| 概要 | It is Serre who pointed out that SL_2(Z) acts on a tree. According to Bass-Serre theory, a group that acts on a tree can be decomposed as an amalgam of its subgroups.
In this talk I will consider actions of groups on quasi-trees -- these are metric spaces (e.g. graphs) quasi-isometric to a tree. Having an action on a quasi-tree is a much more flexible condition than having an action on a tree. There is a general construction of such actions for "rank 1" groups. There are also many groups that do not admit nontrivial actions on quasi-trees, by the work of Manning, Burger-Mozes, N. Ozawa (SL_3(Z) is an example). Applications include a construction of quasi-morphisms and quasi-cocyces on various groups, characterization of elements in mapping class groups with zero stable commutator length, and finiteness of asymptotic dimension of mapping class groups. This is joint work with Ken Bromberg and Koji Fujiwara. |