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# Geometric group theory: Bounded cohomology and norms on groups

Addition of geometric volume classes- James Farre (Yale)

Abstract: We study the algebraic structure of three dimensional bounded cohomology generated by volume classes for infinite co-volume, finitely generated Kleinian groups. While bounded cohomology is generally unwieldy, we show that addition admits a natural geometric interpretation for the volume classes of tame hyperbolic manifolds of infinite volume and bounded geometry: the volume classes of singly degenerate manifolds sum to the volume classes for manifolds with many degenerate ends. It turns out that this generates the linear dependencies among volume classes, giving a complete description of the algebraic structure of some geometrically defined subspaces of bounded cohomology. We will indicate some problems left open by this discussion and give some suggestions for future directions. Definitions, background, and geometric aspects of hyperbolic manifolds homotopy equivalent to a closed surface will be reviewed, but we assume some familiarity with hyperbolic geometry.

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The Spectrum of Simplicial Volume- Nicolaus Heuer (Cambridge)

Abstract: Simplicial volume was first introduced by Gromov to study the minimal volume of manifolds. Since then it has emerged as an active research field with a wide range of applications. In dimensions two and three, the set of possible values of simplicial volume may be fully computed using geometrization, but is hardly understood in higher dimensions. In joint work with Clara Löh (University of Regensburg), we show that the set of simplicial volumes in higher dimensions is dense in the non-negative reals. I will also discuss how the exact set of simplicial volumes in dimension four or higher may look like.

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Computability of the Minimal Genus on Second Homology- Thorben Kastenholz (University of Bonn)

Abstract: Surface representatives of second homology classes can be used to give geometric invariants for second homology classes, the most prominent examples are the genus and the Euler characteristic. In this talk I will explain why determining the minimal genus of a given homology class is in general undecidable, and how to compute it for a large class of "negatively-curved" spaces including 2-dimensional CAT(-1)-complexes. This will need a normal Form result proven by me and Mark Pedron, that extends a theorem by Edmonds on maps between surfaces.

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Quasimorphisms on diffeomorphism groups- Richard Webb (University of Manchester)

Abstract: I will explain how to construct an unbounded quasimorphism on the group of isotopically-trivial diffeomorphisms of a surface of positive genus. As a corollary the commutator length and fragmentation norm are both (stably) unbounded, which solves a problem of Burago--Ivanov--Polterovich. The proof uses a new hyperbolic graph on which these groups act by isometries, which is inspired by techniques from mapping class groups. This is joint work with Jonathan Bowden and Sebastian Hensel.

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## 9 replies on “Geometric group theory: Bounded cohomology and norms on groups”

[…] Bounded cohomology and norms on groups […]

Nicolaus_Heuersays:

Hi! I’m Nicolaus Heuer and I look forward to comments on my talk!

james.farresays:

Hi all,

James

ShiWangsays:

Hi James,

Very nice talk! In your talk you mentioned that (at least) in H^3 it is generic to have a sequence of closed geodesics with length tending to zero (i.e. unbounded geometry). Are there any references about this fact?

Shi

james.farresays:

Hi Shi; thanks!

Here are a couple of references.

This is probably not a standard reference, but I like this paper in which Sisto and Taylor show that for random mapping classes, the mapping torus has a curve of short length. This is relevant, because the Z covers corresponding to the fiber of these mapping tori are doubly degenerate manifolds of bounded geometry. But, limits of these manifolds are generic. Since the mapping torus has a curve shorter than epsilon with probability one as the random walk goes to infinity for all epsilon, limits have unbounded geometry with probability one.
https://arxiv.org/abs/1611.07545

See page 4th paragraph from the from bottom of page 146 of https://link.springer.com/content/pdf/10.1007/s002220100163.pdf
for a remark and more references, which are more standard.

From the modern perspective, using Minsky’s model manifold (https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n1-p01-s.pdf), unbounded geometry for surface groups is equivalent to the end invariants having `unbounded combinatorics.’

Recall that the curve graph is a locally infinite, infinite diameter, hyperbolic graph with boundary that can be identified with the geometrically infinite end invariants of hyperbolic 3 manifolds with tame ends. There are subsurface projections: coarsely defined maps to the curve graphs of subsurfaces; projecting an ending lamination to a subsurface amounts to intersecting it with a representative of the subsurface. You get a finite collection of parallelism classes of arcs (there are only finitely many pairwise non-isotopic disjoint proper arcs on a subsurface with boundary at one time).

Given two ending laminations, project each onto a subsurface. Record the distance between these projections in the curve/arc graphs. Minsky proves that if this number is large for a specific subsurface, then the boundary of that subsurface in the corresponding hyperbolic 3 manifold is short. The shortness is bounded by functions of these projection lengths.

So the question of having unbounded geometry for a 3-manifold is reduced to the question of its ending laminations having unbounded combinatorics–that these projection distances are arbitrarily large over all subsurfaces. Hence the connection with the first reference.

Ending laminations with unbounded combinatorics are like irrational numbers that are poorly approximated (in terms of their continued fraction expansions)–this is a precise correspondence when your surface is a 1-times punctured torus or a 4-times punctured sphere. Poorly approximated irrational numbers are generic.

James

ShiWangsays:

Hi James,

Thanks a lot for the detailed references. I’ll take a look and might bug you with more questions.

Shi

ThorbenKastenholzsays:

Hey, this is Thorben Kastenholz. I am also looking forward to your questions.

lvzhou.chensays:

Hi Thorben, thanks for the interesting talk. Is there a characterization/description of those extremal vertices in the attainable set?

ThorbenKastenholzsays:

Hi Lvzshou,
that’s a great question. If one considers the convex hull of the attainable set, then extremal vertices of the convex hull represent minimizers for invariants that are linear combinations of the genus and chi^-. One example of such a linear combination that is interesting is chi^-+2g which is the minimal number of triangles needed to represent the class (i.e. something like the integral simplicial volume). In general there is not much we can say, because the structure of the attainable set in itself is kind of mysterious at the moment. For example it would be interesting to investigate whether every set fulfilling the proposition I explained in the talk occurs as the attainable set of some class. A good starting point for that would be to construct examples which have arbitrarily “big steps” in the attainable set.