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…
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48 results for “foliated simplicial volume”
Integral volume vanishes for manifolds with circle foliations.
problem Integral foliated simplicial volume calculation.
method Regular foliation by circles analysis.
result Integral foliated simplicial volume vanishes.
Upper bounds for fundamental groups via foliated simplicial volume.
problem Bounding fundamental groups of manifolds.
method Integral foliated simplicial volume approach.
result Integral foliated simplicial volume gives upper bounds for fundamental groups.
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…
Formula connects foliated simplicial volume with group cost.
problem Calculating integral foliated simplicial volume.
method Ergodic decomposition formula for simplicial volume.
result Integration formula linking foliated simplicial volume and group cost.
3-manifolds' volumes match stable integral values.
problem Determining 3-manifold volumes accurately.
method Integral foliated simplicial volume and ergodic theory.
result 3-manifolds' volumes equal stable integral simplicial volumes.
Integral foliated simplicial volume is zero for certain amenable covers.
problem Calculating the integral foliated simplicial volume of specific manifolds.
method Using open amenable covers and fixed price property of fundamental groups.
result Integral foliated simplicial volume is zero for manifolds with multiplicity at most n.
The simplicial volume of oriented closed connected smooth manifolds that admit a non-trivial smooth -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…
Study simplicial volume via foliated simplices and duality.
problem Calculate simplicial volume using foliated simplices and duality.
method Defined real singular foliated homology, constructed foliated fundamental class, and established isometric isomorphism with measurable bounded cohomology.
result Norm of foliated fundamental class equals simplicial volume of .
Let N be a manifold (with boundary) of dimension at least 3, such that its interior admits a hyperbolic metric of finite volume. We discuss the possible limits arising from sequences of relative fundamental cycles approximating the simplicial volume. As applications, we extend results of Jungreis and Calegari from clos…
Study of -adic simplicial volumes and their properties.
problem Understanding simplicial volumes over -adic seminormed rings.
method Definition and study of -adic simplicial volumes, homology bounds, and computation of volumes for surfaces.
result Established homology bounds and computed -adic simplicial volumes for surfaces.
We define the ideal simplicial volume for compact manifolds with boundary. Roughly speaking, the ideal simplicial volume of a manifold measures the minimal size of possibly ideal triangulations of "with real coefficients", thus providing a variation of the ordinary simplicial volume defined by Gromov in 1982, t…
Study shows small transcendental simplicial volumes exist on manifolds.
problem Understanding the limits of simplicial volumes on manifolds.
method Examined closed manifolds and transcendental numbers.
result Found manifolds with arbitrarily small transcendental simplicial volumes.
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.
problem Understanding simplicial volume in fiber bundles.
method Investigated fiber bundles with connected structure groups.
result Simplicial volume of total space matches trivial bundle under certain conditions.
We provide sharp lower bounds for the simplicial volume of compact -manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of -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.
problem Conditions for minimal volume entropy to be zero or positive.
method Analyzes topological conditions related to fiber growth of maps.
result Examples of finite simplicial complexes with zero simplicial volume and large minimal volume entropy.
Positive simplicial volume implies locally symmetric space structure.
problem Understanding simplicial volume in locally homogeneous spaces.
method Analyzing properties of locally homogeneous Riemannian manifolds.
result Closed locally homogeneous manifolds with positive simplicial volume are locally symmetric.
The study examines conditions for minimal volume entropy of simplicial complexes.
problem Conditions for minimal volume entropy of simplicial complexes.
method Topological conditions and growth of fundamental groups.
result Examples of simplicial complexes with zero simplicial volume and large minimal volume entropy.
The simplicial volume of non-R^3 contractible 3-manifolds is infinite.
problem Characterizing contractible 3-manifolds based on their simplicial volume.
method Analyzing the simplicial volume of contractible 3-manifolds and open 3-manifolds.
result The Euclidean space is the unique contractible 3-manifold with vanishing minimal volume.
Investigates simplicial volume over finite fields and compares it with other coefficients.
problem Examines simplicial volume over coefficients.
method Analyzes simplicial volume and gradient invariants over coefficients, comparing with other coefficient rings.
result Compares simplicial volumes and Betti numbers over different coefficient rings.
We prove that cubical simplicial volume of oriented closed 3-manifolds is equal to one fifth of ordinary simplicial volume.
Volume entropy semi-normmath.GT
Innovates volume entropy semi-norm, proving equivalence to simplicial volume.
problem Equivalence of volume entropy and simplicial volume in real homology.
method Introduces volume entropy semi-norm and proves its equivalence to simplicial volume.
result Equivalence of volume entropy semi-norm and simplicial volume in every dimension.
Study shows simplicial volume of certain fiber bundles is zero.
problem Understanding simplicial volume in fiber bundles with nonpositive curvature.
method Proved simplicial volume is zero for specific fiber bundles.
result Simplicial volume of certain fiber bundles is zero.
Benedetti and Guadagnini have conjectured that the marked lenght spectrum of the constant mean curvature foliation in a 2+1 dimensional flat spacetime with compact hyperbolic Cauchy surfaces converges, in the direction of the singularity, to that of the marked measure spectrum of the R-tree dual to the measur…
Degree theorems and Lipschitz simplicial volume for non-positively curved manifolds of finite volumemath.GT
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…
Graph manifolds' simplicial volume approximated by covering volumes.
problem Proving simplicial volume of graph manifolds is zero.
method Approximating simplicial volume of graph manifolds by integral simplicial volumes of their finite coverings.
result Uniform proof of rank, Betti number, and torsion homology gradients for graph manifolds.
Let be the interior of a connected, oriented, compact manifold of dimension at least 2. If each path component of has amenable fundamental group, then we prove that the simplicial volume of is equal to the relative simplicial volume of and also to the geometric (Lipschitz) simplicial volume…
The paper proves simplicial volume positivity for certain nonpositively curved 4-manifolds with nonzero Euler characteristic.
problem Proving positivity of simplicial volume for specific types of 4-manifolds.
method Using the Gauss-Bonnet theorem for Riemannian simplices, the paper shows that for closed nonpositively curved 4-manifolds with nonzero Euler characteristic, simplicial volume is positive.
result The paper confirms conjectures about the relationship between simplicial volume and Euler characteristic for four-dimensional manifolds.
Study shows range of simplicial volumes for open manifolds.
problem Understanding simplicial volumes of open manifolds.
method Analyzes locally finite simplicial volumes in dimensions at least 4.
result Set of simplicial volumes is [0, ∞] for open manifolds.
The study shows simplicial volume finiteness for certain manifolds with amenable fundamental groups.
problem Determining the simplicial volume of manifolds with specific properties.
method Analyzing the fundamental group and using amenability properties.
result Simplicial volume is finite for certain manifolds with amenable fundamental groups.
Study simplicial volume for fixed fundamental groups, finding gaps.
problem Understanding simplicial volume for manifolds with fixed fundamental group.
method Relate gap problem to rationality questions in bounded (co)homology.
result Show existence of gaps in simplicial volume spectrum at zero.
Study simplicial volume of manifolds from reflection group trick.
problem Characterize manifolds with positive simplicial volume.
method Define a partial order on triangulations and solve explicitly for minimal elements.
result Explicitly solved triangulations of the two-dimensional sphere and performed extensive analysis for three-dimensional case.
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 shows complete affine manifolds have zero simplicial volume.
problem Estimating the amenable category of affine manifolds.
method Construction of manifolds with infinite amenable normal subgroups.
result Zero simplicial volume for all such manifolds.
Integral filling volume, complexity and integral simplicial volume of 3-dimensional mapping torimath.GT
Integral filling volume of mapping tori grows sublinearly with complexity.
problem Characterizing mapping classes with vanishing integral filling volume.
method Analyzing Dehn twists and mapping tori, using simplicial volume and complexity.
result Integral simplicial volume of mapping tori grows sublinearly with respect to the monodromy power.
Study integral simplicial volume of cyclic covers of torus bundles.
problem Asymptotic behavior of integral simplicial volume of cyclic covers.
method Integral filling volume invariant for monodromy, bounds established.
result Established both lower and upper bounds for the limit of integral simplicial volume.
Positive simplicial volume found for certain non-positively curved manifolds with specific submanifolds.
problem Determining conditions for positive simplicial volume in non-positively curved manifolds.
method Analyzing isolated, closed totally geodesic submanifolds of codimension one and their impact on simplicial volume.
result Positive simplicial volume for certain non-positively curved manifolds with specific submanifolds.
Study essentiality and simplicial volume of manifolds fibered over spheres.
problem When manifolds fibered over spheres are essential or have positive simplicial volume.
method Analyzing mapping tori and fiber bundles over spheres, using results on macroscopic dimension and characteristic classes.
result Mapping tori of odd-dimensional manifolds with non-zero simplicial volume are essential, while fiber bundles over spheres of dimension d > 1 have zero simplicial volume.
Estimates simplicial volume for complex hyperbolic surfaces.
problem Bounding the simplicial volume of complex hyperbolic surfaces.
method Estimates Gromov norm and uses top dimensional class in .
result Explicit upper bound for simplicial volume.
Alternative proof of simplicial volume bound using area-minimizing sets.
problem Bounding simplicial volume of manifolds with restricted ball volumes.
method Area-minimizing separating sets method.
result Explicit constants for simplicial volume bound derived.
Proportionality principle for the simplicial volume of families of Q-rank 1 locally symmetric spacesmath.GT
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…
New constructions show manifold volumes are dense in non-negative reals.
problem Understanding the spectrum of simplicial volumes in manifolds.
method Group homology constructions and manifold constructions using cross-products and Thom realisation.
result The set of simplicial volumes of orientable closed connected manifolds is dense in for dimensions > 3, and every non-negative rational number is a simplicial volume for dimension 4.
Study shows simplicial volume is superadditive under specific conditions.
problem Understanding superadditivity of simplicial volume under gluings.
method New results in relative bounded cohomology and pairs of multicomplexes.
result Simplicial volume is superadditive for boundary connected sums and 1-handle attachments.
The study shows that nonpositively curved 4-manifolds with zero Euler characteristic have degenerating Ricci curvature.
problem Characterizing nonpositively curved 4-manifolds with zero Euler characteristic.
method Analyzing the Ricci curvature and foliations in neighborhoods of points.
result Nonpositively curved 4-manifolds with zero Euler characteristic have degenerating Ricci curvature.
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 they are additive with respect to connected sums and gluings along -injective, amenable aspherical boundary components…
Study on simplicial volume and Euler characteristic of aspherical manifolds.
problem Whether vanishing simplicial volume implies vanishing Euler characteristic.
method Various strategies for both affirmative and negative answers, context with other problems, and comparative analysis of additivity properties.
result Found counterexamples among aspherical spaces that are homology equivalent to manifolds but not manifolds themselves.
Simplicial volume vanishes for 4-manifolds with open book decompositions.
problem Characterizing open book decompositions in dimension 4.
method Adapted proof by Bucher and Neofytidis.
result Simplicial volume vanishes for 4-manifolds with open book decompositions.