Geometric density for invariant random subgroups of groups acting on CAT(0) spaces

TL;DR

This paper establishes that invariant random subgroups (IRS) of groups acting on CAT(0) spaces inherit geometric density properties under certain conditions, extending Borel density concepts to IRSs.

Contribution

It proves that IRSs of groups with geometrically dense actions on CAT(0) spaces are also geometrically dense, under finite telescopic dimension or finite-dimensional Tits boundary assumptions.

Findings

IRS inherits geometric density in CAT(0) spaces

Extension of Borel density theorem to IRSs

Conditions include finite telescopic dimension or finite-dimensional Tits boundary

Abstract

We prove that an IRS of a group with a geometrically dense action on a CAT(0) space also acts geometrically densely; assuming the space is either of finite telescopic dimension or locally compact with finite dimensional Tits boundary. This can be thought of as a Borel density theorem for IRSs.