By: Danny Calegari

Danny Calegari — Sat, 24 Oct 2009 04:21:45 +0000

Dilatation satisfies for positive , but unfortunately not for negative ; it is more like a norm (as your comment more or less observes). One can think of dilatation as a bit like translation length (in the Teichmuller metric); by contrast, a quasimorphism is less like the length of a geodesic than the integral of some -form over that geodesic. This idea leads to the concept of counting quasimorphisms, one of the few known general methods for constructing quasimorphisms on groups (under suitable negative curvature hypotheses). The Bestvina-Fujiwara construction of quasimorphisms on mapping class groups is of this nature.

Stable norms — those satisfying — and quasimorphisms are related; for instance, there is a duality between stable commutator length (which is a (pseudo)-norm on the space of homologically trivial group 1-chains) and homogeneous quasimorphisms modulo homomorphisms (with the ”defect” norm). Another interesting relationship between norms and quasimorphisms is discussed in a paper of Burago-Ivanov-Polterovich (http://arxiv.org/abs/0710.1412).

By: JSE

JSE — Sat, 24 Oct 2009 03:36:46 +0000

Naive question. The dilatation lambda is a map from the mapping class group to R (or at least on pseudo-Anosovs, so replace the mapping class group with a subgroup consisting entirely of p-As if you want.) It satisfies lambda(g^n) = n lambda(g). Is there any chance this is a quasimorphism?

My guess is no, just on grounds that lambda is a bit like a function that sends matrices to the largest absolute value of an eigenvalue, and this isn’t a quasimorphism at all. (Indeed, for Gamma_{1,1} this really IS what lambda is so I guess that ends the question in at least one case.)

Comments on: Quasimorphisms from knot invariants

By: Danny Calegari

By: JSE