# A short proof of Thoma's theorem on type I groups

## TL;DR

This paper provides a new, simpler proof of Thoma's theorem, characterizing countable discrete groups of type I as those containing a finite index abelian subgroup, using elementary methods.

## Contribution

It offers a more accessible proof of Thoma's theorem, avoiding complex techniques and relying on elementary methods.

## Key findings

- Countable discrete groups of type I contain a finite index abelian subgroup
- The new proof simplifies understanding of Thoma's theorem
- Elementary methods suffice for the proof

## Abstract

In the theory of unitary group representations, a group is called type I if all factor representations are of type I, and by a celebrated theorem of James Glimm [Gli61b], the type I groups are precisely those groups for which the irreducible unitary representations are what descriptive set theorists now call "concretely classifiable". Elmar Thoma [Tho64] proved the following surprising characterization of the countable discrete groups of type I: They are precisely those that contain a finite index abelian subgroup. In this paper we give a new, simpler proof of Thoma's theorem, which relies only on relatively elementary methods. [Gli61b] James Glimm, Type I $C^{\ast} $-algebras, Ann. of Math. (2) 73 (1961), 572--612. MR 0124756 [Tho64] Elmar Thoma, \"Uber unit\"are Darstellungen abz\"ahlbarer, diskreter Gruppen, Math. Ann. 153 (1964), 111--138. MR 0160118