Journal of Operator Theory

Volume 95, Issue 1, Winter 2026 pp. 3-20.

The linear $\operatorname{SL}_2(\mathbb{Z})$-action on $\mathbb{T}^n$: ergodic and von Neumann algebraic aspects

Authors: Paul Jolissaint (1), Alain Valette (2)
Author institution: (1) Institut de Mathematiques, Universite de Neuchatel, E.-Argand 11, 2000 Neuchatel, Switzerland
(2) Institut de Mathematiques, Universite de Neuchatel, E.-Argand 11, 2000 Neuchatel, Switzerland

Summary: The linear action of $\operatorname{SL}_2(\mathbb{R})$ on $\mathbb{R}^n$ corresponding to its unique irreducible representation induces an action $\operatorname{SL}_2(\mathbb{Z})\curvearrowright \mathbb{T}^n$ for every $n\geqslant 2$ that factors through $\operatorname{PSL}_2(\mathbb{Z})$ for $n$ odd. Thus, setting $G_n=\operatorname{SL}_2(\mathbb{Z})$ (respectively $G_n=\operatorname{PSL}_2(\mathbb{Z})$) for $n$ even (respectively $n$ odd), $G_n\curvearrowright \mathbb{T}^n$ is free and ergodic, every ergodic sub-equivalence relation of the orbital equivalence relation is either amenable or rigid, and the fundamental group of the II$_1$ factor $N_n:= L^\infty(\mathbb{T}^n)\rtimes G_n$ is trivial. For $n$ even, $L^\infty(\mathbb{T}^n)\rtimes H$ is a maximal Haagerup subalgebra of $N_n$ for every suitable maximal amenable subgroup $H$ of $\operatorname{SL}_2(\mathbb{Z})$.

DOI: http://dx.doi.org/10.7900/jot.2023dec07.2496
Keywords: ergodic p.m.p. action, amenable equivalence relation, group measure space construction, rigid inclusion, Cartan subalgebra, type {II}$_1$ factor

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