Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Arithmetic trialitarian hyperbolic lattices are not LERF

Nikolay Bogachev, Leone Slavich, Hongbin Sun

Abstract

A group is LERF (locally extended residually finite) if all its finitely generated subgroups are separable. We prove that the trialitarian arithmetic lattices in $\mathbf{PSO}_{7,1}(\mathbb{R})$ are not LERF. This result, together with previous work by the third author, implies that all arithmetic lattices in $\mathbf{PO}_{n,1}(\mathbb{R})$, $n>3$, are not LERF.

Arithmetic trialitarian hyperbolic lattices are not LERF

Abstract

A group is LERF (locally extended residually finite) if all its finitely generated subgroups are separable. We prove that the trialitarian arithmetic lattices in are not LERF. This result, together with previous work by the third author, implies that all arithmetic lattices in , , are not LERF.
Paper Structure (4 sections, 7 theorems, 8 equations, 1 figure)

This paper contains 4 sections, 7 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Trialitarian lattices and manifolds
  3. Geodesic subspaces
  4. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

Arithmetic trialitarian lattices in $\mathrm{Isom}(\mathbb{H}^7)$ are not LERF.

Figures (1)

  • Figure 1: The Dynkin diagram of the $D_4$ root system, with roots labeled.

Theorems & Definitions (9)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • Proposition 3.5
  • proof : Proof of Theorem \ref{['main']}