Charmenability of arithmetic groups of product type. (English) Zbl 07569645
Let \(\Gamma\) be a countable group. The set of positive definite functions \(\varphi\) on \(\Gamma\) normalized with \(\varphi(e)=1\) is a compact convex \(\Gamma\)-set on \(\ell^\infty(\Gamma)\) with the weak-\(*\) topology. Thanks to the GNS construction, any such function is obtained as a matrix coefficient \(\varphi(g)=\langle \pi(g)u,u\rangle \) with \(u\) is a unit vector of a Hilbert space \(\mathcal{H}\) and \(\pi: \Gamma\to \mathrm{U}(\mathcal{H})\) is a unitary representation.
Amongst these positive definite functions, the ones invariant under conjugation are called characters. The set of characters is also a compact convex invariant subset. Examples are given by normalized traces of finite dimensional unitary representations.
In previous papers, the authors introduced two properties for \(\Gamma\) related to its characters. Let us recall that the amenable radical of \(\Gamma\) is the largest amenable normal subgroup of \(\Gamma\).
The group \(\Gamma\) is charmenable if it satisfies the following two properties:
The group \(\Gamma\) is charfinite if it is charmenable and moreover it satisfies the following conditions:
The second and the third author proved that lattices in higher rank connected simple Lie groups with finite center are charfinite [Publ. Math., Inst. Hautes Étud. Sci. 133, 1–46 (2021; Zbl 1504.22009)]. The first author and A. Furman also proved that irreducible lattices in products of (at least two) simple algebraic groups are charmenable (resp. charfinite if one of the factors has property (T)) [in: Proceedings of the international congress of mathematicians (ICM 2014), Seoul, Korea, August 13–21, 2014. Vol. III: Invited lectures. Seoul: KM Kyung Moon Sa. 71–96 (2014; Zbl 1378.37004); Compos. Math. 156, No. 1, 158–178 (2020; Zbl 1435.22010)].
In this paper, they pursue the study of charmenability of lattices in any characteristic. Their main result is the following one. Let \(k\) be a local field. Let \(\mathrm{G}\) be an almost \(k\)-simple connected algebraic \(k\)-group such that rank at least \(2\). Then every lattice of \(\mathrm{G}(k)\) is charfinite.
From that result, they deduce consequences for associated von Neumann algebras, invariant random subgroups and uniformly recurrent subgroups.
Amongst these positive definite functions, the ones invariant under conjugation are called characters. The set of characters is also a compact convex invariant subset. Examples are given by normalized traces of finite dimensional unitary representations.
In previous papers, the authors introduced two properties for \(\Gamma\) related to its characters. Let us recall that the amenable radical of \(\Gamma\) is the largest amenable normal subgroup of \(\Gamma\).
The group \(\Gamma\) is charmenable if it satisfies the following two properties:
- 1.
- Every nonempty \(\Gamma\)-invariant weak-\(\ast\) compact convex subset of the set of positive definite functions contains a character.
- 2.
- Every extremal character is either supported on the amenable radical of \(\Gamma\) or its GNS von Neumann algebra is amenable.
The group \(\Gamma\) is charfinite if it is charmenable and moreover it satisfies the following conditions:
- 1.
- The amenable radical of \(\Gamma\) is finite.
- 2.
- \(\Gamma\) has a finite number of isomorphism classes of unitary representations in each given finite dimension.
- 3.
- Every extremal character is either supported on the amenable radical or its GNS von Neumann algebra is finite dimensional.
The second and the third author proved that lattices in higher rank connected simple Lie groups with finite center are charfinite [Publ. Math., Inst. Hautes Étud. Sci. 133, 1–46 (2021; Zbl 1504.22009)]. The first author and A. Furman also proved that irreducible lattices in products of (at least two) simple algebraic groups are charmenable (resp. charfinite if one of the factors has property (T)) [in: Proceedings of the international congress of mathematicians (ICM 2014), Seoul, Korea, August 13–21, 2014. Vol. III: Invited lectures. Seoul: KM Kyung Moon Sa. 71–96 (2014; Zbl 1378.37004); Compos. Math. 156, No. 1, 158–178 (2020; Zbl 1435.22010)].
In this paper, they pursue the study of charmenability of lattices in any characteristic. Their main result is the following one. Let \(k\) be a local field. Let \(\mathrm{G}\) be an almost \(k\)-simple connected algebraic \(k\)-group such that rank at least \(2\). Then every lattice of \(\mathrm{G}(k)\) is charfinite.
From that result, they deduce consequences for associated von Neumann algebras, invariant random subgroups and uniformly recurrent subgroups.
Reviewer: Bruno Duchesne (Paris)
MSC:
| 22E40 | Discrete subgroups of Lie groups |
| 22D10 | Unitary representations of locally compact groups |
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 46L10 | General theory of von Neumann algebras |
| 46L30 | States of selfadjoint operator algebras |
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