Things we can learn by considering random locally symmetric manifolds

TL;DR

This paper reviews probabilistic methods involving random subgroups to derive results about locally symmetric manifolds, highlighting the evolution from finitely supported measures to advanced theories like IRS and SRS.

Contribution

It provides an overview of recent developments in using random subgroup measures to study locally symmetric manifolds, including new approaches with non-stationary random subgroups.

Findings

Progression from finitely supported measures to IRS and SRS theories

Application of random subgroup measures to derive geometric results

Recent unpublished results using non-stationary random subgroups

Abstract

In recent years various results about locally symmetric manifolds were proven using probabilistic approaches. One of the approaches is to consider random manifolds by associating a probability measure to the space of discrete subgroups of the isometry Lie group. The main goals are to prove results about deterministic groups and manifolds by considering appropriate measures. In this overview paper we describe several such results, observing the evolution process of the measures involved. Starting with a result whose proof considered finitely supported measures (more precisely, measures supported on finitely many conjugacy classes) and proceeding with results which were outcome of the successful and popular theory of IRS (invariant random subgroups). In the last couple of years the theory has expanded to SRS (stationary random subgroups) allowing to deal with a lot more problems and establish stronger results. In the last section we shall review a very recent (yet unpublished) result whose proof make use of random subgroups which are not even stationary.