Bounded cohomology and the Cheeger isoperimetric constant

Kim, Sungwoon; Ким, Инканг

doi:10.1007/s10711-015-0064-x

Cited by 3 publications

(2 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then [4] shows that the hyperbolic n-manifold H n / im ρ has positive Cheeger constant. Moreover, [20] show that positivity of the Cheeger constant is equivalent to the vanishing of the bounded class ρ * Vol n ∈ H n b (S; R). We arrive at the claim in the beginning of the remark.…”

Section: Introductionmentioning

confidence: 94%

Relations in bounded cohomology

Farre

2020

Journal of Topology

View full text Add to dashboard Cite

We explain some interesting relations in the degree 3 bounded cohomology of surface groups. Specifically, we show that if two faithful Kleinian surface group representations are quasi‐isometric, then their bounded fundamental classes are the same in bounded cohomology. This is novel in the setting that one end is degenerate, while the other end is geometrically finite. We also show that a difference of two singly degenerate classes with bounded geometry is boundedly cohomologous to a doubly degenerate class, which has a nice geometric interpretation. Finally, we explain that the above relations completely describe the linear dependencies between the ‘geometric’ bounded classes defined by the volume cocycle with bounded geometry. We obtain a mapping class group invariant Banach subspace of the reduced degree 3 bounded cohomology with explicit topological generating set and describe all linear relations.

show abstract

Section: Introductionmentioning

confidence: 94%

Relations in bounded cohomology

Farre

2020

Journal of Topology

View full text Add to dashboard Cite

show abstract

“…Then [Bow15] shows that the hyperbolic n-manifold H n / im ρ has positive Cheeger constant. Moreover, [KK15] show that positivity of the Cheeger constant is equivalent to the vanishing of the bounded class ρ * Vol n ∈ H n b (S; R). We arrive at the claim in the beginning of the remark.…”

Section: Introductionmentioning

confidence: 94%

Relations in Bounded Cohomology

Farre

2018

Preprint

View full text Add to dashboard Cite

We explain some interesting relations in the degree three bounded cohomology of surface groups. Specifically, we show that if two faithful Kleinian surface group representations are quasi-isometric, then their bounded fundamental classes are the same in bounded cohomology. This is novel in the setting that one end is degenerate, while the other end is geometrically finite. We also show that a difference of two singly degenerate classes with bounded geometry is boundedly cohomologous to a doubly degenerate class, which has a nice geometric interpretation. Finally, we explain that the above relations completely describe the linear dependences between the 'geometric' bounded classes defined by the volume co-cycle with bounded geometry. We obtain a mapping class group invariant Banach sub-space of the reduced degree three bounded cohomology with explicit Banach space basis.

show abstract

Borel and Volume Classes for Dense Representations of Discrete Groups

Farre

2021

International Mathematics Research Notices

View full text Add to dashboard Cite

We show that the bounded Borel class of any dense representation $\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $G$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$ is uniformly separated in semi-norm from any other representation $\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$ for which there is a subgroup $H\le G$ on which $\rho $ is still dense but $\rho ^{\prime}$ is discrete or indiscrete but stabilizes a point, line, or plane in ${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of ${\operatorname{PSL}}_2{\mathbb{R}}$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties.

show abstract

Bounded cohomology and the Cheeger isoperimetric constant

Cited by 3 publications

References 30 publications

Relations in bounded cohomology

Relations in bounded cohomology

Relations in Bounded Cohomology

Borel and Volume Classes for Dense Representations of Discrete Groups

Contact Info

Product

Resources

About