On Ulam stability

TL;DR

This paper investigates the stability of approximate group representations by unitary operators, introducing Ulam stability, and demonstrates that certain lattices are stable while groups with free subgroups are not, impacting understanding of rigidity.

Contribution

It defines Ulam stability for groups and proves that higher rank lattices are stable, while groups with free subgroups are not, advancing the theory of representation stability.

Findings

Higher rank lattices are Ulam stable.

Groups with free subgroups are not strongly Ulam stable.

Groups with free subgroups are not deformation rigid.

Abstract

We study $\epsilon$-representations of discrete groups by unitary operators on a Hilbert space. We define the notion of Ulam stability of a group which loosely means that finite-dimensional $\epsilon$-represendations are uniformly close to unitary representations. One of our main results is that certain lattices in connected semi-simple Lie groups of higher rank are Ulam stable. For infinite-dimensional $\epsilon$-representations, the similarly defined notion of strong Ulam stability is defined and it is shown that groups with free subgroups are not strongly Ulam stable. We also study deformation rigidity of unitary representations and show that groups containing a free subgroup are not deformation rigid.