Can lattices in SL(n,R) act on the circle?

TL;DR

This paper reviews various mathematical approaches to the conjecture that lattices in SL(n,R) for n > 2 cannot act faithfully on the circle, highlighting partial results and key techniques.

Contribution

It provides an expository overview of methods and partial results related to the conjecture on lattices in SL(n,R) acting on the circle for n > 2.

Findings

Partial results supporting the conjecture

Connections between amenability, property (T), and actions on the circle

Application of bounded cohomology and stability theorems

Abstract

This expository paper describes the various methods that have yielded partial results on the conjecture that if n > 2, then no lattice in SL(n,R) has a faithful action on the circle (by homeomorphisms). Topics include amenability, Kazhdan's property (T), bounded cohomology, bounded generation, and the Reeb-Thurston Stability Theorem.