral sequence…

The thesis shows how automorphisms of hyperbolic groups can be represented by train track maps.

The paper studies the space of Gauss maps of complete minimal surfaces and their homotopy types.

Let X be a locally symmetric space associated to a reductive algebraic group G defined over Q. L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of X; they were introduced in [math.RT/0112251]. That paper also introduced the micro-support of an L-module, a com…

The Dixmier-Douady class connects homeomorphisms and foliations.