Co-spectral radius of intersections

TL;DR

This paper investigates how the co-spectral radius of subgroups behaves under intersections, showing it remains stable for invariant random subgroups intersected with co-amenable subgroups, with implications for co-amenability of intersections.

Contribution

It proves that the co-spectral radius does not decrease when intersecting invariant random subgroups with co-amenable subgroups, extending understanding of subgroup properties.

Findings

Co-spectral radius remains unchanged under intersection with co-amenable subgroups.

Intersection of independent co-amenable invariant random subgroups is co-amenable.

Main result applies to invariant random subgroups in discrete groups.

Abstract

We study the behavior of the co-spectral radius of a subgroup $H$ of a discrete group $\Gamma$ under taking intersections. Our main result is that the co-spectral radius of an invariant random subgroup does not drop upon intersecting with a deterministic co-amenable subgroup. As an application, we find that the intersection of independent co-amenable invariant random subgroups is co-amenable.