# Bounded cohomology of transformation groups

## TL;DR

This paper introduces a new method for constructing classes in the bounded cohomology of transformation groups of Riemannian manifolds, demonstrating infinite dimensionality of their third bounded cohomology under certain conditions.

## Contribution

It presents a novel approach to bounded cohomology of transformation groups and establishes conditions for infinite dimensionality of their third bounded cohomology.

## Key findings

- New method for constructing bounded cohomology classes.
- Infinite dimensionality of third bounded cohomology under certain conditions.
- Applicable to groups like $Homeo_0(M,)$, $Diff_0(M,vol)$, and $Symp_0(M,)$.

## Abstract

Let $M$ be a complete connected Riemannian manifold of finite volume. In this paper we present a new method of constructing classes in bounded cohomology of transformation groups such as $Homeo_0(M,\mu)$, $Diff_0(M,vol)$ and $Symp_0(M,\omega)$ (in case $M$ is symplectic). As an application we show that, under certain conditions on $\pi_1(M)$, the $3^{rd}$ bounded cohomology of these groups is infinite dimensional.