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Structural properties of reduced $C^*$-algebras associated with higher-rank lattices

Itamar Vigdorovich

TL;DR

The paper addresses the structural analysis of reduced C*-algebras for higher-rank lattices, proving that cocompact lattices in $ ext{PSL}_{3}(bK)$ possess selflessness, strict comparison, stable rank one, and unique Jiang–Su embeddings, with an explicit description of the Cuntz semigroup. The authors develop a quantitative freeness theory in linear groups via projective dynamics, proximality, and exponential mixing (Kleinbock–Margulis), combining these with Lafforgue’s rapid decay to transfer freeness from the ambient group to the lattice. The key contribution is the effective construction of $r$-free elements in lattices, yielding selfless $C_{r}^{*}(oldsymbol{\Gamma})$ and enabling the full set of C*-algebraic properties, including a precise Cu-structure. This work extends Elliott’s classification program beyond amenable algebras and provides new avenues toward resolving broader questions about rapid decay and rigidity for higher-rank lattices.

Abstract

We present the first examples of higher-rank lattices whose reduced $C^{*}$-algebras satisfy strict comparison, stable rank one, selflessness, uniqueness of embeddings of the Jiang--Su algebra, and allow explicit computations of the Cuntz semigroup. This resolves a question raised in recent groundbreaking work of Amrutam, Gao, Kunnawalkam Elayavalli, and Patchell, in which they exhibited a large class of finitely generated non-amenable groups satisfying these properties. Our proof relies on quantitative estimates in projective dynamics, crucially using the exponential mixing for diagonalizable flows. As a result, we obtain an effective mixed-identity-freeness property, which, combined with V. Lafforgue's rapid decay theorem, yields the desired conclusions.

Structural properties of reduced $C^*$-algebras associated with higher-rank lattices

TL;DR

The paper addresses the structural analysis of reduced C*-algebras for higher-rank lattices, proving that cocompact lattices in possess selflessness, strict comparison, stable rank one, and unique Jiang–Su embeddings, with an explicit description of the Cuntz semigroup. The authors develop a quantitative freeness theory in linear groups via projective dynamics, proximality, and exponential mixing (Kleinbock–Margulis), combining these with Lafforgue’s rapid decay to transfer freeness from the ambient group to the lattice. The key contribution is the effective construction of -free elements in lattices, yielding selfless and enabling the full set of C*-algebraic properties, including a precise Cu-structure. This work extends Elliott’s classification program beyond amenable algebras and provides new avenues toward resolving broader questions about rapid decay and rigidity for higher-rank lattices.

Abstract

We present the first examples of higher-rank lattices whose reduced -algebras satisfy strict comparison, stable rank one, selflessness, uniqueness of embeddings of the Jiang--Su algebra, and allow explicit computations of the Cuntz semigroup. This resolves a question raised in recent groundbreaking work of Amrutam, Gao, Kunnawalkam Elayavalli, and Patchell, in which they exhibited a large class of finitely generated non-amenable groups satisfying these properties. Our proof relies on quantitative estimates in projective dynamics, crucially using the exponential mixing for diagonalizable flows. As a result, we obtain an effective mixed-identity-freeness property, which, combined with V. Lafforgue's rapid decay theorem, yields the desired conclusions.

Paper Structure

This paper contains 13 sections, 24 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Free independences
  3. Zariski dense subgroups are MIF
  4. Freeness criteria for linear groups
  5. Length function on $G$
  6. Dynamical criterion for $r$-freeness
  7. Geometric criterion
  8. finding $r$-free elements in $G$
  9. Basic inequalities on projective space
  10. Uniform displacement for lattices
  11. Existence of a geometric configuration
  12. Finding an $r$-free element in $\Gamma$
  13. selfless reduced $C^{*}$-algebras

Key Result

Theorem 1.1

Let $\Gamma$ be a cocompact lattice $\mathrm{PSL}_{3}(\mathbb{K})$ where $\mathbb{K}$ is a local field of characteristic $0$ (thus $\mathbb{K}$ is isomorphic to either $\mathbb{R}$, $\mathbb{C}$, or a finite field extension of $\mathbb{Q}_{p}$). Then:

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4: Ping-Pong Lemma
  • proof
  • Example 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 42 more