Metric groups, unitary representations and continuous logic

TL;DR

This paper explores how properties of metric groups and their unitary representations can be expressed in continuous logic, providing axiomatizations for amenability and insights into Kazhdan's property (T).

Contribution

It introduces $L_{oldsymbol{ extomega_1 oldsymbol{ extomega}}}$-axiomatizations for amenability and analyzes the logical expressibility of property (T) in locally compact groups.

Findings

Axiomatization of amenability in continuous logic.

Representation of property (T) negation as a union of axiomatizable classes.

Conditions for properties to be preserved under elementary substructures.

Abstract

We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find $L_{\omega_1 \omega}$-axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan's property {\bf (T)} can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.