Classification of right-angled Coxeter groups with a strongly solid von Neumann algebra (Q6590489)

Originating in the problem of proving absence of Cartan subalgebras in \(\mathrm{II}_1\) factors, the notion of strong solidity address core structure of a von Neumann algebra. Accordingly, the question of which von Neumann algebras are strongly solid has received considerablex attention. Coxeter groups form a natural class of groups, for which there is no complete characterisation yet of which group von Neumann algebras are strongly solid. The present article solves this problem in a subclass and characterises right-angled Coxeter groups whose group von Neumann algebra is strongly solid.\N\NGiven a right-angled Coxeter group \(W\) it shows that \(\mathrm{L}(W)\) is not stronlgy solid if and only if \(W\) contains a subgroup isomorphic with \(\mathbb{Z} \times \mathbb{F}_2\). More precisely, and in the language of Coxeter systems, let \(S \subseteq W\) be a set of Coxeter generators. Then \(\mathrm{L}(W)\) is not strongly solid if and only if there is a subset \(I \subset S\) generating a subgroup isomorphic with \(\mathrm{D}_\infty \times (\mathbb{Z}/2)^{*3}\) or \(\mathrm{D}_\infty \times ((\mathbb{Z}/2 \times \mathbb{Z}/2) * \mathbb{Z}/2)\).