Strong and uniform boundedness of groups

TL;DR

This paper studies the property of boundedness in groups, exploring its strengthenings across various classes like Lie groups and arithmetic groups, with applications to Hamiltonian dynamics.

Contribution

It introduces new strengthenings of boundedness and investigates these properties in diverse group classes, linking algebraic properties to dynamical applications.

Findings

Semisimple Lie groups exhibit strong boundedness properties.

Arithmetic groups are analyzed for boundedness characteristics.

Applications to Hamiltonian dynamics demonstrate practical relevance.

Abstract

A group G is called bounded if every conjugation-invariant norm on G has finite diameter. We introduce various strengthenings of this property and investigate them in several classes of groups including semisimple Lie groups, arithmetic groups and linear algebraic groups. We provide applications to Hamiltonian dynamics.