Infinite projections, Kazhdan compression, and maximal group C*-algebras of S-arithmetic groups.
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Tseng
First public version: August 14, 2026
Let (K) be a number field, let (\Sigma) be a nonempty finite set of nonzero prime ideals of (\mathcal O_K), and let (n \ge 4).
The paper proves that the maximal group C*-algebra of
[ \mathrm{SL}n(\mathcal O{K,\Sigma}) ]
contains an infinite projection and a proper isometry. Consequently, it is not finite, not stably finite, and not MF.
In particular, this applies to
[ \mathrm{SL}_n(\mathbb Z[S^{-1}]) ]
for every nonempty finite set (S) of rational primes and every (n\ge4).
More generally, if a countable discrete group (G) contains a property-(T) subgroup (\Gamma) and some (t\in G) satisfies
[ t\Gamma t^{-1}\subsetneq\Gamma, ]
then the Kazhdan projection associated with (\Gamma) is infinite in (C^*_{\max}(G)).
Let (K) be a number field, let (\Sigma) be a nonempty finite set of nonzero prime ideals of (\mathcal O_K), and let (n\ge4). We prove that (C^*{\max}(\mathrm{SL}_n(\mathcal O{K,\Sigma}))) contains an infinite projection and a proper isometry. Hence it is not finite and is neither stably finite nor MF, although (\mathrm{SL}n(\mathcal O{K,\Sigma})) is a finitely presented, residually finite property-(T) group. In particular, this applies to (\mathrm{SL}_n(\mathbb 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 (\Gamma) and some (t\in G) satisfies (t\Gamma t^{-1}\subsetneq\Gamma), then the Kazhdan projection (p_\Gamma) is infinite in (C^*_{\max}(G)).
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.
Tseng. Infinite Projections in Full Group C-Algebras of S-Arithmetic Groups*. Zenodo, 2026.
This preprint and its LaTeX source are licensed under Creative Commons Attribution 4.0 International (CC BY 4.0).