Actions of infinite simple groups

2014 Fiscal Year Final Research Report

• Project/Area Number: 24654011
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: The University of Tokyo
• Principal Investigator: TSUBOI Takashi 東京大学, 数理(科)学研究科(研究院), 教授 (40114566)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 幾何学 / 代数学 / トポロジー / 微分同相群 / 幾何学的群論 / 無限単純群

Outline of Final Research Achievements

There are many infinite simple groups which have natural actions on spaces. Consider the set X of the unions of the conjugate classes of a nontrivial element and its inverse in an infinite simple group G. For 2 such unions, one union is always contained in a power of the other. This phenominon gives rise to a metric on X. We studied properties of this metric on X by using the action of G on some space. For a nontrivial subset K which is closed under taking inverses, any element of G can be written as a product of conjugates of elements of K. The minimal number of conjugates defines a norm on the group G. We clarified the relationship between the metric on X and the norm and quasimorphisms on G. We also showed that in certain homeomorphism groups, every elemet can be written as one commutator.

Free Research Field

位相幾何学