Anosov groups that are indiscrete in rank one

TL;DR

This paper constructs specific Anosov subgroups in higher rank Lie groups that cannot be embedded discretely into any rank-1 simple Lie group or their finite products, revealing new non-embedding phenomena.

Contribution

It introduces examples of Anosov subgroups that are not discretely embeddable in rank-1 Lie groups, expanding understanding of subgroup embeddings in higher rank Lie groups.

Findings

Constructed Anosov subgroups not discretely embeddable in rank-1 Lie groups

Demonstrated existence of free product subgroups with specific lattice components

Revealed limitations of embedding properties for certain Anosov groups

Abstract

We exhibit Anosov subgroups of $\mathsf{SL}_d(\mathbb{R})$ that do not embed discretely in any rank-$1$ simple Lie group of noncompact type, or indeed, in any finite product of such Lie groups. These subgroups are isomorphic to free products $\Gamma * \Delta$, where $\Gamma$ is a uniform lattice in $\mathsf{F}_4^{(-20)}$ and $\Delta$ is a uniform lattice in $\mathsf{Sp}(m,1)$, $m \geq 51$.