arXiv research

This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M^n with hyperbolic fundamental group is a retract of …

Study minimal volume entropy for free-by-cyclic groups and 2D right-angled Artin groups.

Study on simplicial volume and Euler characteristic of aspherical manifolds.

Study simplicial volume in fiber bundles with connected groups.

Integral foliated simplicial volume is zero for certain amenable covers.

The simplicial volume of oriented closed connected smooth manifolds that admit a non-trivial smooth $S^1$-action vanishes. In the present work we prove a version of this result for the integral foliated simplicial volume of aspherical manifolds: The integral foliated simplicial volume of aspherical oriented closed conn…

The "polyhedral product functor" produces a space from a simplicial complex L and a collection of pairs of spaces, {(A(i),B(i))}, where i ranges over the vertex set of L. We give necessary and sufficient conditions for the resulting space to be aspherical. There are two similar constructions, each of which starts with …

We study random 2-dimensional complexes in the Linial - Meshulam model and find torsion in their fundamental groups at various regimes. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be aspherical; this implies that any aspherical subcomplex of a random 2-complex satisfies…

We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. This provides some further evidence for the veracity of the Auslander Conjecture. Along the way, we provide a simple cohomological criterion fo…

We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated s…

We provide a proof for an inequality between volume and L2-Betti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau's theory of L2-Betti numbers of R-simplicial complexes, and other …

Study simplicial volume via foliated simplices and duality.

New results on relative simplicial volume using bounded acyclicity.

We prove that the locally finite simplicial volume and the Lipschitz simplicial volume are additive with respect to certain gluings of manifolds. In particular, we prove that in dimension $\geq 3$ they are additive with respect to connected sums and gluings along $π_1$-injective, amenable aspherical boundary components…

Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical m…

Study shows simplicial volume is superadditive under specific conditions.

Constructs 4-manifolds with positive Euler characteristic proving a conjecture.

Let the complexity of a closed manifold M be the minimal number of simplices in a triangulation of M. Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the infimum on all finite coverings of M normalized by the covering degree we can promote it to a multiplicative invariant, a…

We show that the aspherical manifolds produced via the relative strict hyperbolization of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automor…

We define a new class of irreducible groups, called groups not infinite-index presentable by products or not IIPP. We prove that certain aspherical manifolds with fundamental groups not IIPP do not admit maps of non-zero degree from direct products. This extends previous results of Kotschick and Loeh, providing new cla…

Paper confirms Whitehead's conjecture for aspherical 2-complexes.

Study simplicial volume of manifolds from reflection group trick.

A topological version of a longstanding conjecture of H. Hopf, originally proposed by W. Thurston, states that the sign of the Euler characteristic of a closed aspherical manifold of dimension $d=2m$ depends only on the parity of $m$. Gromov defined several hyperbolization functors which produce an aspherical manifold …

Mapping class groups of Haken 3-manifolds enjoy many of the homological finiteness properties of mapping class groups of 2-manifolds of finite type. For example, H(M) has a torsionfree subgroup of finite index, which is geometrically finite (i. e. is the fundamental group of a finite aspherical complex). This was prove…

The paper proves the Singer conjecture for aspherical complex surfaces and refines Gromov's inequality.

Study new bounds on TC of spaces with subgroup inclusions.

Study shows some complex shapes don't fit a certain property.

The paper proves macroscopic versions of conjectures about scalar curvature and volume bounds.

The paper describes the structure of injective LOT-complexes and proves they are aspherical.

In this paper we define, for each aspherical orientable 3-manifold $M$ endowed with a \emph{torus splitting} $Ţ$, a 2-dimensional fundamental $l_1$-class $[M]^{Ţ}$ whose $l_1$-norm has similar properties as the Gromov simplicial volume of $M$ (additivity under torus splittings and isometry under finite covering maps). …

Diagrammatic reducibility DR and its generalization vertex asphericity VA are combinatorial tools developed for detecting asphericity of a 2-complex. Here we present tests for a relative version of VA that apply to pairs of 2-complexes $(L,K)$, where $K$ is a subcomplex of $L$. We show that a relative weight test holds…

The study shows that certain cubical presentations lead to aspherical spaces.

Researchers propose a new approach to the Hopf problem for aspherical varieties.

Proves conjecture on graph configuration spaces' complexity.

Proposes a method to learn representations of higher-dimensional simplicial complexes.

Study symplectically aspherical Kähler manifolds with unique properties.

Alexander's conjecture extended to infinite simplicial complexes.

The study examines conditions for minimal volume entropy of simplicial complexes.

Mixes higher-order simplicial complexes for data augmentation.

Let $Γ$ be a Coxeter graph, let $(W,S)$ be its associated Coxeter system, and let $(A,Σ$) be its associated Artin-Tits system. We regard $W$ as a reflection group acting on a real vector space $V$. Let $I$ be the Tits cone, and let $E_Γ$ be the complement in $I +iV$ of the reflecting hyperplanes. Recall that Charney, D…

An additional minimal simplicial n-complex contains a non-splittable link in R^(2n).

Constructs simplified or complexified simplicial complexes.

A notion of up and down Grover walks on simplicial complexes are proposed and their properties are investigated. These are abstract Szegedy walks, which is a special kind of unitary operators on a Hilbert space. The operators introduced in the present paper are usual Grover walks on graphs defined by using combinatoria…

We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle-Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to …

Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.

We describe the second homotopy group of any CW-complex $K$ by analyzing the universal cover of a locally finite model of $K$ using the notion of $G$-coloring of a partially ordered set. As applications we prove a generalization of the Hurewicz theorem, which relates the homotopy and homology of non-necessarily simply-…

The paper improves bounds on topological complexity for certain manifolds.

The study constructs geometrically decomposable aspherical 4-manifolds with non-zero signature and explores their properties.