arXiv research

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…

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 show that non-elliptic prime 3-manifolds satisfy integral approximation for the simplicial volume, i.e., that their simplicial volume equals the stable integral simplicial volume. The proof makes use of integral foliated simplicial volume and tools from ergodic theory.

Formula connects foliated simplicial volume with group cost.

Integral foliated simplicial volume is zero for certain amenable covers.

Integral filling volume of mapping tori grows sublinearly with complexity.

Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal …

Study integral simplicial volume of cyclic covers of torus bundles.

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…

New length functions on mapping class groups linked to simplicial volumes of mapping tori.

We show that integral foliated simplicial volume of closed manifolds gives an upper bound for the cost of the corresponding fundamental groups.

Study shows complete affine manifolds have zero simplicial volume.

We show that the integral foliated simplicial volume of a connected compact oriented smooth manifold with a regular foliation by circles vanishes.

We observe that stable integral simplicial volume of closed manifolds gives an upper bound for the rank gradient of the corresponding fundamental groups.

Graph manifolds are manifolds that decompose along tori into pieces with a tame $S^1$-structure. In this paper, we prove that the simplicial volume of graph manifolds (which is known to be zero) can be approximated by integral simplicial volumes of their finite coverings. This gives a uniform proof of the vanishing of …

Study of $p$-adic simplicial volumes and their properties.

We define the ideal simplicial volume for compact manifolds with boundary. Roughly speaking, the ideal simplicial volume of a manifold $M$ measures the minimal size of possibly ideal triangulations of $M$ "with real coefficients", thus providing a variation of the ordinary simplicial volume defined by Gromov in 1982, t…

The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact lo…

Study simplicial volume in fiber bundles with connected groups.

We provide sharp lower bounds for the simplicial volume of compact $3$-manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of $3$-manifolds, including handlebodies and products of surfaces with the interval. Our results provide the firs…

Minimal volume entropy vanishes or is positive under certain fiber growth conditions.

Positive simplicial volume implies locally symmetric space structure.

We prove that cubical simplicial volume of oriented closed 3-manifolds is equal to one fifth of ordinary simplicial volume.

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

The simplicial volume of non-R^3 contractible 3-manifolds is infinite.

Study shows simplicial volume of certain fiber bundles is zero.

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively c…

Let $M$ be the interior of a connected, oriented, compact manifold $V$ of dimension at least 2. If each path component of $\partial V$ has amenable fundamental group, then we prove that the simplicial volume of $M$ is equal to the relative simplicial volume of $V$ and also to the geometric (Lipschitz) simplicial volume…

The paper proves simplicial volume positivity for certain nonpositively curved 4-manifolds with nonzero Euler characteristic.

Study shows range of simplicial volumes for open manifolds.

The study shows simplicial volume finiteness for certain manifolds with amenable fundamental groups.

Study simplicial volume for fixed fundamental groups, finding gaps.

We show that there exist closed manifolds with arbitrarily small transcendental simplicial volumes. Moreover, we exhibit an explicit uncountable family of (transcendental) real numbers that are not realised as the simplicial volume of a closed manifold.

Study simplicial volume of manifolds from reflection group trick.

Positive simplicial volume found for certain non-positively curved manifolds with specific submanifolds.

Study essentiality and simplicial volume of manifolds fibered over spheres.

Alternative proof of simplicial volume bound using area-minimizing sets.

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for Q-rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the…

Study shows simplicial volume is superadditive under specific conditions.

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…

Simplicial volume vanishes for 4-manifolds with open book decompositions.

Study on simplicial volume and Euler characteristic of aspherical manifolds.

For primes $p$, we investigate an $\mathbb{F}_p$-version of simplicial volume and compare these invariants with their siblings over other coefficient rings. We will also consider the associated gradient invariants, obtained by stabilisation along finite coverings. Throughout, we will discuss the relation between such s…

Paper proves Gromov's conjecture on manifolds with certain group properties.

In this paper, it is shown that for any closed orientable $3$-manifold with positive simplicial volume, the growth of the Seifert volume of its finite covers is faster than the linear rate. In particular, each closed orientable $3$-manifold with positive simplicial volume has virtually positive Seifert volume. The resu…

We show that compact, locally symmetric spaces of non-compact type have positive simplicial volume. This gives a positive answer to a question that was first raised by Gromov in 1982. We provide a summary of results that are known to follow from positivity of the simplicial volume.

Defines and classifies Thurston geometries and connects simplicial volume to Kodaira dimension.

We compute the value of the simplicial volume for closed, oriented Riemannian manifolds covered by $\mathbb{H}^{2}\times\mathbb{H}^{2}$ explicitly, thus in particular for products of closed hyperbolic surfaces. This gives the first exact value of a nonvanishing simplicial volume for a manifold of nonconstant curvature.