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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