Rank gradient versus stable integral simplicial volume

Löh, Clara

doi:10.1007/s10998-017-0212-1

Cited by 9 publications

(16 citation statements)

References 14 publications

(16 reference statements)

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“…Proof In particular, we have

{∥ M ∥}_{double-struckZ}^{\infty} = 0

. The rank gradient estimate hence follows from the fact that stable integral simplicial volume is an upper bound for the rank gradient . For the homology and torsion homology gradients, we consider the descending chain

{false(Γ_{j} false)}_{j \in double-struckN}

of finite index subgroups of

π_{1} false(M false)

from above.…”

Section: Introductionmentioning

confidence: 99%

Integral approximation of simplicial volume of graph manifolds

Fauser

Friedl

Loeh

2019

Bull. London Math. Soc.

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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 rank gradients, Betti number gradients and torsion homology gradients for graph manifolds. Theorem 1.2. Let M be an oriented closed connected graph manifold (in the sense of Definition 2.7) with residually finite fundamental group. Then M = M ∞ Z = 0. More generally, let (Γ j ) j∈N be a descending chain of finite index subgroups of π 1 (M ) with trivial intersection and let (M j ) j∈N be the corresponding tower of finite coverings. Then M = lim j→∞ M j Z [π 1 (M ) : Γ j ] = 0.

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“…Proof In particular, we have

{∥ M ∥}_{double-struckZ}^{\infty} = 0

{false(Γ_{j} false)}_{j \in double-struckN}

of finite index subgroups of

π_{1} false(M false)

from above.…”

Section: Introductionmentioning

confidence: 99%

Integral approximation of simplicial volume of graph manifolds

Fauser

Friedl

Loeh

2019

Bull. London Math. Soc.

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“…Remark 10.5 (growth and gradient invariants). In particular, in these situations, we obtain corresponding vanishing results for homology growth and logarithmic homology torsion growth [8, Theorem 1.6] as well as for the rank gradient [14].…”

Section: Proof We Provementioning

confidence: 92%

Variations on the theme of the uniform boundary condition

Fauser

Löh

2019

J. Topol. Anal.

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The uniform boundary condition in a normed chain complex asks for a uniform linear bound on fillings of null-homologous cycles. For the ℓ 1 -norm on the singular chain complex, Matsumoto and Morita established a characterisation of the uniform boundary condition in terms of bounded cohomology. In particular, spaces with amenable fundamental group satisfy the uniform boundary condition in every degree. We will give an alternative proof of statements of this type, using geometric Følner arguments on the chain level instead of passing to the dual cochain complex. These geometric methods have the advantage that they also lead to integral refinements. In particular, we obtain applications in the context of integral foliated simplicial volume.Dynamical versions of Følner sequences lead to corresponding results for the parametrised ℓ 1 -norm: Theorem 1.3 (parametrised UBC for tori). Let d ∈ N >0 and let M := (S 1 ) d be the d-torus, let Γ := π 1 (M) ∼ = Z d , and let α = Γ (X, µ) be an (essentially) free standard Γ-space. Then C * (M; α) = L ∞ (X; Z) ⊗ ZΓ

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“…In the closed case, stable integral simplicial volume gives an upper bound for L 2 -Betti numbers [19, p. 305] [34], logarithmic torsion growth of homology [13,Theorem 1.6;33], and the rank gradient [26].…”

Section: The Approximation Problem For Simplicial Volumementioning

confidence: 99%

“…If