Infinite Projections in Full Group C*-Algebras of S-Arithmetic Groups

Author: Tseng
Version: v1.0 — August 14, 2026
Archived preprint DOI: https://doi.org/10.5281/zenodo.21939762
All versions DOI: https://doi.org/10.5281/zenodo.21939761

This repository contains the public preprint and its LaTeX source.

Main result

Let K be a number field, let Σ be a nonempty finite set of nonzero prime ideals of O_K, and let n ≥ 4. The paper proves that

C*max(SL_n(O_{K,Σ}))

contains an infinite projection and a proper isometry. Consequently, it is not finite, not stably finite, and not MF.

In particular, this applies to SL_n(Z[S^{-1}]) for every nonempty finite set S of rational primes and every n ≥ 4.

More generally, if a countable discrete group G contains a property-(T) subgroup Γ and some t ∈ G satisfies

tΓt^{-1} ⊊ Γ,

then the Kazhdan projection associated with Γ is infinite in C*max(G).

Files

• s_arithmetic_full_group_cstar.pdf — preprint
• s_arithmetic_full_group_cstar.tex — LaTeX source

Abstract

Let K be a number field, let Σ be a nonempty finite set of nonzero prime ideals of O_K, and let n ≥ 4. We prove that C*max(SL_n(O_{K,Σ})) contains an infinite projection and a proper isometry. Hence it is not finite and is neither stably finite nor MF, although SL_n(O_{K,Σ}) is a finitely presented, residually finite property-(T) group. In particular, this applies to SL_n(Z[S^{-1}]) for every nonempty finite set S of rational primes.

More generally, if a countable discrete group G contains a property-(T) subgroup Γ and some t ∈ G satisfies tΓt^{-1} ⊊ Γ, then the Kazhdan projection p_Γ is infinite in C*max(G); explicitly, u_{t^{-1}}p_Γ + (1 − p_Γ) is a proper isometry.

To the best of our knowledge, these are the first examples of countable discrete groups with a non-finite—and hence non-MF—full group C*-algebra.

The preparation of this manuscript was AI-assisted.

Citation

Please cite the archived v1.0 preprint using:

Tseng. _Infinite Projections in Full Group C-Algebras of S-Arithmetic Groups_. Zenodo, 2026. DOI: 10.5281/zenodo.21939762*

License

The preprint and its LaTeX source are licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0).

See LICENSE for the full license text.

The canonical archival record is available on Zenodo: https://doi.org/10.5281/zenodo.21939762