Abstract
Let G=A *B be a free product of freely indecomposable groups. We explicitly construct quasimorphisms on G which are invariant with respect to all automorphisms of G. We also prove that the space of such quasimorphisms is infinite-dimensional whenever G is not the infinite dihedral group. As an application we prove that an invariant analogue of stable commutator length recently introduced by Kawasaki and Kimura is non-trivial for these groups.
Highlights
The study of quasimorphisms on a given group G is an important branch of geometric group theory [6] with quasimorphisms sharing deep relationships with the underlying structure of the group G
There are numerous applications in symplectic geometry originating from work of Entov and Polterovich [7]
We make use of an explicit description of the automorphism group of a free product found in [8] to see in Proposition 4.11 that our code quasimorphisms are unbounded and invariant with respect to all automorphisms of G if A and B are not infinite cyclic
Summary
The study of quasimorphisms on a given group G is an important branch of geometric group theory [6] with quasimorphisms sharing deep relationships with the underlying structure of the group G. For free groups Fn the so called counting quasimorphisms originating from the work of Brooks in [4] yield a wide variety of examples. His ideas have been developed further by Calegari and Fujiwara who constructed unbounded quasimorphisms on non-elementary hyperbolic groups [5]. There are numerous applications in symplectic geometry originating from work of Entov and Polterovich [7]. Another fundamental paper on the geometry of quasimorphisms and central extensions is [1]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
