极小体积熵与纤维增长

Volume and homology growth of aspherical manifolds

We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. Further, we show that linear growth of mod p Betti numbers or exponential growth of torsion homology imply that a closed aspherical manifold is "large".

Growth of homology torsion in finite coverings and hyperbolic volume

We give an upper bound for the growth of homology torsions of finite coverings of irreducible 3-manifolds with tori boundary in terms of hyperbolic volume.

熵与体积 via 海格劳德图

Entropy versus volume via Heegaard diagrams

该研究通过海格劳德弗洛尔同调和双曲几何，建立了双曲3流形的熵与体积之间的不等式关系，改进了Kojima的结果。

Homological eigenvalues of mapping classes and torsion homology growth for fibered 3--manifolds

Let S be an orientable surface with negative Euler characteristic, let ψ\in\Mod(S) be a mapping class of S, and let T_ψ be the mapping torus of ψ. We study the action of lifts of ψon the homology of finite covers of S via the torsion homology growth of towers of finite covers of T_ψ. We show that ψadmits a lift to a finite cover with a homological eigenvalue of length greater than one if and only if the mapping torus T_ψ admits a finite cover X and a certain tower of abelian covers which have exponential torsion homology growth. We show that the existence of such a lift of ψis intrinsic to T_ψ…

A relative isoperimetric inequality for certain warped product spaces

Given a warped product space $\mathbb{R} \times_{f} N$ with logarithmically convex warping function $f$, we prove a relative isoperimetric inequality for regions bounded between a subset of a vertical fiber and its image under an almost everywhere differentiable mapping in the horizontal direction. In particular, given a $k$--dimensional region $F \subset \{b\} \times N$, and the horizontal graph $C \subset \mathbb{R} \times_{f} N$ of an almost everywhere differentiable map over $F$, we prove that the $k$--volume of $C$ is always at least the $k$--volume of the smooth constant height graph ove…

A remark on torsion growth in homology and volume of 3-manifolds

We show that Lück's conjecture on torsion growth in homology implies that two 3-manifolds have equal volume if the fundamental groups have the same set of finite quotients.

Hyperbolic 5-manifolds that fiber over $S^1$

We exhibit some finite-volume cusped hyperbolic 5-manifolds that fiber over the circle. These include the smallest hyperbolic 5-manifold known, discovered by Ratcliffe and Tschantz. As a consequence, we build a finite type subgroup of a hyperbolic group that is not hyperbolic.

On the degree of a finite cover which fibers over circle

We give a lower bound for the degree of a finite cover of a hyperbolic 3-manifold which fibers over the circle, in terms of volume, the diameter of the manifold and other new invariants.

关于环面映射环的循环覆盖的积分单纯体积

On the integral simplicial volume of cyclic covers of mapping tori

本文研究了环面映射环的循环覆盖的积分单纯体积的渐近行为，通过积分填充体积建立了极限的上下界，并揭示了其与拓扑熵的联系及若干新应用。

关于纤维丛的单纯体积

On the simplicial volumes of fibre bundles

我们证明了基和纤维亏格至少为2的曲面上的曲面丛具有非零单纯体积。

The simplicial volume of mapping tori of 3-manifolds

We prove that any mapping torus of a closed 3-manifold has zero simplicial volume. When the fiber is a prime 3-manifold, classification results can be applied to show vanishing of the simplicial volume, however the case of reducible fibers is by far more subtle. We thus analyse the possible self-homeomorphisms of reducible 3-manifolds, and use this analysis to produce an explicit representative of the fundamental class of the corresponding mapping tori. To this end, we introduce a new technique for understanding self-homeomorphisms of connected sums in arbitrary dimensions on the level of clas…