No Arabic abstract
We prove that $L(SL_2(textbf{k}))$ is a maximal Haagerup von Neumann subalgebra in $L(textbf{k}^2rtimes SL_2(textbf{k}))$ for $textbf{k}=mathbb{Q}$. Then we show how to modify the proof to handle $textbf{k}=mathbb{Z}$. The key step for the proof is a complete description of all intermediate von Neumann subalgebras between $L(SL_2(textbf{k}))$ and $L^{infty}(Y)rtimes SL_2(textbf{k})$, where $SL_2(textbf{k})curvearrowright Y$ denotes the quotient of the algebraic action $SL_2(textbf{k})curvearrowright widehat{textbf{k}^2}$ by modding out the relation $phisim phi$, where $phi$, $phiin widehat{textbf{k}^2}$ and $phi(x, y):=phi(-x, -y)$ for all $(x, y)in textbf{k}^2$. As a by-product, we show $L(PSL_2(mathbb{Q}))$ is a maximal von Neumann subalgebra in $L^{infty}(Y)rtimes PSL_2(mathbb{Q})$; in particular, $PSL_2(mathbb{Q})curvearrowright Y$ is a prime action, i.e. it admits no non-trivial quotient actions.
We initiate a study of maximal subgroups and maximal von Neumann subalgebras which have the Haagerup property. We determine maximal Haagerup subgroups inside $mathbb{Z}^2 rtimes SL_2(mathbb{Z})$ and obtain several explicit instances where maximal Haagerup subgroups yield maximal Haagerup subalgebras. Our techniques are on one hand based on group-theoretic considerations, and on the other on certain results on intermediate von Neumann algebras, in particular these allowing us to deduce that all the intermediate algebras for certain inclusions arise from groups or from group actions. Some remarks and examples concerning maximal non-(T) subgroups and subalgebras are also presented, and we answer two questions of Ge regarding maximal von Neumann subalgebras.
Consider a minimal free topological dynamical system $(X, T, mathbb{Z}^d)$. It is shown that the comparison radius of the crossed product C*-algebra $mathrm{C}(X) rtimes mathbb{Z}^d$ is at most the half of the mean topological dimension of $(X, T, mathbb{Z}^d)$. As a consequence, the C*-algebra $mathrm{C}(X) rtimes mathbb{Z}^d$ is classifiable if $(X, T, mathbb{Z}^d)$ has zero mean dimension.
For a positive integer $g$, let $mathrm{Sp}_{2g}(R)$ denote the group of $2g times 2g$ symplectic matrices over a ring $R$. Assume $g ge 2$. For a prime number $ell$, we give a self-contained proof that any closed subgroup of $mathrm{Sp}_{2g}(mathbb{Z}_ell)$ which surjects onto $mathrm{Sp}_{2g}(mathbb{Z}/ellmathbb{Z})$ must in fact equal all of $mathrm{Sp}_{2g}(mathbb{Z}_ell)$. The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties.
The notion of $SL_2$-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic $SL_2$-tilings that contain a rectangular domain of positive integers. Every such $SL_2$-tiling corresponds to a pair of frieze patterns and a unimodular $2times2$-matrix with positive integer coefficients. We relate this notion to triangulated $n$-gons in the Farey graph.
For every pair of distinct primes $p$, $q$ we prove that $mathbb{Z}_p^3 times mathbb{Z}_q$ is a CI-group with respect to binary relational structures.