Representations of some lattices into the group of analytic diffeomorphisms of the sphere $\mathbb{S}^2$

TL;DR

This paper investigates the representations of certain lattices into the group of analytic diffeomorphisms of the sphere, extending known results for higher dimensions to the challenging case of dimension four.

Contribution

It revisits and modifies existing proofs to establish the finiteness of such representations specifically for the case n=4.

Findings

Finite image for representations when n≥5

Extension of results to the n=4 case with modified arguments

Clarification of the proof strategy for low-dimensional cases

Abstract

In \cite{Ghys} it is proved that any morphism from a subgroup of finite index of $\mathrm{SL}(n,\mathbb{Z})$ to the group of analytic diffeomorphisms of $\mathbb{S}^2$ has a finite image as soon as $n\geq 5$. The case $n=4$ is also claimed to follow along the same arguments; in fact this is not straightforward and this case indeed needs a modification of the argument. In this paper we recall the strategy for $n\geq 5$ and then focus on the case $n=4$.